# Linear Extensions of Posets¶

class sage.combinat.posets.linear_extensions.LinearExtensionOfPoset

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 $$P$$.

When the elements of $$P$$ are indexed by $$\{1,2,\ldots,n\}$$, $$\pi$$ denotes a permutation of the elements of $$P$$ in one-line notation.

INPUT:

• linear_extension – a list of the elements of $$P$$
• poset – the underlying poset $$P$$

Poset, LinearExtensionsOfPosets

EXAMPLES:

sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]), linear_extension=True, facade = False)
sage: p = P.linear_extension([1,4,2,3]); p
[1, 4, 2, 3]
sage: p.parent()
The set of all linear extensions of Finite poset containing 4 elements
sage: p[0], p[1], p[2], p[3]
(1, 4, 2, 3)


Following Schützenberger and later Haiman and Malvenuto-Reutenauer, Stanley [Stanley2009] defined a promotion and evacuation operator on any finite poset $$P$$ using operators $$\tau_i$$ on the linear extensions of $$P$$:

sage: p.promotion()
[1, 2, 3, 4]
sage: Q = p.promotion().to_poset()
sage: Q.cover_relations()
[[1, 3], [1, 4], [2, 3]]
sage: Q == P
True

sage: p.promotion(3)
[1, 4, 2, 3]
sage: Q = p.promotion(3).to_poset()
sage: Q == P
False
sage: Q.cover_relations()
[[1, 2], [1, 4], [3, 4]]


REFERENCES:

 [Stanley2009] (1, 2, 3, 4) Richard Stanley, Promotion and evacuation, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 9, 24 pp.
check()

Checks whether self is indeed a linear extension of the underlying poset.

TESTS:

sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]))
sage: P.linear_extension([1,4,2,3])
[1, 4, 2, 3]
sage: P.linear_extension([4,3,2,1])
Traceback (most recent call last):
...
ValueError: [4, 3, 2, 1] is not a linear extension of Finite poset containing 4 elements

evacuation()

Computes evacuation on the linear extension of a poset.

Evacuation on a linear extension $$\pi$$ of length $$n$$ is defined as $$\pi (\tau_1 \cdots \tau_{n-1}) (\tau_1 \cdots \tau_{n-2}) \cdots (\tau_1)$$. For more details see [Stanley2009].

EXAMPLES:

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 = P.linear_extension([1,2,3,4,5,6,7])
sage: p.evacuation()
[1, 4, 2, 3, 7, 5, 6]
sage: p.evacuation().evacuation() == p
True

poset()

Returns the underlying original poset.

EXAMPLES:

sage: P = Poset(([1,2,3,4], [[1,2],[2,3],[1,4]]))
sage: p = P.linear_extension([1,2,4,3])
sage: p.poset()
Finite poset containing 4 elements

promotion(i=1)

Computes the (generalized) promotion on the linear extension of a poset.

INPUT:

• $$i$$ – an integer between $$1$$ and $$n-1$$, where $$n$$ is the cardinality of the poset (default: $$1$$)

The $$i$$-th generalized promotion operator $$\partial_i$$ on a linear extension $$\pi$$ is defined as $$\pi \tau_i \tau_{i+1} \cdots \tau_{n-1}$$, where $$n$$ is the size of the linear extension (or size of the underlying poset).

For more details see [Stanley2009].

EXAMPLES:

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 = P.linear_extension([1,2,3,4,5,6,7])
sage: q = p.promotion(4); q
[1, 2, 3, 5, 6, 4, 7]
sage: p.to_poset() == q.to_poset()
False
sage: p.to_poset().is_isomorphic(q.to_poset())
True

tau(i)

Returns the operator $$\tau_i$$ on linear extensions self of a poset.

INPUT:

• $$i$$ – an integer between $$1$$ and $$n-1$$, where $$n$$ is the cardinality of the poset.

The operator $$\tau_i$$ on a linear extension $$\pi$$ of a poset $$P$$ interchanges positions $$i$$ and $$i+1$$ if the result is again a linear extension of $$P$$, and otherwise acts trivially. For more details, see [Stanley2009].

EXAMPLES:

sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]), linear_extension=True)
sage: L = P.linear_extensions()
sage: l = L.an_element(); l
[1, 2, 3, 4]
sage: l.tau(1)
[2, 1, 3, 4]
sage: for p in L:
...       for i in range(1,4):
...           print i, p, p.tau(i)
...
1 [1, 2, 3, 4] [2, 1, 3, 4]
2 [1, 2, 3, 4] [1, 2, 3, 4]
3 [1, 2, 3, 4] [1, 2, 4, 3]
1 [1, 2, 4, 3] [2, 1, 4, 3]
2 [1, 2, 4, 3] [1, 4, 2, 3]
3 [1, 2, 4, 3] [1, 2, 3, 4]
1 [1, 4, 2, 3] [1, 4, 2, 3]
2 [1, 4, 2, 3] [1, 2, 4, 3]
3 [1, 4, 2, 3] [1, 4, 2, 3]
1 [2, 1, 3, 4] [1, 2, 3, 4]
2 [2, 1, 3, 4] [2, 1, 3, 4]
3 [2, 1, 3, 4] [2, 1, 4, 3]
1 [2, 1, 4, 3] [1, 2, 4, 3]
2 [2, 1, 4, 3] [2, 1, 4, 3]
3 [2, 1, 4, 3] [2, 1, 3, 4]


TESTS:

sage: type(l.tau(1))
<class 'sage.combinat.posets.linear_extensions.LinearExtensionsOfPoset_with_category.element_class'>
sage: l.tau(2) == l
True

to_poset()

Returns the poset associated to the linear extension self.

This method returns the poset obtained from the original poset $$P$$ by relabelling the ‘i’-th element of self to the $$i$$-th element of the original poset, while keeping the linear extension of the original poset.

For a poset with default linear extension $$1,\dots,n$$, self can be interpreted as a permutation, and the relabelling is done according to the inverse of this permutation.

EXAMPLES:

sage: P = Poset(([1,2,3,4], [[1,2],[1,3],[3,4]]), facade = False)
sage: p = P.linear_extension([1,3,4,2])
sage: Q = p.to_poset(); Q
Finite poset containing 4 elements
sage: P == Q
False


The default linear extension remains the same:

sage: list(P)
[1, 2, 3, 4]
sage: list(Q)
[1, 2, 3, 4]


But the relabelling can be seen on cover relations:

sage: P.cover_relations()
[[1, 2], [1, 3], [3, 4]]
sage: Q.cover_relations()
[[1, 2], [1, 4], [2, 3]]

sage: p = P.linear_extension([1,2,3,4])
sage: Q = p.to_poset()
sage: P == Q
True


The set of all linear extensions of a finite poset

INPUT:

• poset – a poset $$P$$ of size $$n$$
• facade – a boolean (default: False)

EXAMPLES:

sage: elms = [1,2,3,4]
sage: rels = [[1,3],[1,4],[2,3]]
sage: P = Poset((elms, rels), linear_extension=True)
sage: L = P.linear_extensions(); L
The set of all linear extensions of Finite poset containing 4 elements
sage: L.cardinality()
5
sage: L.list()
[[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3]]
sage: L.an_element()
[1, 2, 3, 4]
sage: L.poset()
Finite poset containing 4 elements

Element

alias of LinearExtensionOfPoset

markov_chain_digraph(action='promotion', labeling='identity')

Returns the digraph of the action of generalized promotion or tau on self

INPUT:

• action – ‘promotion’ or ‘tau’ (default: ‘promotion’)
• labeling – ‘identity’ or ‘source’ (default: ‘identity’)

Todo

• generalize this feature by accepting a family of operators as input
• move up in some appropriate category

This method creates a graph with vertices being the linear extensions of a given finite poset and an edge from $$\pi$$ to $$\pi'$$ if $$\pi' = \pi \partial_i$$ where $$\partial_i$$ is the promotion operator (see promotion()) if action is set to promotion and $$\tau_i$$ (see tau()) if action is set to tau. The label of the edge is $$i$$ (resp. $$\pi_i$$) if labeling is set to identity (resp. source).

EXAMPLES:

sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]), linear_extension = True)
sage: L = P.linear_extensions()
sage: G = L.markov_chain_digraph(); G
Looped multi-digraph on 5 vertices
sage: sorted(G.vertices(), key = repr)
[[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3]]
sage: sorted(G.edges(), key = repr)
[([1, 2, 3, 4], [1, 2, 3, 4], 4), ([1, 2, 3, 4], [1, 2, 4, 3], 2), ([1, 2, 3, 4], [1, 2, 4, 3], 3),
([1, 2, 3, 4], [2, 1, 4, 3], 1), ([1, 2, 4, 3], [1, 2, 3, 4], 3), ([1, 2, 4, 3], [1, 2, 4, 3], 4),
([1, 2, 4, 3], [1, 4, 2, 3], 2), ([1, 2, 4, 3], [2, 1, 3, 4], 1), ([1, 4, 2, 3], [1, 2, 3, 4], 1),
([1, 4, 2, 3], [1, 2, 3, 4], 2), ([1, 4, 2, 3], [1, 4, 2, 3], 3), ([1, 4, 2, 3], [1, 4, 2, 3], 4),
([2, 1, 3, 4], [1, 2, 4, 3], 1), ([2, 1, 3, 4], [2, 1, 3, 4], 4), ([2, 1, 3, 4], [2, 1, 4, 3], 2),
([2, 1, 3, 4], [2, 1, 4, 3], 3), ([2, 1, 4, 3], [1, 4, 2, 3], 1), ([2, 1, 4, 3], [2, 1, 3, 4], 2),
([2, 1, 4, 3], [2, 1, 3, 4], 3), ([2, 1, 4, 3], [2, 1, 4, 3], 4)]

sage: G = L.markov_chain_digraph(labeling = 'source')
sage: sorted(G.vertices(), key = repr)
[[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3]]
sage: sorted(G.edges(), key = repr)
[([1, 2, 3, 4], [1, 2, 3, 4], 4), ([1, 2, 3, 4], [1, 2, 4, 3], 2), ([1, 2, 3, 4], [1, 2, 4, 3], 3),
([1, 2, 3, 4], [2, 1, 4, 3], 1), ([1, 2, 4, 3], [1, 2, 3, 4], 4), ([1, 2, 4, 3], [1, 2, 4, 3], 3),
([1, 2, 4, 3], [1, 4, 2, 3], 2), ([1, 2, 4, 3], [2, 1, 3, 4], 1), ([1, 4, 2, 3], [1, 2, 3, 4], 1),
([1, 4, 2, 3], [1, 2, 3, 4], 4), ([1, 4, 2, 3], [1, 4, 2, 3], 2), ([1, 4, 2, 3], [1, 4, 2, 3], 3),
([2, 1, 3, 4], [1, 2, 4, 3], 2), ([2, 1, 3, 4], [2, 1, 3, 4], 4), ([2, 1, 3, 4], [2, 1, 4, 3], 1),
([2, 1, 3, 4], [2, 1, 4, 3], 3), ([2, 1, 4, 3], [1, 4, 2, 3], 2), ([2, 1, 4, 3], [2, 1, 3, 4], 1),
([2, 1, 4, 3], [2, 1, 3, 4], 4), ([2, 1, 4, 3], [2, 1, 4, 3], 3)]


The edges of the graph are by default colored using blue for edge 1, red for edge 2, green for edge 3, and yellow for edge 4:

sage: view(G) #optional - dot2tex graphviz


Alternatively, one may get the graph of the action of the tau operator:

sage: G = L.markov_chain_digraph(action='tau'); G
Looped multi-digraph on 5 vertices
sage: sorted(G.vertices(), key = repr)
[[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3]]
sage: sorted(G.edges(), key = repr)
[([1, 2, 3, 4], [1, 2, 3, 4], 2), ([1, 2, 3, 4], [1, 2, 4, 3], 3), ([1, 2, 3, 4], [2, 1, 3, 4], 1),
([1, 2, 4, 3], [1, 2, 3, 4], 3), ([1, 2, 4, 3], [1, 4, 2, 3], 2), ([1, 2, 4, 3], [2, 1, 4, 3], 1),
([1, 4, 2, 3], [1, 2, 4, 3], 2), ([1, 4, 2, 3], [1, 4, 2, 3], 1), ([1, 4, 2, 3], [1, 4, 2, 3], 3),
([2, 1, 3, 4], [1, 2, 3, 4], 1), ([2, 1, 3, 4], [2, 1, 3, 4], 2), ([2, 1, 3, 4], [2, 1, 4, 3], 3),
([2, 1, 4, 3], [1, 2, 4, 3], 1), ([2, 1, 4, 3], [2, 1, 3, 4], 3), ([2, 1, 4, 3], [2, 1, 4, 3], 2)]
sage: view(G) #optional - dot2tex graphviz


markov_chain_transition_matrix(), promotion(), tau()

TESTS:

sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]), linear_extension = True, facade = True)
sage: L = P.linear_extensions()
sage: G = L.markov_chain_digraph(labeling = 'source'); G
Looped multi-digraph on 5 vertices

markov_chain_transition_matrix(action='promotion', labeling='identity')

Returns the transition matrix of the Markov chain for the action of generalized promotion or tau on self

INPUT:

• action – ‘promotion’ or ‘tau’ (default: ‘promotion’)
• labeling – ‘identity’ or ‘source’ (default: ‘identity’)

This method yields the transition matrix of the Markov chain defined by the action of the generalized promotion operator $$\partial_i$$ (resp. $$\tau_i$$) on the set of linear extensions of a finite poset. Here the transition from the linear extension $$\pi$$ to $$\pi'$$, where $$\pi' = \pi \partial_i$$ (resp. $$\pi'= \pi \tau_i$$) is counted with weight $$x_i$$ (resp. $$x_{\pi_i}$$ if labeling is set to source).

EXAMPLES:

sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]), linear_extension = True)
sage: L = P.linear_extensions()
sage: L.markov_chain_transition_matrix()
[-x0 - x1 - x2            x2       x0 + x1             0             0]
[      x1 + x2 -x0 - x1 - x2             0            x0             0]
[            0            x1      -x0 - x1             0            x0]
[            0            x0             0 -x0 - x1 - x2       x1 + x2]
[           x0             0             0       x1 + x2 -x0 - x1 - x2]

sage: L.markov_chain_transition_matrix(labeling = 'source')
[-x0 - x1 - x2            x3       x0 + x3             0             0]
[      x1 + x2 -x0 - x1 - x3             0            x1             0]
[            0            x1      -x0 - x3             0            x1]
[            0            x0             0 -x0 - x1 - x2       x0 + x3]
[           x0             0             0       x0 + x2 -x0 - x1 - x3]

sage: L.markov_chain_transition_matrix(action = 'tau')
[     -x0 - x2            x2             0            x0             0]
[           x2 -x0 - x1 - x2            x1             0            x0]
[            0            x1           -x1             0             0]
[           x0             0             0      -x0 - x2            x2]
[            0            x0             0            x2      -x0 - x2]

sage: L.markov_chain_transition_matrix(action = 'tau', labeling = 'source')
[     -x0 - x2            x3             0            x1             0]
[           x2 -x0 - x1 - x3            x3             0            x1]
[            0            x1           -x3             0             0]
[           x0             0             0      -x1 - x2            x3]
[            0            x0             0            x2      -x1 - x3]


markov_chain_digraph(), promotion(), tau()

poset()

Returns the underlying original poset.

EXAMPLES:

sage: P = Poset(([1,2,3,4], [[1,2],[2,3],[1,4]]))
sage: L = P.linear_extensions()
sage: L.poset()
Finite poset containing 4 elements


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