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Identifier
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