# Generating Series¶

This file makes a number of extensions to lazy power series by endowing them with some semantic content for how they’re to be interpreted.

This code is based on the work of Ralf Hemmecke and Martin Rubey’s Aldor-Combinat, which can be found at http://www.risc.uni-linz.ac.at/people/hemmecke/aldor/combinat/index.html. In particular, the relevant section for this file can be found at http://www.risc.uni-linz.ac.at/people/hemmecke/AldorCombinat/combinatse10.html. One notable difference is that we use power-sum symmetric functions as the coefficients of our cycle index series.

TESTS:

sage: from sage.combinat.species.stream import Stream, _integers_from
sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: p = SymmetricFunctions(QQ).power()
sage: CIS = CycleIndexSeriesRing(QQ)

sage: geo1 = CIS((p([1])^i  for i in _integers_from(0)))
sage: geo2 = CIS((p([2])^i  for i in _integers_from(0)))
sage: s = geo1 * geo2
sage: s[0]
p[]
sage: s[1]
p[1] + p[2]
sage: s[2]
p[1, 1] + p[2, 1] + p[2, 2]
sage: s[3]
p[1, 1, 1] + p[2, 1, 1] + p[2, 2, 1] + p[2, 2, 2]


Whereas the coefficients of the above test are homogeneous with respect to total degree, the following test groups with respect to weighted degree where each variable x_i has weight i.

sage: def g():
....:     for i in _integers_from(0):
....:         yield p([2])^i
....:         yield p(0)
sage: geo1 = CIS((p([1])^i  for i in _integers_from(0)))
sage: geo2 = CIS(g())
sage: s = geo1 * geo2
sage: s[0]
p[]
sage: s[1]
p[1]
sage: s[2]
p[1, 1] + p[2]
sage: s[3]
p[1, 1, 1] + p[2, 1]
sage: s[4]
p[1, 1, 1, 1] + p[2, 1, 1] + p[2, 2]

class sage.combinat.species.generating_series.CycleIndexSeries(A, stream=None, order=None, aorder=None, aorder_changed=True, is_initialized=False, name=None)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: f = L()
Uninitialized lazy power series

arithmetic_product(g, check_input=True)

Return the arithmetic product of self with g.

For species $$M$$ and $$N$$ such that $$M[\varnothing] = N[\varnothing] = \varnothing$$, their arithmetic product is the species $$M \boxdot N$$ of “$$M$$-assemblies of cloned $$N$$-structures”. This operation is defined and several examples are given in [MM].

The cycle index series for $$M \boxdot N$$ can be computed in terms of the component series $$Z_M$$ and $$Z_N$$, as implemented in this method.

INPUT:

• g – a cycle index series having the same parent as self.
• check_input – (default: True) a Boolean which, when set to False, will cause input checks to be skipped.

OUTPUT:

The arithmetic product of self with g. This is a cycle index series defined in terms of self and g such that if self and g are the cycle index series of two species $$M$$ and $$N$$, their arithmetic product is the cycle index series of the species $$M \boxdot N$$.

EXAMPLES:

For $$C$$ the species of (oriented) cycles and $$L_{+}$$ the species of nonempty linear orders, $$C \boxdot L_{+}$$ corresponds to the species of “regular octopuses”; a $$(C \boxdot L_{+})$$-structure is a cycle of some length, each of whose elements is an ordered list of a length which is consistent for all the lists in the structure.

sage: C = species.CycleSpecies().cycle_index_series()
sage: Lplus = species.LinearOrderSpecies(min=1).cycle_index_series()
sage: RegularOctopuses = C.arithmetic_product(Lplus)
sage: RegOctSpeciesSeq = RegularOctopuses.generating_series().counts(8)
sage: RegOctSpeciesSeq
[0, 1, 3, 8, 42, 144, 1440, 5760]


It is shown in [MM] that the exponential generating function for regular octopuses satisfies $$(C \boxdot L_{+}) (x) = \sum_{n \geq 1} \sigma (n) (n - 1)! \frac{x^{n}}{n!}$$ (where $$\sigma (n)$$ is the sum of the divisors of $$n$$).

sage: RegOctDirectSeq = [0] + [sum(divisors(i))*factorial(i-1) for i in range(1,8)]
sage: RegOctDirectSeq == RegOctSpeciesSeq
True


AUTHORS:

• Andrew Gainer-Dewar (2013)

REFERENCES:

 [MM] (1, 2) M. Maia and M. Mendez. “On the arithmetic product of combinatorial species”. Discrete Mathematics, vol. 308, issue 23, 2008, pp. 5407-5427. Arxiv math/0503436v2.
coefficient_cycle_type(t)

Returns the coefficient of a cycle type t.

EXAMPLES:

sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: p = SymmetricFunctions(QQ).power()
sage: CIS = CycleIndexSeriesRing(p)
sage: f = CIS([0, p([1]), 2*p([1,1]),3*p([2,1])])
sage: f.coefficient_cycle_type([1])
1
sage: f.coefficient_cycle_type([1,1])
2
sage: f.coefficient_cycle_type([2,1])
3

compositional_inverse()

Return the compositional inverse of self if possible.

(Specifically, if self is of the form $$0 + p_{1} + \dots$$.)

The compositional inverse is the inverse with respect to plethystic substitution. This is the operation on cycle index series which corresponds to substitution, a.k.a. partitional composition, on the level of species. See Section 2.2 of [BLL] for a definition of this operation.

EXAMPLES:

sage: Eplus = species.SetSpecies(min=1).cycle_index_series()
sage: Eplus(Eplus.compositional_inverse()).coefficients(8)
[0, p[1], 0, 0, 0, 0, 0, 0]


TESTS:

sage: Eplus = species.SetSpecies(min=2).cycle_index_series()
sage: Eplus.compositional_inverse()
Traceback (most recent call last):
...
ValueError: not an invertible series


ALGORITHM:

Let $$F$$ be a species satisfying $$F = 0 + X + F_2 + F_3 + \dots$$ for $$X$$ the species of singletons. (Equivalently, $$\lvert F[\varnothing] \rvert = 0$$ and $$\lvert F[\{1\}] \rvert = 1$$.) Then there exists a (virtual) species $$G$$ satisfying $$F \circ G = G \circ F = X$$.

It follows that $$(F - X) \circ G = F \circ G - X \circ G = X - G$$. Rearranging, we obtain the recursive equation $$G = X - (F - X) \circ G$$, which can be solved using iterative methods.

Warning

This algorithm is functional but can be very slow. Use with caution!

The compositional inverse $$\Omega$$ of the species $$E_{+}$$ of nonempty sets can be handled much more efficiently using specialized methods. These are implemented in CombinatorialLogarithmSeries.

AUTHORS:

• Andrew Gainer-Dewar
count(t)

Returns the number of structures corresponding to a certain cycle type.

EXAMPLES:

sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: p = SymmetricFunctions(QQ).power()
sage: CIS = CycleIndexSeriesRing(p)
sage: f = CIS([0, p([1]), 2*p([1,1]),3*p([2,1])])
sage: f.count([1])
1
sage: f.count([1,1])
4
sage: f.count([2,1])
6

expand_as_sf(n, alphabet='x')

Returns the expansion of a cycle index series as a symmetric function in n variables.

Specifically, this returns a LazyPowerSeries whose ith term is obtained by calling expand() on the ith term of self.

This relies on the (standard) interpretation of a cycle index series as a symmetric function in the power sum basis.

INPUT:

• self – a cycle index series
• n – a positive integer
• alphabet – a variable for the expansion (default: $$x$$)

EXAMPLES:

sage: from sage.combinat.species.set_species import SetSpecies
sage: SetSpecies().cycle_index_series().expand_as_sf(2).coefficients(4)
[1, x0 + x1, x0^2 + x0*x1 + x1^2, x0^3 + x0^2*x1 + x0*x1^2 + x1^3]

functorial_composition(g)

Returns the functorial composition of self and g.

If $$F$$ and $$G$$ are species, their functorial composition is the species $$F \Box G$$ obtained by setting $$(F \Box G) [A] = F[ G[A] ]$$. In other words, an $$(F \Box G)$$-structure on a set $$A$$ of labels is an $$F$$-structure whose labels are the set of all $$G$$-structures on $$A$$.

It can be shown (as in section 2.2 of [BLL]) that there is a corresponding operation on cycle indices:

$Z_{F} \Box Z_{G} = \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_{n}} \operatorname{fix} F[ (G[\sigma])_{1}, (G[\sigma])_{2}, \dots ] \, p_{1}^{\sigma_{1}} p_{2}^{\sigma_{2}} \dots.$

This method implements that operation on cycle index series.

EXAMPLES:

The species $$G$$ of simple graphs can be expressed in terms of a functorial composition: $$G = \mathfrak{p} \Box \mathfrak{p}_{2}$$, where $$\mathfrak{p}$$ is the SubsetSpecies. This is how it is implemented in SimpleGraphSpecies():

sage: S = species.SimpleGraphSpecies()
sage: S.cycle_index_series().coefficients(5)
[p[],
p[1],
p[1, 1] + p[2],
4/3*p[1, 1, 1] + 2*p[2, 1] + 2/3*p[3],
8/3*p[1, 1, 1, 1] + 4*p[2, 1, 1] + 2*p[2, 2] + 4/3*p[3, 1] + p[4]]

generating_series()

EXAMPLES:

sage: P = species.PartitionSpecies()
sage: cis = P.cycle_index_series()
sage: f = cis.generating_series()
sage: f.coefficients(5)
[1, 1, 1, 5/6, 5/8]

isotype_generating_series()

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: cis = P.cycle_index_series()
sage: f = cis.isotype_generating_series()
sage: f.coefficients(10)
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]

stretch(k)

Returns the stretch of a cycle index series by a positive integer $$k$$.

If

$f = \sum_{n=0}^{\infty} f_n(x_1, x_2, \ldots, x_n),$

then the stretch $$g$$ of $$f$$ by $$k$$ is

$g = \sum_{n=0}^{\infty} f_n(x_k, x_{2k}, \ldots, x_{nk}).$

EXAMPLES:

sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: p = SymmetricFunctions(QQ).power()
sage: CIS = CycleIndexSeriesRing(p)
sage: f = CIS([p([1])])
sage: f.stretch(3).coefficients(10)
[p[3], 0, 0, p[3], 0, 0, p[3], 0, 0, p[3]]

weighted_composition(y_species)

Returns the composition of this cycle index series with the cycle index series of y_species where y_species is a weighted species.

Note that this is basically the same algorithm as composition except we can not use the optimization that the powering of cycle index series commutes with ‘stretching’.

EXAMPLES:

sage: E = species.SetSpecies(); C = species.CycleSpecies()
sage: E_cis = E.cycle_index_series()
sage: E_cis.weighted_composition(C).coefficients(4)
[p[], p[1], p[1, 1] + p[2], p[1, 1, 1] + p[2, 1] + p[3]]
sage: E(C).cycle_index_series().coefficients(4)
[p[], p[1], p[1, 1] + p[2], p[1, 1, 1] + p[2, 1] + p[3]]

sage.combinat.species.generating_series.CycleIndexSeriesRing(R)

Returns the ring of cycle index series. Note that is is just a LazyPowerSeriesRing whose elements have some extra methods.

EXAMPLES:

sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: R = CycleIndexSeriesRing(QQ); R
Cycle Index Series Ring over Symmetric Functions over Rational Field in the powersum basis
sage: R([1]).coefficients(4)
[1, 1, 1, 1]


TESTS: We test to make sure that caching works.

sage: R is CycleIndexSeriesRing(QQ)
True

class sage.combinat.species.generating_series.CycleIndexSeriesRing_class(R)

EXAMPLES:

sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: R = CycleIndexSeriesRing(QQ); R
Cycle Index Series Ring over Symmetric Functions over Rational Field in the powersum basis
True

class sage.combinat.species.generating_series.ExponentialGeneratingSeries(A, stream=None, order=None, aorder=None, aorder_changed=True, is_initialized=False, name=None)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: f = L()
Uninitialized lazy power series

count(n)

Returns the number of structures of size n.

EXAMPLES:

sage: from sage.combinat.species.generating_series import ExponentialGeneratingSeriesRing
sage: R = ExponentialGeneratingSeriesRing(QQ)
sage: f = R([1])
sage: [f.count(i) for i in range(7)]
[1, 1, 2, 6, 24, 120, 720]

counts(n)

Returns the number of structures on a set for size i for i in range(n).

EXAMPLES:

sage: from sage.combinat.species.generating_series import ExponentialGeneratingSeriesRing
sage: R = ExponentialGeneratingSeriesRing(QQ)
sage: f = R(range(20))
sage: f.counts(5)
[0, 1, 4, 18, 96]

functorial_composition(y)

Returns the exponential generating series which is the functorial composition of self with y.

If $$f = \sum_{n=0}^{\infty} f_n \frac{x^n}{n!}$$ and $$g = \sum_{n=0}^{\infty} f_n \frac{x^n}{n!}$$, then functorial composition $$f \Box g$$ is defined as

$f \Box g = \sum_{n=0}^{\infty} f_{g_n} \frac{x^n}{n!}$

REFERENCES:

EXAMPLES:

sage: G = species.SimpleGraphSpecies()
sage: g = G.generating_series()
sage: g.coefficients(10)
[1, 1, 1, 4/3, 8/3, 128/15, 2048/45, 131072/315, 2097152/315, 536870912/2835]

sage.combinat.species.generating_series.ExponentialGeneratingSeriesRing(R)

Returns the ring of ordinary generating series. Note that is is just a LazyPowerSeriesRing whose elements have some extra methods.

EXAMPLES:

sage: from sage.combinat.species.generating_series import ExponentialGeneratingSeriesRing
sage: R = ExponentialGeneratingSeriesRing(QQ); R
Lazy Power Series Ring over Rational Field
sage: R([1]).coefficients(4)
[1, 1, 1, 1]
sage: R([1]).counts(4)
[1, 1, 2, 6]


TESTS: We test to make sure that caching works.

sage: R is ExponentialGeneratingSeriesRing(QQ)
True

class sage.combinat.species.generating_series.ExponentialGeneratingSeriesRing_class(R)

EXAMPLES:

sage: from sage.combinat.species.generating_series import ExponentialGeneratingSeriesRing
sage: R = ExponentialGeneratingSeriesRing(QQ)
True

class sage.combinat.species.generating_series.OrdinaryGeneratingSeries(A, stream=None, order=None, aorder=None, aorder_changed=True, is_initialized=False, name=None)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: f = L()
Uninitialized lazy power series

count(n)

Returns the number of structures on a set of size n.

EXAMPLES:

sage: from sage.combinat.species.generating_series import OrdinaryGeneratingSeriesRing
sage: R = OrdinaryGeneratingSeriesRing(QQ)
sage: f = R(range(20))
sage: f.count(10)
10

counts(n)

Returns the number of structures on a set for size i for i in range(n).

EXAMPLES:

sage: from sage.combinat.species.generating_series import OrdinaryGeneratingSeriesRing
sage: R = OrdinaryGeneratingSeriesRing(QQ)
sage: f = R(range(20))
sage: f.counts(10)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

sage.combinat.species.generating_series.OrdinaryGeneratingSeriesRing(R)

Returns the ring of ordinary generating series.

Note that is is just a LazyPowerSeriesRing whose elements have some extra methods.

EXAMPLES:

sage: from sage.combinat.species.generating_series import OrdinaryGeneratingSeriesRing
sage: R = OrdinaryGeneratingSeriesRing(QQ); R
Lazy Power Series Ring over Rational Field
sage: R([1]).coefficients(4)
[1, 1, 1, 1]
sage: R([1]).counts(4)
[1, 1, 1, 1]


TESTS: We test to make sure that caching works.

sage: R is OrdinaryGeneratingSeriesRing(QQ)
True

class sage.combinat.species.generating_series.OrdinaryGeneratingSeriesRing_class(R)

EXAMPLES:

sage: from sage.combinat.species.generating_series import OrdinaryGeneratingSeriesRing
sage: R = OrdinaryGeneratingSeriesRing(QQ)
True

sage.combinat.species.generating_series.factorial_gen()

A generator for the factorials starting at 0.

EXAMPLES:

sage: from sage.combinat.species.generating_series import factorial_gen
sage: g = factorial_gen()
sage: [next(g) for i in range(5)]
[1, 1, 2, 6, 24]


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