# Ticket #14542: trac_14542_cycle_index_arithmetic_product.patch

File trac_14542_cycle_index_arithmetic_product.patch, 4.9 KB (added by agd, 9 years ago)

Patch to implement arithmetic product of cycle indices

• ## sage/combinat/species/generating_series.py

# HG changeset patch
# User Andrew Gainer-Dewar <andrew dot gainer dot dewar at gmail.com>
# Date 1367877237 18000
# Node ID bd8b4321be5a6615e6e5680449af6d75fe8a5868
# Parent  9c903bae39b955af5d4e5fa1a33bdc9e4c21b7bd
Implement arithmetic product of cycle index series

diff --git a/sage/combinat/species/generating_series.py b/sage/combinat/species/generating_series.py
 a class CycleIndexSeries(LazyPowerSeries): yield res n += 1 def arithmetic_product(self, g): """ Returns 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. 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] M. Maia and M. Mendez. "On the arithmetic product of combinatorial species." Discrete Mathematics, vol. 308, issue 23, 2008, pp. 5407-5427. http://arxiv.org/abs/math/0503436. """ from sage.combinat.partition import Partition, Partitions from sage.combinat.species.stream import Stream, _integers_from from sage.rings.arith import gcd, lcm, divisors from itertools import product, repeat, chain p = self.base_ring() assert self.coefficient(0) == p.zero() assert g.coefficient(0) == p.zero() # We first define an operation \\boxtimes on partitions as in Lemma 2.1 of [MM]_. def arith_prod_of_partitions (l1, l2): # Given two partitions l_1 and l_2, we construct a new partition l_1 \\boxtimes l_2 by # the following procedure: each pair of parts a \\in l_1 and b \\in l_2 contributes a part # \\lcm (a, b) to l_1 \\boxtimes l_2 with multiplicity \\gcm (l_1, l_2). term_iterable = chain.from_iterable( repeat(lcm(pair), times=gcd(pair)) for pair in product(l1, l2) ) term_list = sorted(term_iterable, reverse=True) res = Partition(term_list) return res # We then extend this to an operation on symmetric functions as per eq. 52 of [MM]_. def arith_prod_sf (x, y): ap_sf_wrapper = lambda l1, l2: p(arith_prod_of_partitions(l1, l2)) return p._apply_multi_module_morphism(x, y, ap_sf_wrapper) # Sage stores cycle index series by degree. # Thus, to compute the arithmetic product Z_M \\boxdot Z_N it is useful # to compute all terms of a given degree n` at once. def arith_prod_coeff (n): if n == 0: res = p.zero() else: index_set = ((d, n/d) for d in divisors(n)) res = sum(arith_prod_sf(self.coefficient(i), g.coefficient(j)) for i,j in index_set) # Build a list which has res in the nth slot and 0's before and after # to feed to sum_generator res_in_seq = [p.zero()]*n + [res, p.zero()] return self.parent(res_in_seq) # Finally, we use the sum_generator method to assemble these results into a single # LazyPowerSeries object. return self.parent().sum_generator(arith_prod_coeff(n) for n in _integers_from(0)) def _cycle_type(self, s): """ EXAMPLES::