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Identifier
Values
[] => 1
[1] => 1
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 3
[1,1,1] => 1
[4] => 1
[3,1] => 1
[2,2] => 1
[2,1,1] => 5
[1,1,1,1] => 1
[5] => 1
[4,1] => 5
[3,2] => 1
[3,1,1] => 1
[2,2,1] => 5
[2,1,1,1] => 7
[1,1,1,1,1] => 1
[6] => 3
[5,1] => 1
[4,2] => 1
[4,1,1] => 9
[3,3] => 1
[3,2,1] => 3
[3,1,1,1] => 1
[2,2,2] => 9
[2,2,1,1] => 17
[2,1,1,1,1] => 9
[1,1,1,1,1,1] => 1
[7] => 1
[6,1] => 3
[5,2] => 1
[5,1,1] => 1
[4,3] => 1
[4,2,1] => 7
[4,1,1,1] => 13
[3,3,1] => 19
[3,2,2] => 1
[3,2,1,1] => 5
[3,1,1,1,1] => 1
[2,2,2,1] => 15
[2,2,1,1,1] => 37
[2,1,1,1,1,1] => 11
[1,1,1,1,1,1,1] => 1
[8] => 1
[7,1] => 1
[6,2] => 3
[6,1,1] => 3
[5,3] => 1
[5,2,1] => 3
[5,1,1,1] => 1
[4,4] => 1
[4,3,1] => 5
[4,2,2] => 1
[4,2,1,1] => 29
[4,1,1,1,1] => 17
[3,3,2] => 1
[3,3,1,1] => 37
[3,2,2,1] => 5
[3,2,1,1,1] => 7
[3,1,1,1,1,1] => 1
[2,2,2,2] => 33
[2,2,2,1,1] => 45
[2,2,1,1,1,1] => 65
[2,1,1,1,1,1,1] => 13
[1,1,1,1,1,1,1,1] => 1
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Description
The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation.
Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that
$$ \alpha\pi\alpha^{-1} = \pi^2.$$
References
[1] Homolya, S., Szigeti, Jenő Solving equations in the symmetric group arXiv:2104.03593
Code
def statistic(la):
    total = 0
    cycles = []
    for p in la:
        cycles.append(tuple(range(total+1, total+p+1)))
        total += p
    a = Permutation(cycles)
    return sum(1 for pi in Permutations(len(a)) if a*pi == pi^2*a)
Created
Apr 09, 2021 at 10:57 by Martin Rubey
Updated
Apr 09, 2021 at 10:57 by Martin Rubey