*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
*****************************************************************************
-----------------------------------------------------------------------------
Statistic identifier: St001385
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition.
Equivalently, given an integer partition $\lambda$, this is the number of molecular combinatorial species that decompose into a product of atomic species of sizes $\lambda_1,\lambda_2,\dots$. In particular, the value on the partition $(n)$ is the number of atomic species of degree $n$, [2].
-----------------------------------------------------------------------------
References: [1] Naughton, L., Pfeiffer, G. Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group [[arXiv:1211.1911]]
[2] Number of atomic species of degree n; also number of connected permutation groups of degree n. [[OEIS:A005226]]
-----------------------------------------------------------------------------
Code:
@cached_function
def conjugacy_classes_subgroups(n):
if n == 1:
return 1
return len(SymmetricGroup(n).conjugacy_classes_subgroups())
def statistic(la):
def aux(n):
return n*conjugacy_classes_subgroups(n) - sum(aux(k)*conjugacy_classes_subgroups(n-k) for k in range(1,n))
def connected(n):
return sum(moebius(n//d)*aux(d) for d in divisors(n))//n
return prod(connected(n) for n in la)
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 1
[1,1] => 1
[3] => 2
[2,1] => 1
[1,1,1] => 1
[4] => 6
[3,1] => 2
[2,2] => 1
[2,1,1] => 1
[1,1,1,1] => 1
[5] => 6
[4,1] => 6
[3,2] => 2
[3,1,1] => 2
[2,2,1] => 1
[2,1,1,1] => 1
[1,1,1,1,1] => 1
[6] => 27
[5,1] => 6
[4,2] => 6
[4,1,1] => 6
[3,3] => 4
[3,2,1] => 2
[3,1,1,1] => 2
[2,2,2] => 1
[2,2,1,1] => 1
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 1
[7] => 20
[6,1] => 27
[5,2] => 6
[5,1,1] => 6
[4,3] => 12
[4,2,1] => 6
[4,1,1,1] => 6
[3,3,1] => 4
[3,2,2] => 2
[3,2,1,1] => 2
[3,1,1,1,1] => 2
[2,2,2,1] => 1
[2,2,1,1,1] => 1
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 1
[8] => 130
[7,1] => 20
[6,2] => 27
[6,1,1] => 27
[5,3] => 12
[5,2,1] => 6
[5,1,1,1] => 6
[4,4] => 36
[4,3,1] => 12
[4,2,2] => 6
[4,2,1,1] => 6
[4,1,1,1,1] => 6
[3,3,2] => 4
[3,3,1,1] => 4
[3,2,2,1] => 2
[3,2,1,1,1] => 2
[3,1,1,1,1,1] => 2
[2,2,2,2] => 1
[2,2,2,1,1] => 1
[2,2,1,1,1,1] => 1
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 1
[9] => 124
[8,1] => 130
[7,2] => 20
[7,1,1] => 20
[6,3] => 54
[6,2,1] => 27
[6,1,1,1] => 27
[5,4] => 36
[5,3,1] => 12
[5,2,2] => 6
[5,2,1,1] => 6
[5,1,1,1,1] => 6
[4,4,1] => 36
[4,3,2] => 12
[4,3,1,1] => 12
[4,2,2,1] => 6
[4,2,1,1,1] => 6
[4,1,1,1,1,1] => 6
[3,3,3] => 8
[3,3,2,1] => 4
[3,3,1,1,1] => 4
[3,2,2,2] => 2
[3,2,2,1,1] => 2
[3,2,1,1,1,1] => 2
[3,1,1,1,1,1,1] => 2
[2,2,2,2,1] => 1
[2,2,2,1,1,1] => 1
[2,2,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 1
[10] => 598
[9,1] => 124
[8,2] => 130
[8,1,1] => 130
[7,3] => 40
[7,2,1] => 20
[7,1,1,1] => 20
[6,4] => 162
[6,3,1] => 54
[6,2,2] => 27
[6,2,1,1] => 27
[6,1,1,1,1] => 27
[5,5] => 36
[5,4,1] => 36
[5,3,2] => 12
[5,3,1,1] => 12
[5,2,2,1] => 6
[5,2,1,1,1] => 6
[5,1,1,1,1,1] => 6
[4,4,2] => 36
[4,4,1,1] => 36
[4,3,3] => 24
[4,3,2,1] => 12
[4,3,1,1,1] => 12
[4,2,2,2] => 6
[4,2,2,1,1] => 6
[4,2,1,1,1,1] => 6
[4,1,1,1,1,1,1] => 6
[3,3,3,1] => 8
[3,3,2,2] => 4
[3,3,2,1,1] => 4
[3,3,1,1,1,1] => 4
[3,2,2,2,1] => 2
[3,2,2,1,1,1] => 2
[3,2,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1] => 2
[2,2,2,2,2] => 1
[2,2,2,2,1,1] => 1
[2,2,2,1,1,1,1] => 1
[2,2,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 1
-----------------------------------------------------------------------------
Created: Apr 22, 2019 at 20:05 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Apr 22, 2019 at 20:05 by Martin Rubey