# Finite posets¶

This module implements finite partially ordered sets. It defines:

 FinitePoset A class for finite posets FinitePosets_n A class for finite posets up to isomorphism (i.e. unlabeled posets) Poset() Construct a finite poset from various forms of input data. is_poset() Tests whether a directed graph is acyclic and transitively reduced.

List of Poset methods

 antichains_iterator() Returns an iterator over the antichains of the poset. antichains() Returns the antichains of the poset. bottom() Returns the bottom element of the poset, if it exists. cardinality() Returns the number of elements in the poset. chains() Returns all the chains of self chain_polytope() Returns the chain polytope of the poset. chain_polynomial() Returns the chain polynomial of the poset. closed_interval() Returns a list of the elements $$z$$ such that $$x \le z \le y$$. compare_elements() Compares $$x$$ and $$y$$ in the poset. comparability_graph() Returns the comparability graph of the poset. completion_by_cuts() Returns the Dedekind-MacNeille completion of the poset. cover_relations_iterator() Returns an iterator for the cover relations of the poset. cover_relations() Returns the list of pairs $$[u,v]$$ which are cover relations cover_relations_graph() Return the graph of cover relations covers() Returns True if y covers x and False otherwise. coxeter_transformation() Returns the matrix of the Auslander-Reiten translation acting on the Grothendieck group of the derived category of modules. cuts() Returns the cuts of the given poset. dilworth_decomposition() Returns a partition of the points into the minimal number of chains. disjoint_union() Return the disjoint union of the poset with other. dual() Returns the dual poset of the given poset. evacuation() Computes evacuation on the linear extension associated to the poset self. f_polynomial() Returns the f-polynomial of a bounded poset. flag_f_polynomial() Returns the flag f-polynomial of a bounded and ranked poset. flag_h_polynomial() Returns the flag h-polynomial of a bounded and ranked poset. frank_network() Returns Frank’s network (a DiGraph along with a cost function on its edges) associated to self. graphviz_string() Returns a representation in the DOT language, ready to render in graphviz. greene_shape() Computes the Greene-Kleitman partition aka Greene shape of the poset self. h_polynomial() Returns the h-polynomial of a bounded poset. has_bottom() Returns True if the poset has a unique minimal element. hasse_diagram() Returns the Hasse diagram of self as a Sage DiGraph. has_isomorphic_subposet() Return True if the poset contains a subposet isomorphic to another poset, and False otherwise. has_top() Returns True if the poset contains a unique maximal element, and False otherwise. height() Return the height (number of elements in the longest chain) of the poset. incomparability_graph() Returns the incomparability graph of the poset. interval() Returns a list of the elements $$z$$ such that $$x \le z \le y$$. intervals() Returns a list of all intervals of the poset. intervals_iterator() Returns an iterator for all the intervals of the poset. intervals_number() Returns the number of intervals in the poset. is_bounded() Returns True if the poset contains a unique maximal element and a unique minimal element, and False otherwise. is_chain() Returns True if the poset is totally ordered, and False otherwise. is_connected() Return True if the poset is connected, and False otherwise. is_EL_labelling() Returns whether f is an EL labelling of self is_gequal() Returns True if $$x$$ is greater than or equal to $$y$$ in the poset, and False otherwise. is_graded() Returns whether this poset is graded. is_greater_than() Returns True if $$x$$ is greater than but not equal to $$y$$ in the poset, and False otherwise. is_isomorphic() Returns True if both posets are isomorphic. is_join_semilattice() Returns True is the poset has a join operation, and False otherwise. is_lequal() Returns True if $$x$$ is less than or equal to $$y$$ in the poset, and False otherwise. is_less_than() Returns True if $$x$$ is less than but not equal to $$y$$ in the poset, and False otherwise. is_linear_extension() Returns whether l is a linear extension of self is_meet_semilattice() Returns True if self has a meet operation, and False otherwise. isomorphic_subposets_iterator() Return an iterator over the subposets isomorphic to another poset. isomorphic_subposets() Return all subposets isomorphic to another poset. is_incomparable_chain_free() Returns whether the poset is $$(m+n)$$-free. is_ranked() Returns whether this poset is ranked. is_slender() Returns whether the poset self is slender or not. lequal_matrix() Computes the matrix whose (i,j) entry is 1 if self.linear_extension()[i] < self.linear_extension()[j] and 0 otherwise level_sets() Returns a list l such that l[i+1] is the set of minimal elements of the poset obtained by removing the elements in l[0], l[1], ..., l[i]. linear_extension() Returns a linear extension of this poset. linear_extensions() Returns the enumerated set of all the linear extensions of this poset list() List the elements of the poset. This just returns the result of linear_extension(). lower_covers_iterator() Returns an iterator for the lower covers of the element y. An lower cover of y is an element x such that y x is a cover relation. lower_covers() Returns a list of lower covers of the element y. An lower cover of y is an element x such that y x is a cover relation. maximal_antichains() Return all maximal antichains of the poset. maximal_chains() Returns all maximal chains of this poset. Each chain is listed in increasing order. maximal_elements() Returns a list of the maximal elements of the poset. minimal_elements() Returns a list of the minimal elements of the poset. mobius_function_matrix() Returns a matrix whose (i,j) entry is the value of the Mobius function evaluated at self.linear_extension()[i] and self.linear_extension()[j]. mobius_function() Returns the value of the Mobius function of the poset on the elements x and y. open_interval() Returns a list of the elements $$z$$ such that $$x < z < y$$. The order is that induced by the ordering in order_complex() Returns the order complex associated to this poset. order_filter() Returns the order filter generated by a list of elements. order_ideal() Returns the order ideal generated by a list of elements. order_polynomial() Returns the order polynomial of the poset. order_polytope() Returns the order polytope of the poset. ordinal_product() Return the ordinal product of the poset with other. ordinal_sum() Return the ordinal sum of the poset with other. p_partition_enumerator() Returns a $$P$$-partition enumerator of the poset. plot() Returns a Graphic object corresponding the Hasse diagram of the poset. product() Returns the cartesian product of self and other. promotion() Computes the (extended) promotion on the linear extension of the poset self random_subposet() Return a random subposet that contains each element with probability p. rank_function() Returns a rank function of the poset, if it exists. rank() Returns the rank of an element, or the rank of the poset if element is None. relabel() Returns a copy of this poset with its elements relabelled relations() Returns a list of all relations of the poset. relations_iterator() Returns an iterator for all the relations of the poset. relations_number() Returns the number of relations in the poset. show() Displays the Hasse diagram of the poset. subposet() Returns the poset containing elements with partial order induced by that of self. top() Returns the top element of the poset, if it exists. unwrap() Unwraps an element of this poset upper_covers_iterator() Returns an iterator for the upper covers of the element y. An upper cover of y is an element x such that y x is a cover relation. upper_covers() Returns a list of upper covers of the element y. An upper cover of y is an element x such that y x is a cover relation. width() Returns the width of the poset (the size of its longest antichain). with_linear_extension() Returns a copy of self with a different default linear extension. zeta_polynomial() Returns the zeta polynomial of the poset.

## Classes and functions¶

class sage.combinat.posets.posets.FinitePoset(hasse_diagram, elements, category, facade, key)

A (finite) $$n$$-element poset constructed from a directed acyclic graph.

INPUT:

• hasse_diagram – an instance of FinitePoset, or a DiGraph that is transitively-reduced, acyclic, loop-free, and multiedge-free.
• elements – an optional list of elements, with element[i] corresponding to vertex i. If elements is None, then it is set to be the vertex set of the digraph. Note that if this option is set, then elements is considered as a specified linear extension of the poset and the $$linear_extension$$ attribute is set.
• categoryFinitePosets, or a subcategory thereof.
• facade – a boolean or None (default); whether the FinitePoset‘s elements should be wrapped to make them aware of the Poset they belong to.
• If facade = True, the FinitePoset‘s elements are exactly those given as input.
• If facade = False, the FinitePoset‘s elements will become PosetElement objects.
• If facade = None (default) the expected behaviour is the behaviour of facade = True, unless the opposite can be deduced from the context (i.e. for instance if a FinitePoset is built from another FinitePoset, itself built with facade = False)
• key – any hashable value (default: None).

EXAMPLES:

sage: uc = [[2,3], [], [1], [1], [1], [3,4]]
sage: from sage.combinat.posets.posets import FinitePoset
sage: P = FinitePoset(DiGraph(dict([[i,uc[i]] for i in range(len(uc))])), facade=False); P
Finite poset containing 6 elements
sage: P.cover_relations()
[[5, 4], [5, 3], [4, 1], [0, 2], [0, 3], [2, 1], [3, 1]]
sage: TestSuite(P).run()
sage: P.category()
Join of Category of finite posets and Category of finite enumerated sets
sage: P.__class__
<class 'sage.combinat.posets.posets.FinitePoset_with_category'>

sage: Q = sage.combinat.posets.posets.FinitePoset(P, facade = False); Q
Finite poset containing 6 elements

sage: Q is P
True


We keep the same underlying Hasse diagram, but change the elements:

sage: Q = sage.combinat.posets.posets.FinitePoset(P, elements=[1,2,3,4,5,6], facade=False); Q
Finite poset containing 6 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 5], [2, 6], [3, 4], [3, 5], [4, 6], [5, 6]]


sage: P = Poset(DiGraph({'a':['b'],'b':['c'],'c':['d']}), facade=False)
sage: P.category()
Join of Category of finite posets and Category of finite enumerated sets
sage: parent(P[0]) is P
True

sage: Q.category()
Join of Category of finite posets
and Category of finite enumerated sets
sage: parent(Q[0]) is str
True
sage: TestSuite(Q).run(skip = ['_test_an_element']) # is_parent_of is not yet implemented


sage: PQ = Poset(P, facade=True)
sage: PQ.category()
Join of Category of finite posets
and Category of finite enumerated sets
sage: parent(PQ[0]) is str
True
sage: PQ is Q
True


sage: QP = Poset(Q, facade = False)
sage: QP.category()
Join of Category of finite posets
and Category of finite enumerated sets
sage: parent(QP[0]) is QP
True


Note

A class that inherits from this class needs to define Element. This is the class of the elements that the inheriting class contains. For example, for this class, FinitePoset, Element is PosetElement. It can also define _dual_class which is the class of dual posets of this class. E.g. FiniteMeetSemilattice._dual_class is FiniteJoinSemilattice.

TESTS:

Equality is derived from UniqueRepresentation. We check that this gives consistent results:

sage: P = Poset([[1,2],[3],[3]])
sage: P == P
True
sage: Q = Poset([[1,2],[],[1]])
sage: Q == P
False
sage: p1, p2 = Posets(2).list()
sage: p2 == p1, p1 != p2
(False, True)
sage: [[p1.__eq__(p2) for p1 in Posets(2)] for p2 in Posets(2)]
[[True, False], [False, True]]
sage: [[p2.__eq__(p1) for p1 in Posets(2)] for p2 in Posets(2)]
[[True, False], [False, True]]
sage: [[p2 == p1 for p1 in Posets(3)] for p2 in Posets(3)]
[[True, False, False, False, False],
[False, True, False, False, False],
[False, False, True, False, False],
[False, False, False, True, False],
[False, False, False, False, True]]

sage: [[p1.__ne__(p2) for p1 in Posets(2)] for p2 in Posets(2)]
[[False, True], [True, False]]
sage: P = Poset([[1,2,4],[3],[3]])
sage: Q = Poset([[1,2],[],[1],[4]])
sage: P != Q
True
sage: P != P
False
sage: Q != Q
False
sage: [[p1.__ne__(p2) for p1 in Posets(2)] for p2 in Posets(2)]
[[False, True], [True, False]]

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
sage: Q = Poset(P)
sage: Q == P
False
sage: Q = Poset(P, linear_extension=True)
sage: Q == P
True

Element

alias of PosetElement

antichains(element_constructor=<type 'list'>)

Returns the antichains of the poset.

INPUT:

• element_constructor – a function taking an iterable as argument (default: list)

OUTPUT: an enumerated set

An antichain of a poset is a collection of elements of the poset that are pairwise incomparable.

EXAMPLES:

sage: A = Posets.PentagonPoset().antichains(); A
Set of antichains of Finite lattice containing 5 elements
sage: list(A)
[[], [0], [1], [1, 2], [1, 3], [2], [3], [4]]
sage: A.cardinality()
8
sage: A[3]
[1, 2]
sage: list(Posets.AntichainPoset(3).antichains())
[[], [2], [2, 1], [2, 1, 0], [2, 0], [1], [1, 0], [0]]
sage: list(Posets.ChainPoset(3).antichains())
[[], [0], [1], [2]]


To get the antichains of a given size one can currently use:

sage: list(A.elements_of_depth_iterator(2))
[[1, 2], [1, 3]]


Eventually the following syntax will be accepted:

sage: A.subset(size = 2) # todo: not implemented


To get the antichains as, say, sets, one may use the element_constructor option:

sage: list(Posets.ChainPoset(3).antichains(element_constructor = set))
[set(), {0}, {1}, {2}]


Note

Internally, this uses sage.combinat.subsets_pairwise.PairwiseCompatibleSubsets and SearchForest. At this point, iterating through this set is about twice slower than using antichains_iterator() (tested on posets.AntichainPoset(15)). The algorithm is the same (depth first search through the tree), but antichains_iterator() manually inlines things which apparently avoids some infrastructure overhead.

On the other hand, this returns a full featured enumerated set, with containment testing, etc.

antichains_iterator()

Returns an iterator over the antichains of the poset.

EXAMPLES:

sage: Posets.PentagonPoset().antichains_iterator()
<generator object antichains_iterator at ...>


antichains()

bottom()

Returns the bottom element of the poset, if it exists.

EXAMPLES:

sage: P = Poset({0:[3],1:[3],2:[3],3:[4],4:[]})
sage: P.bottom() is None
True
sage: Q = Poset({0:[1],1:[]})
sage: Q.bottom()
0

canonical_label()

Return the unique poset on the labels $$\{0, \ldots, n-1\}$$ (where $$n$$ is the number of elements in self) that is isomorphic to self and invariant in the isomorphism class.

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True, facade=False)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: P.cover_relations()
[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
sage: Q = P.canonical_label()
sage: Q.list()
[0, 1, 2, 3, 4, 5]
sage: Q.cover_relations()
[[0, 1], [0, 2], [1, 4], [2, 3], [2, 4], [3, 5], [4, 5]]


sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: P.cover_relations()
[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
sage: Q = P.canonical_label()
sage: Q.list()
[0, 1, 2, 3, 4, 5]
sage: Q.cover_relations()
[[0, 1], [0, 2], [1, 4], [2, 3], [2, 4], [3, 5], [4, 5]]


TESTS:

sage: P = Poset(digraphs.Path(10), linear_extension = True)
sage: Q = P.canonical_label()
sage: Q.linear_extension()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: Q.cover_relations()
[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]]
sage: P = Poset(digraphs.Path(10))
sage: Q = P.canonical_label()
sage: Q.linear_extension()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: Q.cover_relations()
[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]]

cardinality()

Returns the number of elements in the poset.

EXAMPLES:

sage: Poset([[1,2,3],[4],[4],[4],[]]).cardinality()
5

chain_polynomial()

Return the chain polynomial of self.

The coefficient of $$q^k$$ is the number of chains of length $$k$$ in self. The length of a chain is the number of elements.

Warning

This is not what has been called the chain polynomial in [St1986]. The latter is identical with the order polynomial (order_polynomial()).

EXAMPLES:

sage: P = Posets.ChainPoset(3)
sage: t = P.chain_polynomial(); t
q^3 + 3*q^2 + 3*q + 1
sage: t(1) == len(list(P.chains()))
True

sage: P = Posets.BooleanLattice(3)
sage: P.chain_polynomial()
6*q^4 + 18*q^3 + 19*q^2 + 8*q + 1

sage: P = Posets.AntichainPoset(5)
sage: P.chain_polynomial()
5*q + 1

sage: P = Poset({})
sage: P.chain_polynomial()
1
sage: parent(P.chain_polynomial())
Univariate Polynomial Ring in q over Integer Ring

sage: R = Poset({1: []})
sage: R.chain_polynomial()
q + 1

chain_polytope()

Return the chain polytope of the poset self.

The chain polytope of a finite poset $$P$$ is defined as the subset of $$\RR^P$$ consisting of all maps $$x : P \to \RR$$ satisfying

$x(p) \geq 0 \mbox{ for all } p \in P,$

and

$\begin{split}x(p_1) + x(p_2) + \ldots + x(p_k) \leq 1 \mbox{ for all chains } p_1 < p_2 < \ldots < p_k \mbox{ in } P.\end{split}$

This polytope was defined and studied in [St1986].

EXAMPLES:

sage: P = posets.AntichainPoset(3)
sage: Q = P.chain_polytope();Q
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices
sage: P = posets.PentagonPoset()
sage: Q = P.chain_polytope();Q
A 5-dimensional polyhedron in QQ^5 defined as the convex hull of 8 vertices

chains(element_constructor=<type 'list'>, exclude=None)

Return all the chains of self.

INPUT:

• element_constructor – a function taking an iterable as

argument (default: list)

• exclude – elements of the poset to be excluded (default: None)

OUTPUT:

The enumerated set of all chains of self, each of which is given as an element_constructor.

A chain of a poset is a set of elements of the poset that are pairwise comparable.

EXAMPLES:

sage: A = Posets.PentagonPoset().chains(); A
Set of chains of Finite lattice containing 5 elements
sage: list(A)
[[], [0], [0, 1], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]


To get the chains of a given size one can currently use:

sage: list(A.elements_of_depth_iterator(2))
[[0, 1], [0, 2], [0, 3], [0, 4], [1, 4], [2, 3], [2, 4], [3, 4]]


For bounded posets, one can exclude the bounds as follows:

sage: P = Posets.DiamondPoset(5)
sage: list(P.chains(exclude=[0, 4]))
[[], [1], [2], [3]]


Another example of exclusion of vertices:

sage: P = Poset({1: [2, 3], 2: [4], 3: [4, 5]})
sage: list(P.chains(element_constructor=tuple, exclude=[3]))
[(), (1,), (1, 2), (1, 2, 4), (1, 4), (1, 5), (2,), (2, 4), (4,), (5,)]


Eventually the following syntax will be accepted:

sage: A.subset(size = 2) # todo: not implemented


characteristic_polynomial()

Return the characteristic polynomial of a graded poset self.

If $$P$$ is a graded poset with rank $$n$$ and a unique minimal element $$\hat{0}$$, then the characteristic polynomial of $$P$$ is defined to be

$\sum_{x \in P} \mu(\hat{0}, x) q^{n-\rho(x)} \in \ZZ[q],$

where $$\rho$$ is the rank function, and $$\mu$$ is the Moebius function of $$P$$.

See section 3.10 of [EnumComb1].

EXAMPLES:

sage: P = Posets.DiamondPoset(5)
sage: P.characteristic_polynomial()
q^2 - 3*q + 2
sage: P = Poset({1:[2,3],2:[4],3:[5],4:[6],5:[6],6:[7]})
sage: P.characteristic_polynomial()
q^4 - 2*q^3 + q
sage: P = Poset({1: []})
sage: P.characteristic_polynomial()
1

closed_interval(x, y)

Return a list of the elements $$z$$ such that $$x \le z \le y$$.

EXAMPLES:

sage: uc = [[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]]
sage: dag = DiGraph(dict(zip(range(len(uc)),uc)))
sage: P = Poset(dag)
sage: I = set(map(P,[2,5,6,4,7]))
sage: I == set(P.closed_interval(2,7))
True

comparability_graph()

Returns the comparability graph of self.

EXAMPLES:

sage: p = posets.ChainPoset(4)
sage: p.comparability_graph().is_isomorphic(graphs.CompleteGraph(4))
True

sage: p = posets.DiamondPoset(5)
sage: g = p.comparability_graph(); g
Comparability graph on 5 vertices
sage: g.size()
7

compare_elements(x, y)

Compare $$x$$ and $$y$$ in the poset.

If x = y, then 0 is returned; if x < y, then -1 is returned; if x > y, then 1 is returned; and if x and y are not comparable, then None is returned.

EXAMPLES:

sage: P = Poset([[1,2],[4],[3],[4],[]])
sage: P.compare_elements(0,0)
0
sage: P.compare_elements(0,4)
-1
sage: P.compare_elements(4,0)
1
sage: P.compare_elements(1,2)

completion_by_cuts()

Return the completion by cuts of self.

This is a lattice, also called the Dedekind-MacNeille completion.

OUTPUT:

• a finite lattice

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: P.completion_by_cuts().is_isomorphic(P)
True

sage: P = posets.AntichainPoset(3)
sage: Q = P.completion_by_cuts()
sage: Q.is_isomorphic(posets.DiamondPoset(5))
True

sage: P = posets.SymmetricGroupBruhatOrderPoset(3)
sage: Q = P.completion_by_cuts(); Q
Finite lattice containing 7 elements


cuts()

cover_relations()

Returns the list of pairs [u,v] of elements of the poset such that u v is a cover relation (that is, u v and there does not exist z such that u z v).

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: Q.cover_relations()
[[1, 2], [0, 2], [2, 3], [3, 4]]

cover_relations_graph()

Return the graph of cover relations.

EXAMPLES:

sage: P = Poset({0:[1,2],1:[3],2:[3],3:[]})
sage: G = P.cover_relations_graph(); G
Graph on 4 vertices
sage: S = Poset()
sage: H = S.cover_relations_graph(); H
Graph on 0 vertices


Check that it is hashable and coincides with the Hasse diagram as a graph:

sage: hash(G) == hash(G)
True
sage: G == Graph(P.hasse_diagram())
True

cover_relations_iterator()

Returns an iterator for the cover relations of the poset.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: type(Q.cover_relations_iterator())
<type 'generator'>
sage: [z for z in Q.cover_relations_iterator()]
[[1, 2], [0, 2], [2, 3], [3, 4]]

covers(x, y)

Returns True if y covers x and False otherwise.

EXAMPLES:

sage: Q = Poset([[1,5],[2,6],[3],[4],[],[6,3],[4]])
sage: Q.covers(Q(1),Q(6))
True
sage: Q.covers(Q(1),Q(4))
False

coxeter_transformation()

Returns the matrix of the Auslander-Reiten translation acting on the Grothendieck group of the derived category of modules on the poset self, in the basis of simple modules. This matrix is usually called the Coxeter transformation.

EXAMPLES:

sage: Posets.PentagonPoset().coxeter_transformation()
[ 0  0  0  0 -1]
[ 0  0  0  1 -1]
[ 0  1  0  0 -1]
[-1  1  1  0 -1]
[-1  1  0  1 -1]


TESTS:

sage: M = Posets.PentagonPoset().coxeter_transformation()
sage: M**8 == 1
True

cuts()

Return the list of cuts of the poset self.

A cut is a subset $$A$$ of self such that the set of lower bounds of the set of upper bounds of $$A$$ is exactly $$A$$.

The cuts are computed here using the maximal independent sets in the auxiliary graph defined as $$P \times [0,1]$$ with an edge from $$(x, 0)$$ to $$(y, 1)$$ if and only if $$x \not\geq_P y$$. See the end of section 4 in [JRJ94].

EXAMPLES:

sage: P = posets.AntichainPoset(3)
sage: Pc = P.cuts()
sage: [list(c) for c in Pc]
[[0], [0, 1, 2], [], [1], [2]]
sage: Pc[0]
frozenset({0})


REFERENCES:

 [JRJ94] Jourdan, Guy-Vincent; Rampon, Jean-Xavier; Jard, Claude (1994), “Computing on-line the lattice of maximal antichains of posets”, Order 11 (3) p. 197-210, doi:10.1007/BF02115811
dilworth_decomposition()

Return a partition of the points into the minimal number of chains.

According to Dilworth’s theorem, the points of a poset can be partitioned into $$\alpha$$ chains, where $$\alpha$$ is the cardinality of its largest antichain. This method returns such a partition.

width() – return the width of the poset.

ALGORITHM:

We build a bipartite graph in which a vertex $$v$$ of the poset is represented by two vertices $$v^-,v^+$$. For any two $$u,v$$ such that $$u<v$$ in the poset we add an edge $$v^+u^-$$.

A matching in this graph is equivalent to a partition of the poset into chains: indeed, a chain $$v_1...v_k$$ gives rise to the matching $$v_1^+v_2^-,v_2^+v_3^-,...$$, and from a matching one can build the union of chains.

According to Dilworth’s theorem, the number of chains is equal to
$$\alpha$$ (the posets’ width).

EXAMPLE:

sage: p = posets.BooleanLattice(4)
sage: p.width()
6
sage: p.dilworth_decomposition()  # random
[[7, 6, 4], [11, 3], [12, 8, 0], [13, 9, 1], [14, 10, 2], [15, 5]]


TESTS:

sage: p = posets.IntegerCompositions(5)
sage: d = p.dilworth_decomposition()
sage: for chain in d:
....:    for i in range(len(chain)-1):
....:        assert p.is_greater_than(chain[i],chain[i+1])
sage: set(p) == set().union(*d)
True

disjoint_union(other, labels='pairs')

Return a poset isomorphic to disjoint union (also called direct sum) of the poset with other.

The disjoint union of $$P$$ and $$Q$$ is a poset that contains every element and relation from both $$P$$ and $$Q$$, and where every element of $$P$$ is incomparable to every element of $$Q$$.

Mathematically, it is only defined when $$P$$ and $$Q$$ have no common element; here we force that by giving them different names in the resulting poset.

INPUT:

• other, a poset.
• labels - (defaults to ‘pairs’) If set to ‘pairs’, each element v in this poset will be named (0,v) and each element u in other will be named (1,u) in the result. If set to ‘integers’, the elements of the result will be relabeled with consecutive integers.

EXAMPLES:

sage: P1 = Poset( (['a', 'b'], [['a', 'b']]) )
sage: P2 = Poset( (['c', 'd'], [['c', 'd']]) )
sage: P = P1.disjoint_union(P2); P
Finite poset containing 4 elements
sage: sorted(P.cover_relations())
[[(0, 'a'), (0, 'b')], [(1, 'c'), (1, 'd')]]
sage: P = P1.disjoint_union(P2, labels='integers');
sage: P.cover_relations()
[[2, 3], [0, 1]]

sage: N5 = Posets.PentagonPoset(); N5
Finite lattice containing 5 elements
sage: N5.disjoint_union(N5)  # Union of lattices is not a lattice
Finite poset containing 10 elements


We show how to get literally direct sum with elements untouched:

sage: P = P1.disjoint_union(P2).relabel(lambda x: x[1])
sage: sorted(P.cover_relations())
[['a', 'b'], ['c', 'd']]

dual()

Return the dual poset of the given poset.

EXAMPLES:

sage: P = Poset(([1,2,3],[[1,2],[1,3]]))
sage: P.cover_relations()
[[1, 2], [1, 3]]
sage: Q = P.dual()
sage: Q.cover_relations()
[[3, 1], [2, 1]]

sage: P = LatticePoset([[1,2],[3],[3]], facade = True)
sage: P.cover_relations()
[[0, 1], [0, 2], [1, 3], [2, 3]]
sage: Q = P.dual()
sage: Q.cover_relations()
[[3, 2], [3, 1], [2, 0], [1, 0]]
sage: Q.category()
Join of Category of finite lattice posets
and Category of finite enumerated sets
sage: Q.__class__
<class 'sage.combinat.posets.lattices.FiniteLatticePoset_with_category'>

sage: P = MeetSemilattice([[1,2],[3],[3]])
sage: P.dual().__class__
<class 'sage.combinat.posets.lattices.FiniteJoinSemilattice_with_category'>
sage: P = JoinSemilattice([[1,2],[3],[3]])
sage: P.dual().__class__
<class 'sage.combinat.posets.lattices.FiniteMeetSemilattice_with_category'>

evacuation()

Compute evacuation on the linear extension associated to the poset self.

OUTPUT:

• an isomorphic poset, with the same default linear extension

Evacuation is defined on a poset self of size $$n$$ by applying the evacuation operator $$(\tau_1 \cdots \tau_{n-1}) (\tau_1 \cdots \tau_{n-2}) \cdots (\tau_1)$$, to the default linear extension $$\pi$$ of self (see evacuation()), and relabeling self accordingly. For more details see [Stan2009].

REFERENCES:

 [Stan2009] (1, 2, 3) Richard Stanley, Promotion and evacuation, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 9, 24 pp.

EXAMPLES:

sage: P = Poset(([1,2], [[1,2]]), linear_extension=True, facade=False)
sage: P.evacuation()
Finite poset containing 2 elements with distinguished linear extension
sage: P.evacuation() == P
True

sage: P = Poset(([1,2,3,4,5,6,7], [[1,2],[1,4],[2,3],[2,5],[3,6],[4,7],[5,6]]), linear_extension=True, facade=False)
sage: P.list()
[1, 2, 3, 4, 5, 6, 7]
sage: Q = P.evacuation(); Q
Finite poset containing 7 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 3], [2, 5], [3, 4], [3, 6], [4, 7], [6, 7]]


Note that the results depend on the linear extension associated to the poset:

sage: P = Poset(([1,2,3,4,5,6,7], [[1,2],[1,4],[2,3],[2,5],[3,6],[4,7],[5,6]]))
sage: P.list()
[1, 2, 3, 5, 6, 4, 7]
sage: Q = P.evacuation(); Q
Finite poset containing 7 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 5], [2, 3], [5, 6], [5, 4], [6, 7], [4, 7]]


Here is an example of a poset where the vertices are not labelled by $$\{1,2,\ldots,n\}$$:

sage: P = Poset((divisors(15), attrcall("divides")), linear_extension = True)
sage: P.list()
[1, 3, 5, 15]
sage: Q = P.evacuation(); Q
Finite poset containing 4 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 3], [1, 5], [3, 15], [5, 15]]


AUTHOR:

• Anne Schilling (2012-02-18)
f_polynomial()

Return the $$f$$-polynomial of a bounded poset self.

This is the $$f$$-polynomial of the order complex of the poset minus its bounds.

The coefficient of $$q^i$$ is the number of chains of $$i+1$$ elements containing both bounds of the poset.

Warning

This is slightly different from the fPolynomial method in Macaulay2.

EXAMPLES:

sage: P = Posets.DiamondPoset(5)
sage: P.f_polynomial()
3*q^2 + q
sage: P = Poset({1:[2,3],2:[4],3:[5],4:[6],5:[7],6:[7]})
sage: P.f_polynomial()
q^4 + 4*q^3 + 5*q^2 + q

sage: P = Poset({2: []})
sage: P.f_polynomial()
1

flag_f_polynomial()

Return the flag $$f$$-polynomial of a bounded and ranked poset self.

This is the sum, over all chains containing both bounds, of a monomial encoding the ranks of the elements of the chain.

More precisely, if $$P$$ is a bounded ranked poset, then the flag $$f$$-polynomial of $$P$$ is defined as the polynomial

$\begin{split}\sum_{\substack{p_0 < p_1 < \ldots < p_k, \\ p_0 = \min P, \ p_k = \max P}} x_{\rho(p_1)} x_{\rho(p_2)} \cdots x_{\rho(p_k)} \in \ZZ[x_1, x_2, \cdots, x_n]\end{split}$

where $$\min P$$ and $$\max P$$ are (respectively) the minimum and the maximum of $$P$$, where $$\rho$$ is the rank function of $$P$$ (normalized to satisfy $$\rho(\min P) = 0$$), and where $$n$$ is the rank of $$\max P$$. (Note that the indeterminate $$x_0$$ doesn’t actually appear in the polynomial.)

For technical reasons, the polynomial is returned in the slightly larger ring $$\ZZ[x_0, x_1, x_2, \cdots, x_{n+1}]$$ by this method.

EXAMPLES:

sage: P = Posets.DiamondPoset(5)
sage: P.flag_f_polynomial()
3*x1*x2 + x2

sage: P = Poset({1:[2,3],2:[4],3:[5],4:[6],5:[6]})
sage: fl = P.flag_f_polynomial(); fl
2*x1*x2*x3 + 2*x1*x3 + 2*x2*x3 + x3
sage: q = polygen(ZZ,'q')
sage: fl(q,q,q,q) == P.f_polynomial()
True

sage: P = Poset({1:[2,3,4],2:[5],3:[5],4:[5],5:[6]})
sage: P.flag_f_polynomial()
3*x1*x2*x3 + 3*x1*x3 + x2*x3 + x3

sage: P = Poset({2: [3]})
sage: P.flag_f_polynomial()
x1

sage: P = Poset({2: []})
sage: P.flag_f_polynomial()
1

flag_h_polynomial()

Return the flag $$h$$-polynomial of a bounded and ranked poset self.

If $$P$$ is a bounded ranked poset whose maximal element has rank $$n$$ (where the minimal element is set to have rank $$0$$), then the flag $$h$$-polynomial of $$P$$ is defined as the polynomial

$\prod_{k=1}^n (1-x_k) \cdot f \left(\frac{x_1}{1-x_1}, \frac{x_2}{1-x_2}, \cdots, \frac{x_n}{1-x_n}\right) \in \ZZ[x_1, x_2, \cdots, x_n],$

where $$f$$ is the flag $$f$$-polynomial of $$P$$ (see flag_f_polynomial()).

For technical reasons, the polynomial is returned in the slightly larger ring $$\QQ[x_0, x_1, x_2, \cdots, x_{n+1}]$$ by this method.

EXAMPLES:

sage: P = Posets.DiamondPoset(5)
sage: P.flag_h_polynomial()
2*x1*x2 + x2

sage: P = Poset({1:[2,3],2:[4],3:[5],4:[6],5:[6]})
sage: fl = P.flag_h_polynomial(); fl
-x1*x2*x3 + x1*x3 + x2*x3 + x3
sage: q = polygen(ZZ,'q')
sage: fl(q,q,q,q) == P.h_polynomial()
True

sage: P = Poset({1:[2,3,4],2:[5],3:[5],4:[5],5:[6]})
sage: P.flag_h_polynomial()
2*x1*x3 + x3

sage: P = posets.ChainPoset(4)
sage: P.flag_h_polynomial()
x3

sage: P = Poset({2: [3]})
sage: P.flag_h_polynomial()
x1

sage: P = Poset({2: []})
sage: P.flag_h_polynomial()
1

frank_network()

Computes Frank’s network of the poset self. This is defined in Section 8 of [BF1999].

OUTPUT:

A pair $$(G, e)$$, where $$G$$ is Frank’s network of $$P$$ encoded as a DiGraph, and $$e$$ is the cost function on its edges encoded as a dictionary (indexed by these edges, which in turn are encoded as tuples of 2 vertices).

Note

Frank’s network of $$P$$ is a certain directed graph with $$2|P| + 2$$ vertices, defined in Section 8 of [BF1999]. Its set of vertices consists of two vertices $$(0, p)$$ and $$(1, p)$$ for each element $$p$$ of $$P$$, as well as two vertices $$(-1, 0)$$ and $$(2, 0)$$. (These notations are not the ones used in [BF1999]; see the table below for their relation.) The edges are:

• for each $$p$$ in $$P$$, an edge from $$(-1, 0)$$ to $$(0, p)$$;
• for each $$p$$ in $$P$$, an edge from $$(1, p)$$ to $$(2, 0)$$;
• for each $$p$$ and $$q$$ in $$P$$ such that $$x \geq y$$, an edge from $$(0, p)$$ to $$(1, q)$$.

We make this digraph into a network in the sense of flow theory as follows: The vertex $$(-1, 0)$$ is considered as the source of this network, and the vertex $$(2, 0)$$ as the sink. The cost function is defined to be $$1$$ on the edge from $$(0, p)$$ to $$(1, p)$$ for each $$p \in P$$, and to be $$0$$ on every other edge. The capacity is $$1$$ on each edge. Here is how to translate this notations into that used in [BF1999]:

our notations                    [BF1999]
(-1, 0)                          s
(0, p)                          x_p
(1, p)                          y_p
(2, 0)                           t
a[e]                           a(e)


REFERENCES:

 [BF1999] (1, 2, 3, 4, 5) Thomas Britz, Sergey Fomin, Finite posets and Ferrers shapes, Advances in Mathematics 158, pp. 86-127 (2001), Arxiv math/9912126 (the arXiv version has less errors).

EXAMPLES:

sage: ps = [[16,12,14,-13],[[12,14],[14,-13],[12,16],[16,-13]]]
sage: G, e = Poset(ps).frank_network()
sage: G.edges()
[((-1, 0), (0, -13), None), ((-1, 0), (0, 12), None), ((-1, 0), (0, 14), None), ((-1, 0), (0, 16), None), ((0, -13), (1, -13), None), ((0, -13), (1, 12), None), ((0, -13), (1, 14), None), ((0, -13), (1, 16), None), ((0, 12), (1, 12), None), ((0, 14), (1, 12), None), ((0, 14), (1, 14), None), ((0, 16), (1, 12), None), ((0, 16), (1, 16), None), ((1, -13), (2, 0), None), ((1, 12), (2, 0), None), ((1, 14), (2, 0), None), ((1, 16), (2, 0), None)]
sage: e
{((-1, 0), (0, -13)): 0,
((-1, 0), (0, 12)): 0,
((-1, 0), (0, 14)): 0,
((-1, 0), (0, 16)): 0,
((0, -13), (1, -13)): 1,
((0, -13), (1, 12)): 0,
((0, -13), (1, 14)): 0,
((0, -13), (1, 16)): 0,
((0, 12), (1, 12)): 1,
((0, 14), (1, 12)): 0,
((0, 14), (1, 14)): 1,
((0, 16), (1, 12)): 0,
((0, 16), (1, 16)): 1,
((1, -13), (2, 0)): 0,
((1, 12), (2, 0)): 0,
((1, 14), (2, 0)): 0,
((1, 16), (2, 0)): 0}
sage: qs = [[1,2,3,4,5,6,7,8,9],[[1,3],[3,4],[5,7],[1,9],[2,3]]]
sage: Poset(qs).frank_network()
(Digraph on 20 vertices,
{((-1, 0), (0, 1)): 0,
((-1, 0), (0, 2)): 0,
((-1, 0), (0, 3)): 0,
((-1, 0), (0, 4)): 0,
((-1, 0), (0, 5)): 0,
((-1, 0), (0, 6)): 0,
((-1, 0), (0, 7)): 0,
((-1, 0), (0, 8)): 0,
((-1, 0), (0, 9)): 0,
((0, 1), (1, 1)): 1,
((0, 2), (1, 2)): 1,
((0, 3), (1, 1)): 0,
((0, 3), (1, 2)): 0,
((0, 3), (1, 3)): 1,
((0, 4), (1, 1)): 0,
((0, 4), (1, 2)): 0,
((0, 4), (1, 3)): 0,
((0, 4), (1, 4)): 1,
((0, 5), (1, 5)): 1,
((0, 6), (1, 6)): 1,
((0, 7), (1, 5)): 0,
((0, 7), (1, 7)): 1,
((0, 8), (1, 8)): 1,
((0, 9), (1, 1)): 0,
((0, 9), (1, 9)): 1,
((1, 1), (2, 0)): 0,
((1, 2), (2, 0)): 0,
((1, 3), (2, 0)): 0,
((1, 4), (2, 0)): 0,
((1, 5), (2, 0)): 0,
((1, 6), (2, 0)): 0,
((1, 7), (2, 0)): 0,
((1, 8), (2, 0)): 0,
((1, 9), (2, 0)): 0})


AUTHOR:

• Darij Grinberg (2013-05-09)
ge(x, y)

Returns True if $$x$$ is greater than or equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = Q(0),Q(1),Q(4)
sage: Q.is_gequal(x,y)
False
sage: Q.is_gequal(y,x)
False
sage: Q.is_gequal(x,z)
False
sage: Q.is_gequal(z,x)
True
sage: Q.is_gequal(z,y)
True
sage: Q.is_gequal(z,z)
True

graphviz_string(graph_string='graph', edge_string='--')

Returns a representation in the DOT language, ready to render in graphviz.

REFERENCES:

EXAMPLES:

sage: P = Poset({'a':['b'],'b':['d'],'c':['d'],'d':['f'],'e':['f'],'f':[]})
sage: print P.graphviz_string()
graph {
"f";"d";"b";"a";"c";"e";
"f"--"e";"d"--"c";"b"--"a";"d"--"b";"f"--"d";
}

greene_shape()

Return the Greene-Kleitman partition of self.

The Greene-Kleitman partition of a finite poset $$P$$ is the partition $$(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$$, where $$c_k$$ is the maximum cardinality of a union of $$k$$ chains of $$P$$. Equivalently, this is the conjugate of the partition $$(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$$, where $$a_k$$ is the maximum cardinality of a union of $$k$$ antichains of $$P$$.

See many sources, e. g., [BF1999], for proofs of this equivalence.

EXAMPLES:

sage: P = Poset([[3,2,1],[[3,1],[2,1]]])
sage: P.greene_shape()
[2, 1]
sage: P = Poset([[1,2,3,4],[[1,4],[2,4],[4,3]]])
sage: P.greene_shape()
[3, 1]
sage: P = Poset([[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[[1,4],[2,4],[4,3]]])
sage: P.greene_shape()
[3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: P = Poset([[],[]])
sage: P.greene_shape()
[]


AUTHOR:

• Darij Grinberg (2013-05-09)
gt(x, y)

Returns True if $$x$$ is greater than but not equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = Q(0),Q(1),Q(4)
sage: Q.is_greater_than(x,y)
False
sage: Q.is_greater_than(y,x)
False
sage: Q.is_greater_than(x,z)
False
sage: Q.is_greater_than(z,x)
True
sage: Q.is_greater_than(z,y)
True
sage: Q.is_greater_than(z,z)
False

h_polynomial()

Return the $$h$$-polynomial of a bounded poset self.

This is the $$h$$-polynomial of the order complex of the poset minus its bounds.

This is related to the $$f$$-polynomial by a simple change of variables:

$h(q) = (1-q)^{\deg f} f \left( \frac{q}{1-q} \right),$

where $$f$$ and $$h$$ denote the $$f$$-polynomial and the $$h$$-polynomial, respectively.

Warning

This is slightly different from the hPolynomial method in Macaulay2.

EXAMPLES:

sage: P = Posets.AntichainPoset(3).order_ideals_lattice()
sage: P.h_polynomial()
q^3 + 4*q^2 + q
sage: P = Posets.DiamondPoset(5)
sage: P.h_polynomial()
2*q^2 + q
sage: P = Poset({1: []})
sage: P.h_polynomial()
1

has_bottom()

Returns True if the poset has a unique minimal element.

EXAMPLES:

sage: P = Poset({0:[3],1:[3],2:[3],3:[4],4:[]})
sage: P.has_bottom()
False
sage: Q = Poset({0:[1],1:[]})
sage: Q.has_bottom()
True

has_isomorphic_subposet(other)

Return True if the poset contains a subposet isomorphic to other.

By subposet we mean that there exist a set X of elements such that self.subposet(X) is isomorphic to other.

INPUT:

• other – a finite poset

EXAMPLES:

sage: D = Poset({1:[2,3], 2:[4], 3:[4]})
sage: T = Poset({1:[2,3], 2:[4,5], 3:[6,7]})
sage: N5 = Posets.PentagonPoset()

sage: N5.has_isomorphic_subposet(T)
False
sage: N5.has_isomorphic_subposet(D)
True

sage: len([P for P in Posets(5) if P.has_isomorphic_subposet(D)])
11

has_top()

Returns True if the poset contains a unique maximal element, and False otherwise.

EXAMPLES:

sage: P = Poset({0:[3],1:[3],2:[3],3:[4,5],4:[],5:[]})
sage: P.has_top()
False
sage: Q = Poset({0:[1],1:[]})
sage: Q.has_top()
True

hasse_diagram(wrapped=True)

Return the Hasse diagram of self as a Sage DiGraph. If dot2tex is installed, then this sets the Hasse diagram’s latex options to use the dot2tex formatting.

Todo

Should the vertices of the diagram have the poset as parent?

EXAMPLES:

sage: Q = Poset({5:[2,3], 1:[3,4], 2:[0], 3:[0], 4:[0]}, facade = False)
sage: Q.hasse_diagram()
Digraph on 6 vertices

sage: P = Poset({'a':['b'],'b':['d'],'c':['d'],'d':['f'],'e':['f'],'f':[]}, facade = False)
sage: H = P.hasse_diagram()
sage: P.cover_relations()
[[e, f], [c, d], [a, b], [b, d], [d, f]]
sage: H.edges()
[(a, b, None), (c, d, None), (b, d, None), (e, f, None), (d, f, None)]

sage: P = Poset((divisors(15), attrcall("divides")), facade = False)
sage: H = P.hasse_diagram()
sage: H.vertices()
[1, 5, 3, 15]
sage: H.edges()
[(1, 3, None), (1, 5, None), (5, 15, None), (3, 15, None)]
sage: H.set_latex_options(format = "dot2tex")   # optional - dot2tex
sage: view(H, tight_page=True) # optional - dot2tex

height()

Return the height (number of elements in the longest chain) of the poset.

EXAMPLES:

sage: P = Poset({0:[1],2:[3,4],4:[5,6]})
sage: P.height()
3
sage: Posets.PentagonPoset().height()
4
sage: Poset({}).height()
0

incomparability_graph()

Returns the incomparability graph of self.

This is the complement of the comparability graph.

EXAMPLES:

sage: p = posets.ChainPoset(4)
sage: p.incomparability_graph().size()
0

sage: p = posets.DiamondPoset(5)
sage: g = p.incomparability_graph(); g
Incomparability graph on 5 vertices
sage: g.size()
3

interval(x, y)

Return a list of the elements $$z$$ such that $$x \le z \le y$$.

INPUT:

• x – any element of the poset
• y – any element of the poset

EXAMPLES:

sage: uc = [[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]]
sage: dag = DiGraph(dict(zip(range(len(uc)),uc)))
sage: P = Poset(dag)
sage: I = set(map(P,[2,5,6,4,7]))
sage: I == set(P.interval(2,7))
True

sage: dg = DiGraph({"a":["b","c"], "b":["d"], "c":["d"]})
sage: P = Poset(dg, facade = False)
sage: P.interval("a","d")
[a, b, c, d]

intervals()

Returns a list of all relations of the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x\leq y$$ in self.

Relations are also often called intervals. The number of intervals is the dimension of the incidence algebra.

OUTPUT:

A list of pairs (each pair is a list), where the first element of the pair is less than or equal to the second element.

Pairs are produced in a rough sort of lexicographic order, where earlier elements are from lower levels of the poset.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: Q.relations()
[[1, 1], [1, 2], [1, 3], [1, 4], [0, 0], [0, 2], [0, 3],
[0, 4], [2, 2], [2, 3], [2, 4], [3, 3], [3, 4], [4, 4]]


AUTHOR:

• Rob Beezer (2011-05-04)
intervals_iterator(strict=False)

Returns an iterator for all the relations of the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x\leq y$$ in self.

Relations are also often called intervals. The number of intervals is the dimension of the incidence algebra.

INPUT:

• strict – boolean (default False) if True, returns an iterator over relations $$x < y$$, excluding all relations $$x \leq x$$.

OUTPUT:

A generator that produces pairs (each pair is a list), where the first element of the pair is less than or equal to the second element.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: type(Q.relations_iterator())
<type 'generator'>
sage: [z for z in Q.relations_iterator()]
[[1, 1], [1, 2], [1, 3], [1, 4], [0, 0], [0, 2], [0, 3],
[0, 4], [2, 2], [2, 3], [2, 4], [3, 3], [3, 4], [4, 4]]

sage: P = posets.PentagonPoset()
sage: list(P.relations_iterator(strict=True))
[[0, 1], [0, 2], [0, 4], [0, 3], [1, 4], [2, 3], [2, 4], [3, 4]]

sage: len(list(P.relations_iterator()))
13


AUTHOR:

• Rob Beezer (2011-05-04)
intervals_number()

Return the number of relations in the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x\leq y$$ in self.

Relations are also often called intervals. The number of intervals is the dimension of the incidence algebra.

EXAMPLES:

sage: from sage.combinat.tamari_lattices import TamariLattice
sage: TamariLattice(4).relations_number()
68

sage: P = posets.BooleanLattice(3)
sage: P.relations_number()
27

is_EL_labelling(f, return_raising_chains=False)

Returns True if f is an EL labelling of self.

A labelling $$f$$ of the edges of the Hasse diagram of a poset is called an EL labelling (edge lexicographic labelling) if for any two elements $$u$$ and $$v$$ with $$u \leq v$$,

• there is a unique $$f$$-raising chain from $$u$$ to $$v$$ in the Hasse diagram, and this chain is lexicographically first among all chains from $$u$$ to $$v$$.

For more details, see [Bj1980].

INPUT:

• f – a function taking two elements a and b in self such that b covers a and returning elements in a totally ordered set.
• return_raising_chains (optional; default:False) if True, returns the set of all raising chains in self, if possible.

EXAMPLES:

Let us consider a Boolean poset:

sage: P = Poset([[(0,0),(0,1),(1,0),(1,1)],[[(0,0),(0,1)],[(0,0),(1,0)],[(0,1),(1,1)],[(1,0),(1,1)]]],facade=True)
sage: label = lambda a,b: min( i for i in [0,1] if a[i] != b[i] )
sage: P.is_EL_labelling(label)
True
sage: P.is_EL_labelling(label,return_raising_chains=True)
{((0, 0), (0, 1)): [1],
((0, 0), (1, 0)): [0],
((0, 0), (1, 1)): [0, 1],
((0, 1), (1, 1)): [0],
((1, 0), (1, 1)): [1]}


REFERENCES:

 [Bj1980] Anders Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), 159-183, doi:10.1090/S0002-9947-1980-0570784-2
is_bounded()

Return True if the poset self is bounded, and False otherwise.

We call a poset bounded if it contains a unique maximal element and a unique minimal element.

EXAMPLES:

sage: P = Poset({0:[3],1:[3],2:[3],3:[4,5],4:[],5:[]})
sage: P.is_bounded()
False
sage: Q = Poset({0:[1],1:[]})
sage: Q.is_bounded()
True

is_chain()

Returns True if the poset is totally ordered, and False otherwise.

EXAMPLES:

sage: L = Poset({0:[1],1:[2],2:[3],3:[4]})
sage: L.is_chain()
True
sage: V = Poset({0:[1,2]})
sage: V.is_chain()
False

is_chain_of_poset(o, ordered=False)

Return whether an iterable o is a chain of self, including a check for o being ordered from smallest to largest element if the keyword ordered is set to True.

INPUT:

• o – an iterable (e. g., list, set, or tuple) containing some elements of self
• ordered – a Boolean (default: False) which decides whether the notion of a chain includes being ordered

OUTPUT:

If ordered is set to False, the truth value of the following assertion is returned: The subset of self formed by the elements of o is a chain in self.

If ordered is set to True, the truth value of the following assertion is returned: Every element of the list o is (strictly!) smaller than its successor in self. (This makes no sense if ordered is a set.)

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")))
sage: sorted(P.list())
[1, 2, 3, 4, 6, 12]
sage: P.is_chain_of_poset([2, 4])
True
sage: P.is_chain_of_poset([12, 6])
True
sage: P.is_chain_of_poset([12, 6], ordered=True)
False
sage: P.is_chain_of_poset([6, 12], ordered=True)
True
sage: P.is_chain_of_poset(())
True
sage: P.is_chain_of_poset((), ordered=True)
True
sage: P.is_chain_of_poset((3, 4, 12))
False
sage: P.is_chain_of_poset((3, 6, 12, 1))
True
sage: P.is_chain_of_poset((3, 6, 12, 1), ordered=True)
False
sage: P.is_chain_of_poset((3, 6, 12), ordered=True)
True
sage: P.is_chain_of_poset((1, 1, 3))
True
sage: P.is_chain_of_poset((1, 1, 3), ordered=True)
False
sage: P.is_chain_of_poset((1, 3), ordered=True)
True
sage: P.is_chain_of_poset((6, 1, 1, 3))
True
sage: P.is_chain_of_poset((2, 1, 1, 3))
False

is_connected()

Return True if the poset is connected, and False otherwise.

Poset is not connected if it can be divided to disjoint parts $$S_1$$ and $$S_2$$ so that every element of $$S_1$$ is incomparable to every element of $$S_2$$.

EXAMPLES:

sage: P=Poset({1:[2,3], 3:[4,5]})
sage: P.is_connected()
True
sage: P=Poset({1:[2,3], 3:[4,5], 6:[7,8]})
sage: P.is_connected()
False

is_gequal(x, y)

Returns True if $$x$$ is greater than or equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = Q(0),Q(1),Q(4)
sage: Q.is_gequal(x,y)
False
sage: Q.is_gequal(y,x)
False
sage: Q.is_gequal(x,z)
False
sage: Q.is_gequal(z,x)
True
sage: Q.is_gequal(z,y)
True
sage: Q.is_gequal(z,z)
True


Returns whether this poset is graded.

A poset is graded if all its maximal chains have the same length. There are various competing definitions for graded posets (see Wikipedia article Graded_poset). This definition is from section 3.1 of Richard Stanley’s Enumerative Combinatorics, Vol. 1 [EnumComb1].

Note that every graded poset is ranked, but the converse is not true.

is_ranked()

EXAMPLES:

sage: P = Poset([[1],[2],[3],[4],[]])
True
sage: Q = Poset([[1,5],[2,6],[3],[4],[],[6,3],[4]])
False
sage: P = Poset( ([1,2,3,4],[[1,2],[2,4],[3,4]] ))
False
sage: P = Poset({1: [2, 3], 4: [5]})
True
sage: P = Poset({1: [2, 3], 3: [4]})
False
sage: P = Poset({1: [2, 3], 4: []})
False
sage: P = Posets.BooleanLattice(4)
True
sage: P = RootSystem(['D',4]).root_poset()
True
sage: P = Poset({})
True


TESTS:

Here we test that the empty poset is graded:

sage: Poset([[],[]]).is_graded()
True

is_greater_than(x, y)

Returns True if $$x$$ is greater than but not equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = Q(0),Q(1),Q(4)
sage: Q.is_greater_than(x,y)
False
sage: Q.is_greater_than(y,x)
False
sage: Q.is_greater_than(x,z)
False
sage: Q.is_greater_than(z,x)
True
sage: Q.is_greater_than(z,y)
True
sage: Q.is_greater_than(z,z)
False

is_incomparable_chain_free(m, n=None)

Returns True if the poset is $$(m+n)$$-free (that is, there is no pair of incomparable chains of lengths $$m$$ and $$n$$), and False if not.

If m is a tuple of pairs of chain lengths, returns True if the poset does not contain a pair of incomparable chains whose lengths comprise one of the chain pairs, and False if not. A poset is $$(m+n)$$-free if it contains no induced subposet that is isomorphic to the poset consisting of two disjoint chains of lengths $$m$$ and $$n$$. See, for example, Exercise 15 in Chapter 3 of [EnumComb1].

INPUT:

• m - tuple of pairs of nonnegative integers
• m, n - nonnegative integers

EXAMPLES:

sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: P.is_incomparable_chain_free(1, 1)
False
sage: P.is_incomparable_chain_free(2, 1)
True

sage: P = Poset(((0, 1, 2, 3, 4), ((0, 1), (1, 2), (0, 3), (4, 2))))
sage: P.is_incomparable_chain_free(((3, 1), (2, 2)))
True

sage: P = Poset((("a", "b", "c", "d", "e", "f", "g", "h", "i", "j"), (("d", "a"), ("e", "a"), ("f", "a"), ("g", "a"), ("h", "b"), ("f", "b"), ("h", "c"), ("g", "c"), ("h", "d"), ("i", "d"), ("h", "e"), ("i", "e"), ("j", "f"), ("i", "f"), ("j", "g"), ("i", "g"), ("j", "h"))))
sage: P.is_incomparable_chain_free(3, 1)
True
sage: P.is_incomparable_chain_free(2, 2)
False

sage: [len([p for p in Posets(n) if p.is_incomparable_chain_free(((3, 1), (2, 2)))]) for n in range(6)] # long time
[1, 1, 2, 5, 14, 42]


TESTS:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: Q.is_incomparable_chain_free(2, 20/10)
True
sage: Q.is_incomparable_chain_free(2, pi)
Traceback (most recent call last):
...
TypeError: 2 and pi must be integers.
sage: Q.is_incomparable_chain_free(2, -1)
Traceback (most recent call last):
...
ValueError: 2 and -1 must be nonnegative integers.
sage: P = Poset(((0, 1, 2, 3, 4), ((0, 1), (1, 2), (0, 3), (4, 2))))
sage: P.is_incomparable_chain_free((3, 1))
Traceback (most recent call last):
...
TypeError: (3, 1) is not a tuple of tuples.
sage: P.is_incomparable_chain_free([3, 1], [2, 2])
Traceback (most recent call last):
...
TypeError: [3, 1] and [2, 2] must be integers.
sage: P.is_incomparable_chain_free([[3, 1], [2, 2]])
True
sage: P.is_incomparable_chain_free(([3, 1], [2, 2]))
True
sage: P.is_incomparable_chain_free([3, 1], 2)
Traceback (most recent call last):
...
TypeError: [3, 1] and 2 must be integers.
sage: P.is_incomparable_chain_free(([3, 1], [2, 2, 2]))
Traceback (most recent call last):
...
ValueError: '([3, 1], [2, 2, 2])' is not a tuple of length-2 tuples.


AUTHOR:

• Eric Rowland (2013-05-28)

REFERENCES:

 [EnumComb1] (1, 2, 3, 4, 5, 6) Richard P. Stanley, Enumerative Combinatorics, volume 1, Second Edition, Cambridge University Press (2011). http://math.mit.edu/~rstan/ec/ec1/
is_isomorphic(other)

Returns True if both posets are isomorphic.

EXAMPLES:

sage: P = Poset(([1,2,3],[[1,3],[2,3]]))
sage: Q = Poset(([4,5,6],[[4,6],[5,6]]))
sage: P.is_isomorphic( Q )
True

is_join_semilattice()

Returns True is the poset has a join operation, and False otherwise.

EXAMPLES:

sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]])
sage: P.is_join_semilattice()
True

sage: P = Poset([[1,2],[3],[3],[]])
sage: P.is_join_semilattice()
True

sage: P = Poset({0:[2,3],1:[2,3]})
sage: P.is_join_semilattice()
False

is_lequal(x, y)

Returns True if $$x$$ is less than or equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = Q(0),Q(1),Q(4)
sage: Q.is_lequal(x,y)
False
sage: Q.is_lequal(y,x)
False
sage: Q.is_lequal(x,z)
True
sage: Q.is_lequal(y,z)
True
sage: Q.is_lequal(z,z)
True

is_less_than(x, y)

Returns True if $$x$$ is less than but not equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = Q(0),Q(1),Q(4)
sage: Q.is_less_than(x,y)
False
sage: Q.is_less_than(y,x)
False
sage: Q.is_less_than(x,z)
True
sage: Q.is_less_than(y,z)
True
sage: Q.is_less_than(z,z)
False

is_linear_extension(l)

Returns whether l is a linear extension of self

INPUT:

• l – a list (or iterable) containing all of the elements of self exactly once

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: P.is_linear_extension([1, 2, 4, 3, 6, 12])
True
sage: P.is_linear_extension([1, 2, 4, 6, 3, 12])
False

sage: [p for p in Permutations(list(P)) if P.is_linear_extension(p)]
[[1, 2, 3, 4, 6, 12],
[1, 2, 3, 6, 4, 12],
[1, 2, 4, 3, 6, 12],
[1, 3, 2, 4, 6, 12],
[1, 3, 2, 6, 4, 12]]
sage: list(P.linear_extensions())
[[1, 2, 3, 4, 6, 12],
[1, 2, 3, 6, 4, 12],
[1, 2, 4, 3, 6, 12],
[1, 3, 2, 4, 6, 12],
[1, 3, 2, 6, 4, 12]]


Note

This is used and systematically tested in LinearExtensionsOfPosets

TESTS:

Check that trac ticket #15313 is fixed:

sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
sage: P.is_linear_extension([1,2,4,3,6,12,1337])
False
sage: P.is_linear_extension([1,2,4,3,6,666,12,1337])
False
sage: P = Poset(DiGraph(5))
sage: P.is_linear_extension(['David', 'McNeil', 'La', 'Lamentable', 'Aventure', 'de', 'Simon', 'Wiesenthal'])
False

is_meet_semilattice()

Returns True if self has a meet operation, and False otherwise.

EXAMPLES:

sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]], facade = False)
sage: P.is_meet_semilattice()
True

sage: P = Poset([[1,2],[3],[3],[]])
sage: P.is_meet_semilattice()
True

sage: P = Poset({0:[2,3],1:[2,3]})
sage: P.is_meet_semilattice()
False

is_parent_of(x)

Returns True if x is an element of the poset.

TESTS:

sage: from sage.combinat.posets.posets import FinitePoset
sage: P5 = FinitePoset(DiGraph({(5,):[(4,1),(3,2)], \
(4,1):[(3,1,1),(2,2,1)], \
(3,2):[(3,1,1),(2,2,1)], \
(3,1,1):[(2,1,1,1)], \
(2,2,1):[(2,1,1,1)], \
(2,1,1,1):[(1,1,1,1,1)], \
(1,1,1,1,1):[]}))
sage: x = P5.list()[3]
sage: x in P5
True


For the sake of speed, an element with the right class and parent is assumed to be in this parent. This can possibly be counterfeited by feeding garbage to the constructor:

sage: x = P5.element_class(P5, "a", 5)
sage: x in P5
True

is_ranked()

Returns whether this poset is ranked.

EXAMPLES:

sage: P = Poset([[1],[2],[3],[4],[]])
sage: P.is_ranked()
True
sage: Q = Poset([[1,5],[2,6],[3],[4],[],[6,3],[4]])
sage: Q.is_ranked()
False
sage: P = Poset( ([1,2,3,4],[[1,2],[2,4],[3,4]] ))
sage: P.is_ranked()
True

is_slender()

Return whether the poset self is slender or not.

It is assumed for this method that self is a finite graded poset. A finite poset $$P$$ is called slender if every rank 2 interval contains three or four elements. See [Stan2009].

EXAMPLES:

sage: P = Poset(([1,2,3,4],[[1,2],[1,3],[2,4],[3,4]]), facade = True)
sage: P.is_slender()
True
sage: P = Poset(([1,2,3,4,5],[[1,2],[1,3],[1,4],[2,5],[3,5],[4,5]]), facade = True)
sage: P.is_slender()
False

sage: W = WeylGroup(['A',2])
sage: G = W.bruhat_poset()
sage: G.is_slender()
True
sage: W = WeylGroup(['A',3])
sage: G = W.bruhat_poset()
sage: G.is_slender()
True

isomorphic_subposets(other)

Return a list of subposets of $$self$$ isomorphic to $$other$$.

By subposet we mean self.subposet(X) which is isomorphic to other and where X is a subset of elements of self.

INPUT:

• other – a finite poset

EXAMPLES:

sage: C2=Poset({0:[1]})
sage: C3=Poset({'a':['b'], 'b':['c']})
sage: for x in C3.isomorphic_subposets(C2): print x.cover_relations()
[['b', 'c']]
[['a', 'c']]
[['a', 'b']]
sage: D = Poset({1:[2,3], 2:[4], 3:[4]})
sage: N5 = Posets.PentagonPoset()
sage: len(N5.isomorphic_subposets(D))
2


Note

If this function takes too much time, try using isomorphic_subposets_iterator().

isomorphic_subposets_iterator(other)

Return an iterator over the subposets of $$self$$ isomorphic to $$other$$.

By subposet we mean self.subposet(X) which is isomorphic to other and where X is a subset of elements of self.

INPUT:

• other – a finite poset

EXAMPLES:

sage: D = Poset({1:[2,3], 2:[4], 3:[4]})
sage: N5 = Posets.PentagonPoset()
sage: for P in N5.isomorphic_subposets_iterator(D):
....:     print P.cover_relations()
[[0, 1], [0, 2], [1, 4], [2, 4]]
[[0, 1], [0, 3], [1, 4], [3, 4]]
[[0, 1], [0, 2], [1, 4], [2, 4]]
[[0, 1], [0, 3], [1, 4], [3, 4]]


Warning

This function will return same subposet as many times as there are automorphism on it. This is due to subgraph_search_iterator() returning labelled subgraphs. On the other hand, this function does not eat memory like isomorphic_subposets() does.

join_matrix()

Deprecated as a function of posets, moved to lattices.

Convert a poset $$P$$ to join-semilattice and use it like JoinSemilattice(P).join_matrix().

le(x, y)

Returns True if $$x$$ is less than or equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = Q(0),Q(1),Q(4)
sage: Q.is_lequal(x,y)
False
sage: Q.is_lequal(y,x)
False
sage: Q.is_lequal(x,z)
True
sage: Q.is_lequal(y,z)
True
sage: Q.is_lequal(z,z)
True

lequal_matrix(ring=Integer Ring, sparse=False)

Computes the matrix whose (i,j) entry is 1 if self.linear_extension()[i] < self.linear_extension()[j] and 0 otherwise.

INPUT:

• ring – the ring of coefficients (default: ZZ)
• sparse – whether the returned matrix is sparse or not (default: True)

EXAMPLES:

sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]], facade = False)
sage: LEQM = P.lequal_matrix(); LEQM
[1 1 1 1 1 1 1 1]
[0 1 0 1 0 0 0 1]
[0 0 1 1 1 0 1 1]
[0 0 0 1 0 0 0 1]
[0 0 0 0 1 0 0 1]
[0 0 0 0 0 1 1 1]
[0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 1]
sage: LEQM[1,3]
1
sage: P.linear_extension()[1] < P.linear_extension()[3]
True
sage: LEQM[2,5]
0
sage: P.linear_extension()[2] < P.linear_extension()[5]
False


We now demonstrate the usage of the optional parameters:

sage: P.lequal_matrix(ring=QQ, sparse=False).parent()
Full MatrixSpace of 8 by 8 dense matrices over Rational Field

level_sets()

Return a list l such that l[i] is the set of minimal elements of the poset obtained from self by removing the elements in l[0], l[1], ..., l[i-1]. (In particular, l[0] is the set of minimal elements of self.)

EXAMPLES:

sage: P = Poset({0:[1,2],1:[3],2:[3],3:[]})
sage: [len(x) for x in P.level_sets()]
[1, 2, 1]

sage: Q = Poset({0:[1,2], 1:[3], 2:[4], 3:[4]})
sage: [len(x) for x in Q.level_sets()]
[1, 2, 1, 1]

linear_extension(linear_extension=None, check=True)

Return a linear extension of this poset.

A linear extension of a finite poset $$P$$ of size $$n$$ is a total ordering $$\pi := \pi_0 \pi_1 \ldots \pi_{n-1}$$ of its elements such that $$i<j$$ whenever $$\pi_i < \pi_j$$ in the poset $$P$$.

INPUT:

• linear_extension – (default: None) a list of the elements of self
• check – a boolean (default: True); whether to check that linear_extension is indeed a linear extension of self.

EXAMPLES:

sage: P = Poset((divisors(15), attrcall("divides")), facade=True)


Without optional argument, the default linear extension of the poset is returned, as a plain list:

sage: P.linear_extension()
[1, 3, 5, 15]


Otherwise, a full-featured linear extension is constructed as an element of P.linear_extensions():

sage: l = P.linear_extension([1,5,3,15]); l
[1, 5, 3, 15]
sage: type(l)
<class 'sage.combinat.posets.linear_extensions.LinearExtensionsOfPoset_with_category.element_class'>
sage: l.parent()
The set of all linear extensions of Finite poset containing 4 elements


By default, the linear extension is checked for correctness:

sage: l = P.linear_extension([1,3,15,5])
Traceback (most recent call last):
...
ValueError: [1, 3, 15, 5] is not a linear extension of Finite poset containing 4 elements


This can be disabled (at your own risks!) with:

sage: P.linear_extension([1,3,15,5], check=False)
[1, 3, 15, 5]


Todo

• Is it acceptable to have those two features for a single method?
• In particular, we miss a short idiom to get the default linear extension

Returns the enumerated set of all the linear extensions of this poset

INPUT:

• facade – a boolean (default: False); whether to return the linear extensions as plain lists

Warning

The facade option is not yet fully functional:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
The set of all linear extensions of Finite poset containing 6 elements with distinguished linear extension
sage: L([1, 2, 3, 4, 6, 12])
Traceback (most recent call last):
...
TypeError: Cannot convert list to sage.structure.element.Element


EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: L = P.linear_extensions(); L
The set of all linear extensions of Finite poset containing 6 elements with distinguished linear extension
sage: l = L.an_element(); l
[1, 2, 3, 4, 6, 12]
sage: L.cardinality()
5
sage: L.list()
[[1, 2, 3, 4, 6, 12],
[1, 2, 3, 6, 4, 12],
[1, 2, 4, 3, 6, 12],
[1, 3, 2, 4, 6, 12],
[1, 3, 2, 6, 4, 12]]


Each element is aware that it is a linear extension of $$P$$:

sage: type(l.parent())
<class 'sage.combinat.posets.linear_extensions.LinearExtensionsOfPoset_with_category'>


sage: L = P.linear_extensions(facade=True)
sage: l = L.an_element()
sage: type(l)
<type 'list'>


Warning

In Sage <= 4.8, this function used to return a plain list of lists. To recover the previous functionality, please use:

sage: L = list(P.linear_extensions(facade=True)); L
[[1, 2, 3, 4, 6, 12],
[1, 2, 3, 6, 4, 12],
[1, 2, 4, 3, 6, 12],
[1, 3, 2, 4, 6, 12],
[1, 3, 2, 6, 4, 12]]
sage: type(L[0])
<type 'list'>


TESTS:

sage: D = Poset({ 0:[1,2], 1:[3], 2:[3,4] })
sage: list(D.linear_extensions())
[[0, 1, 2, 3, 4], [0, 1, 2, 4, 3], [0, 2, 1, 3, 4], [0, 2, 1, 4, 3], [0, 2, 4, 1, 3]]

list()

List the elements of the poset. This just returns the result of linear_extension().

EXAMPLES:

sage: D = Poset({ 0:[1,2], 1:[3], 2:[3,4] }, facade = False)
sage: D.list()
[0, 1, 2, 3, 4]
sage: type(D.list()[0])
<class 'sage.combinat.posets.elements.FinitePoset_with_category.element_class'>

lower_covers(y)

Returns a list of lower covers of the element y. An lower cover of y is an element x such that y x is a cover relation.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: map(Q.lower_covers,Q.list())
[[], [], [1, 0], [2], [3]]

lower_covers_iterator(y)

Returns an iterator for the lower covers of the element y. An lower cover of y is an element x such that y x is a cover relation.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: type(Q.lower_covers_iterator(0))
<type 'generator'>

lt(x, y)

Returns True if $$x$$ is less than but not equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = Q(0),Q(1),Q(4)
sage: Q.is_less_than(x,y)
False
sage: Q.is_less_than(y,x)
False
sage: Q.is_less_than(x,z)
True
sage: Q.is_less_than(y,z)
True
sage: Q.is_less_than(z,z)
False

maximal_antichains()

Return all maximal antichains of the poset.

EXAMPLES:

sage: P=Poset({'a':['b', 'c'], 'b':['d','e']})
sage: P.maximal_antichains()
[['a'], ['b', 'c'], ['c', 'd', 'e']]

sage: Posets.PentagonPoset().maximal_antichains()
[[0], [1, 2], [1, 3], [4]]


maximal_chains(partial=None)

Return all maximal chains of this poset.

Each chain is listed in increasing order.

INPUT:

• partial – list (optional); if present, find all maximal chains starting with the elements in partial

Returns list of the maximal chains of this poset.

This is used in constructing the order complex for the poset.

EXAMPLES:

sage: P = Posets.BooleanLattice(3)
sage: P.maximal_chains()
[[0, 1, 3, 7], [0, 1, 5, 7], [0, 2, 3, 7], [0, 2, 6, 7], [0, 4, 5, 7], [0, 4, 6, 7]]
sage: P.maximal_chains(partial=[0,2])
[[0, 2, 3, 7], [0, 2, 6, 7]]
sage: Q = Posets.ChainPoset(6)
sage: Q.maximal_chains()
[[0, 1, 2, 3, 4, 5]]


maximal_elements()

Returns a list of the maximal elements of the poset.

EXAMPLES:

sage: P = Poset({0:[3],1:[3],2:[3],3:[4],4:[]})
sage: P.maximal_elements()
[4]

meet_matrix()

Deprecated as a function of posets, moved to lattices.

Convert a poset $$P$$ to meet-semilattice and use it like MeetSemilattice(P).join_matrix().

minimal_elements()

Returns a list of the minimal elements of the poset.

EXAMPLES:

sage: P = Poset({0:[3],1:[3],2:[3],3:[4],4:[]})
sage: P(0) in P.minimal_elements()
True
sage: P(1) in P.minimal_elements()
True
sage: P(2) in P.minimal_elements()
True

mobius_function(x, y)

Returns the value of the Mobius function of the poset on the elements x and y.

EXAMPLES:

sage: P = Poset([[1,2,3],[4],[4],[4],[]])
sage: P.mobius_function(P(0),P(4))
2
sage: sum([P.mobius_function(P(0),v) for v in P])
0
sage: sum([abs(P.mobius_function(P(0),v)) \
....:      for v in P])
6
sage: for u,v in P.cover_relations_iterator():
....:     if P.mobius_function(u,v) != -1:
....:         print "Bug in mobius_function!"

sage: Q = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]])
sage: Q.mobius_function(Q(0),Q(7))
0
sage: Q.mobius_function(Q(0),Q(5))
0
sage: Q.mobius_function(Q(2),Q(7))
2
sage: Q.mobius_function(Q(3),Q(3))
1
sage: sum([Q.mobius_function(Q(0),v) for v in Q])
0

mobius_function_matrix(ring=Integer Ring, sparse=False)

Returns a matrix whose (i,j) entry is the value of the Mobius function evaluated at self.linear_extension()[i] and self.linear_extension()[j].

INPUT:

• ring – the ring of coefficients (default: ZZ)
• sparse – whether the returned matrix is sparse or not (default: True)

EXAMPLES:

sage: P = Poset([[4,2,3],[],[1],[1],[1]])
sage: x,y = (P.linear_extension()[0],P.linear_extension()[1])
sage: P.mobius_function(x,y)
-1
sage: M = P.mobius_function_matrix(); M
[ 1 -1 -1 -1  2]
[ 0  1  0  0 -1]
[ 0  0  1  0 -1]
[ 0  0  0  1 -1]
[ 0  0  0  0  1]
sage: M[0,4]
2
sage: M[0,1]
-1


We now demonstrate the usage of the optional parameters:

sage: P.mobius_function_matrix(ring=QQ, sparse=False).parent()
Full MatrixSpace of 5 by 5 dense matrices over Rational Field

open_interval(x, y)

Return a list of the elements $$z$$ such that $$x < z < y$$.

EXAMPLES:

sage: uc = [[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]]
sage: dag = DiGraph(dict(zip(range(len(uc)),uc)))
sage: P = Poset(dag)
sage: I = set(map(P,[5,6,4]))
sage: I == set(P.open_interval(2,7))
True

sage: dg = DiGraph({"a":["b","c"], "b":["d"], "c":["d"]})
sage: P = Poset(dg, facade = False)
sage: P.open_interval("a","d")
[b, c]

order_complex(on_ints=False)

Return the order complex associated to this poset.

The order complex is the simplicial complex with vertices equal to the elements of the poset, and faces given by the chains.

INPUT:

• on_ints – a boolean (default: False)

EXAMPLES:

sage: P = Posets.BooleanLattice(3)
sage: S = P.order_complex(); S
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and 6 facets
sage: S.f_vector()
[1, 8, 19, 18, 6]
sage: S.homology()      # S is contractible
{0: 0, 1: 0, 2: 0, 3: 0}
sage: Q = P.subposet([1,2,3,4,5,6])
sage: Q.order_complex().homology()    # a circle
{0: 0, 1: Z}

sage: P = Poset((divisors(15), attrcall("divides")), facade = True)
sage: P.order_complex()
Simplicial complex with vertex set (1, 3, 5, 15) and facets {(1, 3, 15), (1, 5, 15)}


If on_ints, then the elements of the poset are labelled $$0,1,\dots$$ in the chain complex:

sage: P.order_complex(on_ints=True)
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 3)}

order_filter(elements)

Returns the order filter generated by the elements of an iterable elements.

$$I$$ is an order filter if, for any $$x$$ in $$I$$ and $$y$$ such that $$y \ge x$$, then $$y$$ is in $$I$$.

EXAMPLES:

sage: B = Posets.BooleanLattice(4)
sage: B.order_filter([3,8])
[3, 7, 8, 9, 10, 11, 12, 13, 14, 15]

order_ideal(elements)

Returns the order ideal generated by the elements of an iterable elements.

$$I$$ is an order ideal if, for any $$x$$ in $$I$$ and $$y$$ such that $$y \le x$$, then $$y$$ is in $$I$$.

EXAMPLES:

sage: B = Posets.BooleanLattice(4)
sage: B.order_ideal([7,10])
[0, 1, 2, 3, 4, 5, 6, 7, 8, 10]
sage: B.order_ideal(iter(range(4, 9)))
[0, 1, 2, 3, 4, 5, 6, 7, 8]

order_polynomial()

Return the order polynomial of self.

The order polynomial $$\Omega_P(q)$$ of a poset $$P$$ is defined as the unique polynomial $$S$$ such that for each integer $$m \geq 1$$, $$S(m)$$ is the number of order-preserving maps from $$P$$ to $$\{1,\ldots,m\}$$.

See sections 3.12 and 3.15 of [EnumComb1], and also [St1986].

order_polytope()

EXAMPLES:

sage: P = Posets.AntichainPoset(3)
sage: P.order_polynomial()
q^3

sage: P = Posets.ChainPoset(3)
sage: f = P.order_polynomial(); f
1/6*q^3 + 1/2*q^2 + 1/3*q
sage: [f(i) for i in range(4)]
[0, 1, 4, 10]

order_polytope()

Return the order polytope of the poset self.

The order polytope of a finite poset $$P$$ is defined as the subset of $$\RR^P$$ consisting of all maps $$x : P \to \RR$$ satisfying

$0 \leq x(p) \leq 1 \mbox{ for all } p \in P,$

and

$\begin{split}x(p) \leq x(q) \mbox{ for all } p, q \in P \mbox{ satisfying } p < q.\end{split}$

This polytope was defined and studied in [St1986].

EXAMPLES:

sage: P = posets.AntichainPoset(3)
sage: Q = P.order_polytope();Q
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices
sage: P = posets.PentagonPoset()
sage: Q = P.order_polytope();Q
A 5-dimensional polyhedron in QQ^5 defined as the convex hull of 8 vertices

sage: P = Poset([[1,2,3],[[1,2],[1,3]]])
sage: Q = P.order_polytope()
sage: Q.contains((1,0,0))
False
sage: Q.contains((0,1,1))
True


REFERENCES:

 [St1986] (1, 2, 3, 4) Richard Stanley. Two poset polytopes, Discrete Comput. Geom. (1986), doi:10.1007/BF02187680
ordinal_product(other, labels='pairs')

Return the ordinal product of self and other.

The ordinal product of two posets $$P$$ and $$Q$$ is a partial order on the cartesian product of the underlying sets of $$P$$ and $$Q$$, defined as follows (see [EnumComb1], p. 284).

In the ordinal product, $$(p,q) \leq (p',q')$$ if either $$p \leq p'$$ or $$p = p'$$ and $$q \leq q'$$.

This construction is not symmetric in $$P$$ and $$Q$$.

INPUT:

• other – a poset
• labels – either 'integers' or 'pairs' (default); how the resulting poset will be labeled

EXAMPLES:

sage: P1 = Poset((['a', 'b'], [['a', 'b']]))
sage: P2 = Poset((['c', 'd'], [['c', 'd']]))
sage: P = P1.ordinal_product(P2); P
Finite poset containing 4 elements
sage: sorted(P.cover_relations())
[[('a', 'c'), ('a', 'd')], [('a', 'd'), ('b', 'c')],
[('b', 'c'), ('b', 'd')]]


TESTS:

sage: P1.ordinal_product(24)
Traceback (most recent call last):
...
ValueError: the input is not a finite poset
sage: P1.ordinal_product(P2, labels='camembert')
Traceback (most recent call last):
...
ValueError: labels must be either 'pairs' or 'integers'

ordinal_sum(other, labels='pairs')

Return a poset or (semi)lattice isomorphic to ordinal sum of the poset with other.

The ordinal sum of $$P$$ and $$Q$$ is a poset that contains every element and relation from both $$P$$ and $$Q$$, and where every element of $$P$$ is smaller than every element of $$Q$$.

Mathematically, it is only defined when $$P$$ and $$Q$$ have no common element; here we force that by giving them different names in the resulting poset.

The ordinal sum on lattices is lattice; resp. for meet- and join-semilattices. Hence we check if we can return (semi)lattice instead of plain poset.

INPUT:

• other, a poset.
• labels - (defaults to ‘pairs’) If set to ‘pairs’, each element v in this poset will be named (0,v) and each element u in other will be named (1,u) in the result. If set to ‘integers’, the elements of the result will be relabeled with consecutive integers.

EXAMPLES:

sage: P1 = Poset( ([1, 2, 3, 4], [[1, 2], [1, 3], [1, 4]]) )
sage: P2 = Poset( ([1, 2, 3,], [[2,1], [3,1]]) )
sage: P3 = P1.ordinal_sum(P2); P3
Finite poset containing 7 elements
sage: len(P1.maximal_elements())*len(P2.minimal_elements())
6
sage: len(P1.cover_relations()+P2.cover_relations())
5
sage: len(P3.cover_relations()) # Every element of P2 is greater than elements of P1.
11
sage: P3.list()  # random
[(0, 1), (0, 2), (0, 4), (0, 3), (1, 2), (1, 3), (1, 1)]
sage: P4 = P1.ordinal_sum(P2, labels='integers')
sage: P4.list()  # random
[0, 1, 2, 3, 5, 6, 4]


Return type depends on input types:

sage: P = Poset({1:[2]}); P
Finite poset containing 2 elements
sage: JL = JoinSemilattice({1:[2]}); JL
Finite join-semilattice containing 2 elements
sage: L = LatticePoset({1:[2]}); L
Finite lattice containing 2 elements
sage: P.ordinal_sum(L)
Finite poset containing 4 elements
sage: L.ordinal_sum(JL)
Finite join-semilattice containing 4 elements
sage: L.ordinal_sum(L)
Finite lattice containing 4 elements


p_partition_enumerator(tup, R, check=False)

Return a $$P$$-partition enumerator of self.

Given a total order $$\prec$$ on the elements of a finite poset $$P$$ (the order of $$P$$ and the total order $$\prec$$ can be unrelated; in particular, the latter does not have to extend the former), a $$P$$-partition enumerator is the quasisymmetric function $$\sum_f \prod_{p \in P} x_{f(p)}$$, where the first sum is taken over all $$P$$-partitions $$f$$.

A $$P$$-partition is a function $$f : P \to \{1,2,3,...\}$$ satisfying the following properties for any two elements $$i$$ and $$j$$ of $$P$$ satisfying $$i <_P j$$:

• if $$i \prec j$$ then $$f(i) \leq f(j)$$,
• if $$j \prec i$$ then $$f(i) < f(j)$$.

INPUT:

• tup – the tuple containing all elements of $$P$$ (each of them exactly once), in the order dictated by the total order $$\prec$$
• R – a commutative ring

OUTPUT:

The $$P$$-partition enumerator of self according to tup in the algebra $$QSym$$ of quasisymmetric functions over the base ring $$R$$.

EXAMPLES:

sage: P = Poset([[1,2,3,4],[[1,4],[2,4],[4,3]]])
sage: FP = P.p_partition_enumerator((3,1,2,4), QQ, check=True); FP
2*M[1, 1, 1, 1] + 2*M[1, 2, 1] + M[2, 1, 1] + M[3, 1]

sage: expansion = FP.expand(5)
sage: xs = expansion.parent().gens()
sage: expansion == sum([xs[a]*xs[b]*xs[c]*xs[d] for a in range(5) for b in range(5) for c in range(5) for d in range(5) if a <= b and c <= b and b < d])
True

sage: P = Poset([[],[]])
sage: FP = P.p_partition_enumerator((), QQ, check=True); FP
M[]

plot(label_elements=True, element_labels=None, layout='acyclic', cover_labels=None, **kwds)

Return a Graphic object for the Hasse diagram of the poset.

If the poset is ranked, the plot uses the rank function for the heights of the vertices.

INPUT:

• label_elements (default: True) - whether to display element labels
• element_labels (default: None) - a dictionary of element labels
• cover_labels - a dictionary, list or function representing labels of the covers of self. When set to None (default) no label is displayed on the edges of the Hasse Diagram.
• layout – the type of layout used to display the Diagram. Set to 'acyclic' by default (see GenericGraph.plot for more information).

Note

All options of GenericGraph.plot are also available through this function.

EXAMPLES:

sage: D = Poset({ 1:[2,3], 2:[4], 3:[4,5] })
sage: D.plot(label_elements=False)
Graphics object consisting of 6 graphics primitives
sage: D.plot()
Graphics object consisting of 11 graphics primitives
sage: type(D.plot())
<class 'sage.plot.graphics.Graphics'>
sage: elm_labs = {1:'a', 2:'b', 3:'c', 4:'d', 5:'e'}
sage: D.plot(element_labels=elm_labs)
Graphics object consisting of 11 graphics primitives


Plot of a ranked poset:

sage: P = Poset(DiGraph('E@ACA@?'))
sage: P.is_ranked()
True
sage: P.plot()
Graphics object consisting of 12 graphics primitives


The keyword cover_labels can be used to decorate edges:

sage: P = posets.ChainPoset(3)
sage: P.plot(cover_labels=lambda a, b: a + b)
Graphics object consisting of 8 graphics primitives
sage: P = Poset({0: [1,2]})
sage: P.plot(cover_labels={(0,1): 'here', (0,2): 'there'})
Graphics object consisting of 8 graphics primitives
sage: P = Poset({2: [1], 0: [1]})
sage: P.plot(cover_labels=[(2,1,'da'), (0,1,'niet')])
Graphics object consisting of 8 graphics primitives


TESTS:

We check that label_elements and element_labels are honored:

sage: def get_plot_labels(P): return sorted(t.string for t in P if isinstance(t, sage.plot.text.Text))
sage: P1 = Poset({ 0:[1,2], 1:[3], 2:[3,4] })
sage: P2 = Poset({ 0:[1,2], 1:[3], 2:[3,4] }, facade=True)
sage: get_plot_labels(P1.plot(label_elements=False))
[]
sage: get_plot_labels(P1.plot(label_elements=True))
['0', '1', '2', '3', '4']
sage: element_labels = {0:'a', 1:'b', 2:'c', 3:'d', 4:'e'}
sage: get_plot_labels(P1.plot(element_labels=element_labels))
['a', 'b', 'c', 'd', 'e']
sage: get_plot_labels(P2.plot(element_labels=element_labels))
['a', 'b', 'c', 'd', 'e']


Plot of the empy poset:

sage: P = Poset({})
sage: P.plot()
Graphics object consisting of 0 graphics primitives

product(other)

Return the cartesian product of self and other.

EXAMPLES:

sage: P = Posets.ChainPoset(3)
sage: Q = Posets.ChainPoset(4)
sage: PQ = P.product(Q) ; PQ
Finite lattice containing 12 elements
sage: len(PQ.hasse_diagram().edges())
17
sage: Q.product(P).is_isomorphic(PQ)
True

sage: P = Posets.BooleanLattice(2)
sage: Q = P.product(P)
sage: Q.is_isomorphic(Posets.BooleanLattice(4))
True

promotion(i=1)

Compute the (extended) promotion on the linear extension of the poset self.

INPUT:

• i – an integer between $$1$$ and $$n$$ (default: $$1$$)

OUTPUT:

• an isomorphic poset, with the same default linear extension

The extended promotion is defined on a poset self of size $$n$$ by applying the promotion operator $$\tau_i \tau_{i+1} \cdots \tau_{n-1}$$ to the default linear extension $$\pi$$ of self (see promotion()), and relabeling self accordingly. For more details see [Stan2009].

When the vertices of the poset self are labelled by $$\{1,2,\ldots,n\}$$, the linear extension is the identity, and $$i=1$$, the above algorithm corresponds to the promotion operator on posets defined by Schützenberger as follows. Remove $$1$$ from self and replace it by the minimum $$j$$ of all labels covering $$1$$ in the poset. Then, remove $$j$$ and replace it by the minimum of all labels covering $$j$$, and so on. This process ends when a label is a local maximum. Place the label $$n+1$$ at this vertex. Finally, decrease all labels by $$1$$.

EXAMPLES:

sage: P = Poset(([1,2], [[1,2]]), linear_extension=True, facade=False)
sage: P.promotion()
Finite poset containing 2 elements with distinguished linear extension
sage: P == P.promotion()
True

sage: P = Poset(([1,2,3,4,5,6,7], [[1,2],[1,4],[2,3],[2,5],[3,6],[4,7],[5,6]]))
sage: P.list()
[1, 2, 3, 5, 6, 4, 7]
sage: Q = P.promotion(4); Q
Finite poset containing 7 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 6], [2, 3], [2, 5], [3, 7], [5, 7], [6, 4]]


Note that if one wants to obtain the promotion defined by Schützenberger’s algorithm directly on the poset, one needs to make sure the linear extension is the identity:

sage: P = P.with_linear_extension([1,2,3,4,5,6,7])
sage: P.list()
[1, 2, 3, 4, 5, 6, 7]
sage: Q = P.promotion(4); Q
Finite poset containing 7 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 6], [2, 3], [2, 4], [3, 5], [4, 5], [6, 7]]
sage: Q = P.promotion()
sage: Q.cover_relations()
[[1, 2], [1, 3], [2, 4], [2, 5], [3, 6], [4, 7], [5, 7]]


Here is an example for a poset not labelled by $$\{1, 2, \ldots, n\}$$:

sage: P = Poset((divisors(30), attrcall("divides")), linear_extension=True)
sage: P.list()
[1, 2, 3, 5, 6, 10, 15, 30]
sage: P.cover_relations()
[[1, 2], [1, 3], [1, 5], [2, 6], [2, 10], [3, 6], [3, 15],
[5, 10], [5, 15], [6, 30], [10, 30], [15, 30]]
sage: Q = P.promotion(4); Q
Finite poset containing 8 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 3], [1, 6], [2, 5], [2, 15], [3, 5], [3, 10],
[5, 30], [6, 10], [6, 15], [10, 30], [15, 30]]


AUTHOR:

• Anne Schilling (2012-02-18)
random_subposet(p)

Return a random subposet that contains each element with probability p.

EXAMPLES:

sage: P = Posets.BooleanLattice(3)
sage: set_random_seed(0)
sage: Q = P.random_subposet(0.5)
sage: Q.cover_relations()
[[0, 2], [0, 5], [2, 3], [3, 7], [5, 7]]

rank(element=None)

Return the rank of an element element in the poset self, or the rank of the poset if element is None.

(The rank of a poset is the length of the longest chain of elements of the poset.)

EXAMPLES:

sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]], facade = False)
sage: P.rank(5)
2
sage: P.rank()
3
sage: Q = Poset([[1,2],[3],[],[]])

sage: P = Posets.SymmetricGroupBruhatOrderPoset(4)
sage: [(v,P.rank(v)) for v in P]
[('1234', 0),
('1243', 1),
...
('4312', 5),
('4321', 6)]

rank_function()

Return the (normalized) rank function of the poset, if it exists.

A rank function of a poset $$P$$ is a function $$r$$ that maps elements of $$P$$ to integers and satisfies: $$r(x) = r(y) + 1$$ if $$x$$ covers $$y$$. The function $$r$$ is normalized such that its minimum value on every connected component of the Hasse diagram of $$P$$ is $$0$$. This determines the function $$r$$ uniquely (when it exists).

OUTPUT:

• a lambda function, if the poset admits a rank function
• None, if the poset does not admit a rank function

EXAMPLES:

sage: P = Poset(([1,2,3,4],[[1,4],[2,3],[3,4]]), facade=True)
sage: P.rank_function() is not None
True
sage: P.rank_function() is not None
False
sage: f = P.rank_function(); f is not None
True
sage: f(5)
0
sage: f(2)
0


TESTS:

sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]])
sage: r = P.rank_function()
sage: for u,v in P.cover_relations_iterator():
....:     if r(v) != r(u) + 1:
....:         print "Bug in rank_function!"

sage: Q = Poset([[1,2],[4],[3],[4],[]])
sage: Q.rank_function() is None
True

relabel(relabeling)

Return a copy of this poset with its elements relabelled.

INPUT:

• relabeling – a function or dictionnary

This function should map each (non-wrapped) element of self to some distinct object.

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True, facade = False)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: P.cover_relations()
[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
sage: Q = P.relabel(lambda x: 12/x)
sage: Q.list()
[12, 6, 4, 3, 2, 1]
sage: Q.cover_relations()
[[12, 6], [12, 4], [6, 3], [6, 2], [4, 2], [3, 1], [2, 1]]


Here we relabel the elements of a poset by $$\{0,1,2, ...\}$$, using a dictionary:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True, facade=False)
sage: relabeling = {c.element:i for (i,c) in enumerate(P)}; relabeling
{1: 0, 2: 1, 3: 2, 4: 3, 6: 4, 12: 5}
sage: Q = P.relabel(relabeling)
sage: Q.list()
[0, 1, 2, 3, 4, 5]
sage: Q.cover_relations()
[[0, 1], [0, 2], [1, 3], [1, 4], [2, 4], [3, 5], [4, 5]]


Mind the c.element; this is because the relabeling is applied to the elements of the poset without the wrapping. Thanks to this convention, the same relabeling function can be used both for facade or non facade posets:

sage: P = Poset((divisors(12), attrcall("divides")), facade = True, linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: Q = P.relabel(lambda x: 12/x)
sage: Q.list()
[12, 6, 4, 3, 2, 1]
sage: Q.cover_relations()
[[12, 6], [12, 4], [6, 3], [6, 2], [4, 2], [3, 1], [2, 1]]


Relabeling a (semi)lattice gives a (semi)lattice:

sage: P=JoinSemilattice({0:[1]}) sage: type(P.relabel(lambda n: n+1)) <class ‘sage.combinat.posets.lattices.FiniteJoinSemilattice_with_category’>

Note

As can be seen in the above examples, the default linear extension of Q is that of P after relabeling. In particular, P and Q share the same internal Hasse diagram.

TESTS:

The following checks that trac ticket #14019 has been fixed:

sage: d = DiGraph({2:[1],3:[1]})
sage: p1 = Poset(d)
sage: p2 = p1.relabel({1:1,2:3,3:2})
sage: p1.hasse_diagram() == p2.hasse_diagram()
True
sage: p1 == p2
True

sage: d = DiGraph({2:[1],3:[1]})
sage: p1 = Poset(d)
sage: p2 = p1.relabel({1:2,2:3,3:1})
sage: p3 = p2.relabel({2:1,1:2,3:3})
sage: p1.hasse_diagram() == p3.hasse_diagram()
True
sage: p1 == p3
True

relations()

Returns a list of all relations of the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x\leq y$$ in self.

Relations are also often called intervals. The number of intervals is the dimension of the incidence algebra.

OUTPUT:

A list of pairs (each pair is a list), where the first element of the pair is less than or equal to the second element.

Pairs are produced in a rough sort of lexicographic order, where earlier elements are from lower levels of the poset.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: Q.relations()
[[1, 1], [1, 2], [1, 3], [1, 4], [0, 0], [0, 2], [0, 3],
[0, 4], [2, 2], [2, 3], [2, 4], [3, 3], [3, 4], [4, 4]]


AUTHOR:

• Rob Beezer (2011-05-04)
relations_iterator(strict=False)

Returns an iterator for all the relations of the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x\leq y$$ in self.

Relations are also often called intervals. The number of intervals is the dimension of the incidence algebra.

INPUT:

• strict – boolean (default False) if True, returns an iterator over relations $$x < y$$, excluding all relations $$x \leq x$$.

OUTPUT:

A generator that produces pairs (each pair is a list), where the first element of the pair is less than or equal to the second element.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: type(Q.relations_iterator())
<type 'generator'>
sage: [z for z in Q.relations_iterator()]
[[1, 1], [1, 2], [1, 3], [1, 4], [0, 0], [0, 2], [0, 3],
[0, 4], [2, 2], [2, 3], [2, 4], [3, 3], [3, 4], [4, 4]]

sage: P = posets.PentagonPoset()
sage: list(P.relations_iterator(strict=True))
[[0, 1], [0, 2], [0, 4], [0, 3], [1, 4], [2, 3], [2, 4], [3, 4]]

sage: len(list(P.relations_iterator()))
13


AUTHOR:

• Rob Beezer (2011-05-04)
relations_number()

Return the number of relations in the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x\leq y$$ in self.

Relations are also often called intervals. The number of intervals is the dimension of the incidence algebra.

EXAMPLES:

sage: from sage.combinat.tamari_lattices import TamariLattice
sage: TamariLattice(4).relations_number()
68

sage: P = posets.BooleanLattice(3)
sage: P.relations_number()
27

show(label_elements=True, element_labels=None, cover_labels=None, **kwds)

Displays the Hasse diagram of the poset.

INPUT:

• label_elements (default: True) - whether to display element labels
• element_labels (default: None) - a dictionary of element labels
• cover_labels - a dictionary, list or function representing labels of the covers of self. When set to None (default) no label is displayed on the edges of the Hasse Diagram.

Note

This method also accepts:

EXAMPLES:

sage: D = Poset({ 0:[1,2], 1:[3], 2:[3,4] })
sage: D.plot(label_elements=False)
Graphics object consisting of 6 graphics primitives
sage: D.show()
sage: elm_labs = {0:'a', 1:'b', 2:'c', 3:'d', 4:'e'}
sage: D.show(element_labels=elm_labs)


One more example with cover labels:

sage: P = posets.PentagonPoset()
sage: P.show(cover_labels=lambda a, b: a - b)

subposet(elements)

Return the poset containing elements with partial order induced by that of self.

EXAMPLES:

sage: P = Poset({"a":["c","d"], "b":["d","e"], "c":["f"], "d":["f"], "e":["f"]}, facade = False)
sage: Q = P.subposet(["a","b","f"]); Q
Finite poset containing 3 elements
sage: Q.cover_relations()
[[b, f], [a, f]]


sage: P = Poset({"a":["c","d"], "b":["d","e"], "c":["f"], "d":["f"], "e":["f"]}, facade=True)
sage: Q = P.subposet(["a","b","f"]); Q
Finite poset containing 3 elements
sage: Q.cover_relations()
[['b', 'f'], ['a', 'f']]


One may specified wrapped elements or not:

sage: P = Poset({"a":["c","d"], "b":["d","e"], "c":["f"], "d":["f"], "e":["f"]}, facade = False)
sage: Q = P.subposet([P("a"),P("b"),P("f")]); Q
Finite poset containing 3 elements
sage: Q.cover_relations()
[[b, f], [a, f]]

sage: B = posets.BooleanLattice(2)
sage: above = B.principal_order_filter(0)
sage: Q = B.subposet(above)
sage: above_new = Q.principal_order_filter(Q.list()[0])
sage: Q.subposet(above_new)
Finite poset containing 4 elements


TESTS:

sage: P.subposet(("a","b","f"))
Finite poset containing 3 elements
sage: P.subposet(["a","b","x"])
Traceback (most recent call last):
...
ValueError: <type 'str'> is not an element of this poset
sage: P.subposet(3)
Traceback (most recent call last):
...
TypeError: 'sage.rings.integer.Integer' object is not iterable

to_graph(*args, **kwds)

Deprecated: Use cover_relations_graph() instead. See trac ticket #17449 for details.

top()

Returns the top element of the poset, if it exists.

EXAMPLES:

sage: P = Poset({0:[3],1:[3],2:[3],3:[4,5],4:[],5:[]})
sage: P.top() is None
True
sage: Q = Poset({0:[1],1:[]})
sage: Q.top()
1


TESTS:

sage: R = Poset([[0],[]])
sage: R.list()
[0]
sage: R.top() #Trac #10776
0

unwrap(element)

Return the element element of the poset self in unwrapped form.

INPUT:

• element – an element of self

EXAMPLES:

sage: P = Poset((divisors(15), attrcall("divides")), facade = False)
sage: x = P.an_element(); x
1
sage: x.parent()
Finite poset containing 4 elements
sage: P.unwrap(x)
1
sage: P.unwrap(x).parent()
Integer Ring


For a non facade poset, this is equivalent to using the .element attribute:

sage: P.unwrap(x) is x.element
True


For a facade poset, this does nothing:

sage: P = Poset((divisors(15), attrcall("divides")), facade=True)
sage: x = P.an_element()
sage: P.unwrap(x) is x
True


This method is useful in code where we don’t know if P is a facade poset or not.

upper_covers(y)

Returns a list of upper covers of the element y. An upper cover of y is an element x such that y x is a cover relation.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: map(Q.upper_covers,Q.list())
[[2], [2], [3], [4], []]

upper_covers_iterator(y)

Returns an iterator for the upper covers of the element y. An upper cover of y is an element x such that y x is a cover relation.

EXAMPLES:

sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: type(Q.upper_covers_iterator(0))
<type 'generator'>

width()

Return the width of the poset (the size of its longest antichain).

It is computed through a matching in a bipartite graph. See Wikipedia article Dilworth’s_theorem for more information.

dilworth_decomposition() – return a partition of the poset into the smallest number of chains.

EXAMPLE:

sage: p = posets.BooleanLattice(4)
sage: p.width()
6

with_linear_extension(linear_extension)

Return a copy of self with a different default linear extension.

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
sage: P.cover_relations()
[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
sage: list(P)
[1, 2, 3, 4, 6, 12]
sage: Q = P.with_linear_extension([1,3,2,6,4,12])
sage: list(Q)
[1, 3, 2, 6, 4, 12]
sage: Q.cover_relations()
[[1, 3], [1, 2], [3, 6], [2, 6], [2, 4], [6, 12], [4, 12]]


TESTS:

We check that we can pass in a list of elements of P instead:

sage: Q = P.with_linear_extension(map(P, [1,3,2,6,4,12]))
sage: list(Q)
[1, 3, 2, 6, 4, 12]
sage: Q.cover_relations()
[[1, 3], [1, 2], [3, 6], [2, 6], [2, 4], [6, 12], [4, 12]]


We check that this works for facade posets too:

sage: P = Poset((divisors(12), attrcall("divides")), facade=True)
sage: Q = P.with_linear_extension([1,3,2,6,4,12])
sage: list(Q)
[1, 3, 2, 6, 4, 12]
sage: Q.cover_relations()
[[1, 3], [1, 2], [3, 6], [2, 6], [2, 4], [6, 12], [4, 12]]
sage: sorted(Q.cover_relations()) == sorted(P.cover_relations())
True


Note

With the current implementation, this requires relabeling the internal DiGraph which is $$O(n+m)$$, where $$n$$ is the number of elements and $$m$$ the number of cover relations.

zeta_polynomial()

Return the zeta polynomial of the poset self.

The zeta polynomial of a poset is the unique polynomial $$Z(q)$$ such that for every integer $$m > 1$$, $$Z(m)$$ is the number of weakly increasing sequences $$x_1 \leq x_2 \leq \dots \leq x_{m-1}$$ of elements of the poset.

The polynomial $$Z(q)$$ is integral-valued, but generally doesn’t have integer coefficients. It can be computed as

$Z(q) = \sum_{k \geq 1} \dbinom{q-2}{k-1} c_k,$

where $$c_k$$ is the number of all chains of length $$k$$ in the poset.

In particular, $$Z(2)$$ is the number of vertices and $$Z(3)$$ is the number of intervals.

EXAMPLES:

sage: Posets.ChainPoset(2).zeta_polynomial()
q
sage: Posets.ChainPoset(3).zeta_polynomial()
1/2*q^2 + 1/2*q
sage: P = posets.PentagonPoset()
sage: P.zeta_polynomial()
1/6*q^3 + q^2 - 1/6*q
sage: P = Posets.DiamondPoset(5)
sage: P.zeta_polynomial()
3/2*q^2 - 1/2*q


TESTS:

Checking the simplest cases:

sage: Poset({}).zeta_polynomial()
0
sage: Poset({1: []}).zeta_polynomial()
1
sage: Poset({1: [], 2: []}).zeta_polynomial()
2

class sage.combinat.posets.posets.FinitePosets_n(n)

The finite enumerated set of all posets on $$n$$ vertices, up to an isomorphism.

EXAMPLES:

sage: P = Posets(3)
sage: P.cardinality()
5
sage: for p in P: print p.cover_relations()
[]
[[1, 2]]
[[0, 1], [0, 2]]
[[0, 1], [1, 2]]
[[1, 2], [0, 2]]

cardinality(from_iterator=False)

Return the cardinality of this object.

Note

By default, this returns pre-computed values obtained from the On-Line Encyclopedia of Integer Sequences (OEIS sequence A000112). To override this, pass the argument from_iterator=True.

EXAMPLES:

sage: P = Posets(3)
sage: P.cardinality()
5
sage: P.cardinality(from_iterator=True)
5

sage.combinat.posets.posets.Poset(data=None, element_labels=None, cover_relations=False, linear_extension=False, category=None, facade=None, key=None)

Construct a finite poset from various forms of input data.

INPUT:

• data – different input are accepted by this constructor:

1. A two-element list or tuple (E, R), where E is a collection of elements of the poset and R is a collection of relations x <= y, each represented as a two-element lists/tuples/iterables such as [x, y]. The poset is then the transitive closure of the provided relations. If cover_relations=True, then R is assumed to contain exactly the cover relations of the poset. If E is empty, then E is taken to be the set of elements appearing in the relations R.

2. A two-element list or tuple (E, f), where E is the set of elements of the poset and f is a function such that, for any pair x, y of elements of E, f(x, y) returns whether x <= y. If cover_relations=True, then f(x,y) should return whether x is covered by y.

3. A dictionary, list or tuple of upper covers: data[x] is a list of the elements that cover the element $$x$$ in the poset.

Warning

If data is a list or tuple of length $$2$$, then it is handled by the above case..

4. An acyclic, loop-free and multi-edge free DiGraph. If cover_relations is True, then the edges of the digraph are assumed to correspond to the cover relations of the poset. Otherwise, the cover relations are computed.

5. A previously constructed poset (the poset itself is returned).

• element_labels – (default: None); an optional list or dictionary of objects that label the poset elements.

• cover_relations – a boolean (default: False); whether the data can be assumed to describe a directed acyclic graph whose arrows are cover relations; otherwise, the cover relations are first computed.

• linear_extension – a boolean (default: False); whether to use the provided list of elements as default linear extension for the poset; otherwise a linear extension is computed. If the data is given as the pair (E, f), then E is taken to be the linear extension.

• facade – a boolean or None (default); whether the Poset()‘s elements should be wrapped to make them aware of the Poset they belong to.

• If facade = True, the Poset()‘s elements are exactly those given as input.
• If facade = False, the Poset()‘s elements will become PosetElement objects.
• If facade = None (default) the expected behaviour is the behaviour of facade = True, unless the opposite can be deduced from the context (i.e. for instance if a Poset() is built from another Poset(), itself built with facade = False)

OUTPUT:

FinitePoset – an instance of the FinitePoset class.

If category is specified, then the poset is created in this category instead of FinitePosets.

EXAMPLES:

1. Elements and cover relations:

sage: elms = [1,2,3,4,5,6,7]
sage: rels = [[1,2],[3,4],[4,5],[2,5]]
sage: Poset((elms, rels), cover_relations = True, facade = False)
Finite poset containing 7 elements


Elements and non-cover relations:

sage: elms = [1,2,3,4]
sage: rels = [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]
sage: P = Poset( [elms,rels] ,cover_relations=False); P
Finite poset containing 4 elements
sage: P.cover_relations()
[[1, 2], [2, 3], [3, 4]]

2. Elements and function: the standard permutations of [1, 2, 3, 4] with the Bruhat order:

sage: elms = Permutations(4)
sage: fcn = lambda p,q : p.bruhat_lequal(q)
sage: Poset((elms, fcn))
Finite poset containing 24 elements


With a function that identifies the cover relations: the set partitions of $$\{1, 2, 3\}$$ ordered by refinement:

sage: elms = SetPartitions(3)
sage: def fcn(A, B):
....:     if len(A) != len(B)+1:
....:         return False
....:     for a in A:
....:         if not any(set(a).issubset(b) for b in B):
....:             return False
....:     return True
sage: Poset((elms, fcn), cover_relations=True)
Finite poset containing 5 elements

3. A dictionary of upper covers:

sage: Poset({'a':['b','c'], 'b':['d'], 'c':['d'], 'd':[]})
Finite poset containing 4 elements


A list of upper covers:

sage: Poset([[1,2],[4],[3],[4],[]])
Finite poset containing 5 elements


A list of upper covers and a dictionary of labels:

sage: elm_labs = {0:"a",1:"b",2:"c",3:"d",4:"e"}
sage: P = Poset([[1,2],[4],[3],[4],[]], elm_labs, facade = False)
sage: P.list()
[a, b, c, d, e]


Warning

The special case where the argument data is a list or tuple of length 2 is handled by the above cases. So you cannot use this method to input a 2-element poset.

4. An acyclic DiGraph.

sage: dag = DiGraph({0:[2,3], 1:[3,4], 2:[5], 3:[5], 4:[5]})
sage: Poset(dag)
Finite poset containing 6 elements


Any directed acyclic graph without loops or multiple edges, as long as cover_relations=False:

sage: dig = DiGraph({0:[2,3], 1:[3,4,5], 2:[5], 3:[5], 4:[5]})
sage: dig.allows_multiple_edges()
False
sage: dig.allows_loops()
False
sage: dig.transitive_reduction() == dig
False
sage: Poset(dig, cover_relations=False)
Finite poset containing 6 elements
sage: Poset(dig, cover_relations=True)
Traceback (most recent call last):
...
ValueError: Hasse diagram is not transitively reduced.


Default Linear extension

Every poset $$P$$ obtained with Poset comes equipped with a default linear extension, which is also used for enumerating its elements. By default, this linear extension is computed, and has no particular significance:

sage: P = Poset((divisors(12), attrcall("divides")))
sage: P.list()
[1, 2, 4, 3, 6, 12]
sage: P.linear_extension()
[1, 2, 4, 3, 6, 12]


You may enforce a specific linear extension using the linear_extension option:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: P.linear_extension()
[1, 2, 3, 4, 6, 12]


Depending on popular request, Poset might eventually get modified to always use the provided list of elements as default linear extension, when it is one.

When facade = False, the elements of a poset are wrapped so as to make them aware that they belong to that poset:

sage: P = Poset(DiGraph({'d':['c','b'],'c':['a'],'b':['a']}), facade = False)
sage: d,c,b,a = list(P)
sage: a.parent() is P
True


This allows for comparing elements according to $$P$$:

sage: c < a
True


However, this may have surprising effects:

sage: my_elements = ['a','b','c','d']
sage: any(x in my_elements for x in P)
False


and can be annoying when one wants to manipulate the elements of the poset:

sage: a + b
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for +: 'FinitePoset_with_category.element_class' and 'FinitePoset_with_category.element_class'
sage: a.element + b.element
'ab'


sage: P = Poset(DiGraph({'d':['c','b'],'c':['a'],'b':['a']}))


In this example, the elements of the poset remain plain strings:

sage: d,c,b,a = list(P)
sage: type(a)
<type 'str'>


Of course, those strings are not aware of $$P$$. So to compare two such strings, one needs to query $$P$$:

sage: a < b
True
sage: P.lt(a,b)
False


which models the usual mathematical notation $$a <_P b$$.

Most operations seem to still work, but at this point there is no guarantee whatsoever:

sage: P.list()
['d', 'c', 'b', 'a']
sage: P.principal_order_ideal('a')
['d', 'c', 'b', 'a']
sage: P.principal_order_ideal('b')
['d', 'b']
sage: P.principal_order_ideal('d')
['d']
sage: TestSuite(P).run()


Warning

DiGraph is used to construct the poset, and the vertices of a DiGraph are converted to plain Python int‘s if they are Integer‘s:

sage: G = DiGraph({0:[2,3], 1:[3,4], 2:[5], 3:[5], 4:[5]})
sage: type(G.vertices()[0])
<type 'int'>


This is worked around by systematically converting back the vertices of a poset to Integer‘s if they are int‘s:

sage: P = Poset((divisors(15), attrcall("divides")), facade = False)
sage: type(P.an_element().element)
<type 'sage.rings.integer.Integer'>

sage: P = Poset((divisors(15), attrcall("divides")), facade=True)
sage: type(P.an_element())
<type 'sage.rings.integer.Integer'>


This may be abusive:

sage: P = Poset((range(5), operator.le), facade = True)
sage: P.an_element().parent()
Integer Ring


Unique representation

As most parents, Poset have unique representation (see UniqueRepresentation). Namely if two posets are created from two equal data, then they are not only equal but actually identical:

sage: data1 = [[1,2],[3],[3]]
sage: data2 = [[1,2],[3],[3]]
sage: P1 = Poset(data1)
sage: P2 = Poset(data2)
sage: P1 == P2
True
sage: P1 is P2
True


In situations where this behaviour is not desired, one can use the key option:

sage: P1 = Poset(data1, key = "foo")
sage: P2 = Poset(data2, key = "bar")
sage: P1 is P2
False
sage: P1 == P2
False


key can be any hashable value and is passed down to UniqueRepresentation. It is otherwise ignored by the poset constructor.

TESTS:

sage: P = Poset([[1,2],[3],[3]])
sage: type(hash(P))
<type 'int'>


sage: Poset([1,2,3], lambda x,y : x<y)
Traceback (most recent call last):
...
ValueError: element_labels should be a dict or a list if different from None. (Did you intend data to be equal to a pair ?)


Another kind of bad input, digraphs with oriented cycles:

sage: Poset(DiGraph([[1,2],[2,3],[3,4],[4,1]]))
Traceback (most recent call last):
...
ValueError: The graph is not directed acyclic

sage.combinat.posets.posets.is_poset(dig)

Tests whether a directed graph is acyclic and transitively reduced.

EXAMPLES:

sage: from sage.combinat.posets.posets import is_poset
sage: dig = DiGraph({0:[2,3], 1:[3,4,5], 2:[5], 3:[5], 4:[5]})
sage: is_poset(dig)
False
sage: is_poset(dig.transitive_reduction())
True