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.

REFERENCES:

BLL(1,2,3,4)

F. Bergeron, G. Labelle, and P. Leroux. “Combinatorial species and tree-like structures”. Encyclopedia of Mathematics and its Applications, vol. 67, Cambridge Univ. Press. 1998.

BLL-Intro

Francois Bergeron, Gilbert Labelle, and Pierre Leroux. “Introduction to the Theory of Species of Structures”, March 14, 2008.

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

Bases: sage.combinat.species.series.LazyPowerSeries

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 in self.

EXAMPLES:

sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: p = SymmetricFunctions(QQ).power()
sage: CIS = CycleIndexSeriesRing(QQ)
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 \(0 + p_{1} + \cdots\).)

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]

ALGORITHM:

Let \(F\) be a species satisfying \(F = 0 + X + F_2 + F_3 + \cdots\) 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!

See also

The compositional inverse \(\Omega\) of the species \(E_{+}\) of nonempty sets can be handled much more efficiently using specialized methods. See LogarithmCycleIndexSeries()

AUTHORS:

• Andrew Gainer-Dewar

count(t)

Return the number of structures corresponding to a certain cycle type t.

EXAMPLES:

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

derivative(order=1)

Return the species-theoretic \(n\)-th derivative of self, where \(n\) is order.

For a cycle index series \(F (p_{1}, p_{2}, p_{3}, \ldots)\), its derivative is the cycle index series \(F' = D_{p_{1}} F\) (that is, the formal derivative of \(F\) with respect to the variable \(p_{1}\)).

If \(F\) is the cycle index series of a species \(S\) then \(F'\) is the cycle index series of an associated species \(S'\) of \(S\)-structures with a “hole”.

EXAMPLES:

The species \(E\) of sets satisfies the relationship \(E' = E\):

sage: E = species.SetSpecies().cycle_index_series()
sage: E.coefficients(8) == E.derivative().coefficients(8)
True

The species \(C\) of cyclic orderings and the species \(L\) of linear orderings satisfy the relationship \(C' = L\):

sage: C = species.CycleSpecies().cycle_index_series()
sage: L = species.LinearOrderSpecies().cycle_index_series()
sage: L.coefficients(8) == C.derivative().coefficients(8)
True

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]

exponential()

Return the species-theoretic exponential of self.

For a cycle index \(Z_{F}\) of a species \(F\), its exponential is the cycle index series \(Z_{E} \circ Z_{F}\), where \(Z_{E}\) is the ExponentialCycleIndexSeries().

The exponential \(Z_{E} \circ Z_{F}\) is then the cycle index series of the species \(E \circ F\) of “sets of \(F\)-structures”.

EXAMPLES:

Let \(BT\) be the species of binary trees, \(BF\) the species of binary forests, and \(E\) the species of sets. Then we have \(BF = E \circ BT\):

sage: BT = species.BinaryTreeSpecies().cycle_index_series()
sage: BF = species.BinaryForestSpecies().cycle_index_series()
sage: BT.exponential().isotype_generating_series().coefficients(8) == BF.isotype_generating_series().coefficients(8)
True

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:

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

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]

integral(*args)

Given a cycle index \(G\), it is not in general possible to recover a single cycle index \(F\) such that \(F' = G\) (even up to addition of a constant term).

More broadly, it may be the case that there are many non-isomorphic species \(S\) such that \(S' = T\) for a given species \(T\). For example, the species \(3 C_{3}\) of 3-cycles from three distinct classes and the species \(X^{3}\) of 3-sets are not isomorphic, but \((3 C_{3})' = (X^{3})' = 3 X^{2}\).

EXAMPLES:

sage: C3 = species.CycleSpecies(size=3).cycle_index_series()
sage: X = species.SingletonSpecies().cycle_index_series()
sage: (3*C3).derivative().coefficients(8) == (3*X^2).coefficients(8)
True
sage: (X^3).derivative().coefficients(8) == (3*X^2).coefficients(8)
True

Warning

This method has no implementation and exists only to prevent you from doing something strange. Calling it raises a NotImplementedError!

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]

logarithm()

Return the combinatorial logarithm of self.

For a cycle index \(Z_{F}\) of a species \(F\), its logarithm is the cycle index series \(Z_{\Omega} \circ Z_{F}\), where \(Z_{\Omega}\) is the LogarithmCycleIndexSeries().

The logarithm \(Z_{\Omega} \circ Z_{F}\) is then the cycle index series of the (virtual) species \(\Omega \circ F\) of “connected \(F\)-structures”. In particular, if \(F = E^{+} \circ G\) for \(E^{+}\) the species of nonempty sets and \(G\) some other species, then \(\Omega \circ F = G\).

EXAMPLES:

Let \(G\) be the species of nonempty graphs and \(CG\) be the species of nonempty connected graphs. Then \(G = E^{+} \circ CG\), so \(CG = \Omega \circ G\):

sage: G = species.SimpleGraphSpecies().cycle_index_series() - 1
sage: from sage.combinat.species.generating_series import LogarithmCycleIndexSeries
sage: CG = LogarithmCycleIndexSeries().compose(G)
sage: CG.isotype_generating_series().coefficients(8)
[0, 1, 1, 2, 6, 21, 112, 853]

pointing()

Return the species-theoretic pointing of self.

For a cycle index \(F\), its pointing is the cycle index series \(F^{\bullet} = p_{1} \cdot F'\).

If \(F\) is the cycle index series of a species \(S\) then \(F^{\bullet}\) is the cycle index series of an associated species \(S^{\bullet}\) of \(S\)-structures with a marked “root”.

EXAMPLES:

The species \(E^{\bullet}\) of “pointed sets” satisfies \(E^{\bullet} = X \cdot E\):

sage: E = species.SetSpecies().cycle_index_series()
sage: X = species.SingletonSpecies().cycle_index_series()
sage: E.pointing().coefficients(8) == (X*E).coefficients(8)
True

stretch(k)

Return the stretch of the cycle index series self by a positive integer \(k\).

If

\[f = \sum_{n=0}^{\infty} f_n(p_1, p_2, p_3, \ldots ),\]

then the stretch \(g\) of \(f\) by \(k\) is

\[g = \sum_{n=0}^{\infty} f_n(p_k, p_{2k}, p_{3k}, \ldots ).\]

EXAMPLES:

sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: p = SymmetricFunctions(QQ).power()
sage: CIS = CycleIndexSeriesRing(QQ)
sage: f = CIS([p([]), p([1]), p([2]), p.zero()])
sage: f.stretch(3).coefficients(10)
[p[], 0, 0, p[3], 0, 0, p[6], 0, 0, 0]

weighted_composition(y_species)

Return the composition of this cycle index series with the cycle index series of the weighted species y_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)

Return the ring of cycle index series over R.

This is the ring of formal power series \(\Lambda[x]\), where \(\Lambda\) is the ring of symmetric functions over R in the \(p\)-basis. Its purpose is to house the cycle index series of species (in a somewhat nonstandard notation tailored to Sage): If \(F\) is a species, then the cycle index series of \(F\) is defined to be the formal power series

\[\sum_{n \geq 0} \frac{1}{n!} (\sum_{\sigma \in S_n} \operatorname{fix} F[\sigma] \prod_{z \text{ is a cycle of } \sigma} p_{\text{length of } z}) x^n \in \Lambda_\QQ [x],\]

where \(\operatorname{fix} F[\sigma]\) denotes the number of fixed points of the permutation \(F[\sigma]\) of \(F[n]\). We notice that this power series is “equigraded” (meaning that its \(x^n\)-coefficient is homogeneous of degree \(n\)). A more standard convention in combinatorics would be to use \(x_i\) instead of \(p_i\), and drop the \(x\) (that is, evaluate the above power series at \(x = 1\)); but this would be more difficult to implement in Sage, as it would be an element of a power series ring in infinitely many variables.

Note that it is just a LazyPowerSeriesRing (whose base ring is \(\Lambda\)) 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) # This is not combinatorially
....:                        # meaningful.
[1, 1, 1, 1]

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

Bases: sage.combinat.species.series.LazyPowerSeriesRing

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 == loads(dumps(R))
True

sage.combinat.species.generating_series.ExponentialCycleIndexSeries(R=Rational Field)

Return the cycle index series of the species \(E\) of sets.

This cycle index satisfies

\[Z_{E} = \sum_{n \geq 0} \sum_{\lambda \vdash n} \frac{p_{\lambda}}{z_{\lambda}}.\]

EXAMPLES:

sage: from sage.combinat.species.generating_series import ExponentialCycleIndexSeries
sage: ExponentialCycleIndexSeries().coefficients(5)
[p[], p[1], 1/2*p[1, 1] + 1/2*p[2], 1/6*p[1, 1, 1] + 1/2*p[2, 1]
 + 1/3*p[3], 1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2]
 + 1/3*p[3, 1] + 1/4*p[4]]

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

Bases: sage.combinat.species.series.LazyPowerSeries

count(n)

Return 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)

Return the number of structures on a set for size i for each 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)

Return 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 \(g = \sum_{n=0}^{\infty} g_n \frac{x^n}{n!}\), then functorial composition \(f \Box g\) is defined as

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

REFERENCES:

• Section 2.2 of [BLL].

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)

Return the ring of exponential generating series over R.

Note that it 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]

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

Bases: sage.combinat.species.series.LazyPowerSeriesRing

EXAMPLES:

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

sage.combinat.species.generating_series.LogarithmCycleIndexSeries(R=Rational Field)

Return the cycle index series of the virtual species \(\Omega\), the compositional inverse of the species \(E^{+}\) of nonempty sets.

The notion of virtual species is treated thoroughly in [BLL]. The specific algorithm used here to compute the cycle index of \(\Omega\) is found in [Labelle2008].

EXAMPLES:

The virtual species \(\Omega\) is ‘properly virtual’, in the sense that its cycle index has negative coefficients:

sage: from sage.combinat.species.generating_series import LogarithmCycleIndexSeries
sage: LogarithmCycleIndexSeries().coefficients(4)
[0, p[1], -1/2*p[1, 1] - 1/2*p[2], 1/3*p[1, 1, 1] - 1/3*p[3]]

Its defining property is that \(\Omega \circ E^{+} = E^{+} \circ \Omega = X\) (that is, that composition with \(E^{+}\) in both directions yields the multiplicative identity \(X\)):

sage: Eplus = sage.combinat.species.set_species.SetSpecies(min=1).cycle_index_series()
sage: LogarithmCycleIndexSeries().compose(Eplus).coefficients(4)
[0, p[1], 0, 0]

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

Bases: sage.combinat.species.series.LazyPowerSeries

count(n)

Return 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)

Return the number of structures on a set for size i for each 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)

Return the ring of ordinary generating series over R.

Note that it 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]

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

Bases: sage.combinat.species.series.LazyPowerSeriesRing

EXAMPLES:

sage: from sage.combinat.species.generating_series import OrdinaryGeneratingSeriesRing
sage: R = OrdinaryGeneratingSeriesRing(QQ)
sage: R == loads(dumps(R))
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]