- FindStat

Identifier

St001385: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        []=>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
                    

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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)

Created

Apr 22, 2019 at 20:05 by Martin Rubey

Updated

Apr 22, 2019 at 20:05 by Martin Rubey