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Statistic identifier: St001711

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Collection: Integer partitions

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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
$$ \alpha\pi\alpha^{-1} = \pi^2.$$

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References: [1]   Homolya, S., Szigeti, Jenő Solving equations in the symmetric group [[arXiv:2104.03593]]

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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)


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Statistic 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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Created: Apr 09, 2021 at 10:57 by Martin Rubey

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Last Updated: Apr 09, 2021 at 10:57 by Martin Rubey