Identifier
Identifier
Values
[] generating graphics... => 1
[1] generating graphics... => 1
[2] generating graphics... => 1
[1,1] generating graphics... => 1
[3] generating graphics... => 2
[2,1] generating graphics... => 1
[1,1,1] generating graphics... => 1
[4] generating graphics... => 6
[3,1] generating graphics... => 2
[2,2] generating graphics... => 1
[2,1,1] generating graphics... => 1
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 6
[4,1] generating graphics... => 6
[3,2] generating graphics... => 2
[3,1,1] generating graphics... => 2
[2,2,1] generating graphics... => 1
[2,1,1,1] generating graphics... => 1
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 27
[5,1] generating graphics... => 6
[4,2] generating graphics... => 6
[4,1,1] generating graphics... => 6
[3,3] generating graphics... => 4
[3,2,1] generating graphics... => 2
[3,1,1,1] generating graphics... => 2
[2,2,2] generating graphics... => 1
[2,2,1,1] generating graphics... => 1
[2,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 20
[6,1] generating graphics... => 27
[5,2] generating graphics... => 6
[5,1,1] generating graphics... => 6
[4,3] generating graphics... => 12
[4,2,1] generating graphics... => 6
[4,1,1,1] generating graphics... => 6
[3,3,1] generating graphics... => 4
[3,2,2] generating graphics... => 2
[3,2,1,1] generating graphics... => 2
[3,1,1,1,1] generating graphics... => 2
[2,2,2,1] generating graphics... => 1
[2,2,1,1,1] generating graphics... => 1
[2,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 130
[7,1] generating graphics... => 20
[6,2] generating graphics... => 27
[6,1,1] generating graphics... => 27
[5,3] generating graphics... => 12
[5,2,1] generating graphics... => 6
[5,1,1,1] generating graphics... => 6
[4,4] generating graphics... => 36
[4,3,1] generating graphics... => 12
[4,2,2] generating graphics... => 6
[4,2,1,1] generating graphics... => 6
[4,1,1,1,1] generating graphics... => 6
[3,3,2] generating graphics... => 4
[3,3,1,1] generating graphics... => 4
[3,2,2,1] generating graphics... => 2
[3,2,1,1,1] generating graphics... => 2
[3,1,1,1,1,1] generating graphics... => 2
[2,2,2,2] generating graphics... => 1
[2,2,2,1,1] generating graphics... => 1
[2,2,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 124
[8,1] generating graphics... => 130
[7,2] generating graphics... => 20
[7,1,1] generating graphics... => 20
[6,3] generating graphics... => 54
[6,2,1] generating graphics... => 27
[6,1,1,1] generating graphics... => 27
[5,4] generating graphics... => 36
[5,3,1] generating graphics... => 12
[5,2,2] generating graphics... => 6
[5,2,1,1] generating graphics... => 6
[5,1,1,1,1] generating graphics... => 6
[4,4,1] generating graphics... => 36
[4,3,2] generating graphics... => 12
[4,3,1,1] generating graphics... => 12
[4,2,2,1] generating graphics... => 6
[4,2,1,1,1] generating graphics... => 6
[4,1,1,1,1,1] generating graphics... => 6
[3,3,3] generating graphics... => 8
[3,3,2,1] generating graphics... => 4
[3,3,1,1,1] generating graphics... => 4
[3,2,2,2] generating graphics... => 2
[3,2,2,1,1] generating graphics... => 2
[3,2,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1] generating graphics... => 2
[2,2,2,2,1] generating graphics... => 1
[2,2,2,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 598
[9,1] generating graphics... => 124
[8,2] generating graphics... => 130
[8,1,1] generating graphics... => 130
[7,3] generating graphics... => 40
[7,2,1] generating graphics... => 20
[7,1,1,1] generating graphics... => 20
[6,4] generating graphics... => 162
[6,3,1] generating graphics... => 54
[6,2,2] generating graphics... => 27
[6,2,1,1] generating graphics... => 27
[6,1,1,1,1] generating graphics... => 27
[5,5] generating graphics... => 36
[5,4,1] generating graphics... => 36
[5,3,2] generating graphics... => 12
[5,3,1,1] generating graphics... => 12
[5,2,2,1] generating graphics... => 6
[5,2,1,1,1] generating graphics... => 6
[5,1,1,1,1,1] generating graphics... => 6
[4,4,2] generating graphics... => 36
[4,4,1,1] generating graphics... => 36
[4,3,3] generating graphics... => 24
[4,3,2,1] generating graphics... => 12
[4,3,1,1,1] generating graphics... => 12
[4,2,2,2] generating graphics... => 6
[4,2,2,1,1] generating graphics... => 6
[4,2,1,1,1,1] generating graphics... => 6
[4,1,1,1,1,1,1] generating graphics... => 6
[3,3,3,1] generating graphics... => 8
[3,3,2,2] generating graphics... => 4
[3,3,2,1,1] generating graphics... => 4
[3,3,1,1,1,1] generating graphics... => 4
[3,2,2,2,1] generating graphics... => 2
[3,2,2,1,1,1] generating graphics... => 2
[3,2,1,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1,1] generating graphics... => 2
[2,2,2,2,2] generating graphics... => 1
[2,2,2,2,1,1] generating graphics... => 1
[2,2,2,1,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
click to show generating function       
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