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Statistic identifier: St001926

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Collection: Signed permutations

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Description: Sparre Anderson's `Position of Maximum' statistic.

For $\pi$ a signed permutation of length $n$, first create the tuple $x = (x_1, \dots, x_n)$, where $x_i = c_{|\pi_1|} \pi_{|\pi_1|} + \cdots + c_{|\pi_i|} \pi_{|\pi_i|}$ and $(c_1, \dots ,c_n) = (1, 2, \dots, 2^{n-1})$. The actual value of the c-tuple for Sparre Anderson's statistic does not matter so long as no sums or differences of any subset of the $c_i$'s is zero. The choice of powers of $2$ is just a convenient choice.

This returns the position of the maximum value in the $x$-tuple. This is related to the ''discrete arcsin distribution''. The number of signed permutations with value equal to $k$ is given by $\binom{2k}{k} \binom{2n-2k}{n-k}$. This statistic is equidistributed with Sparre Anderson's `Number of Positives' statistic.

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References: [1]   Jacobs, K. Discrete Stochastics [[DOI:10.1007/978-3-0348-8645-1]]
[2]  [[oeis:A059366]]
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Code:


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Statistic values:

[1,2]      => 2
[1,-2]     => 1
[-1,2]     => 2
[-1,-2]    => 0
[2,1]      => 2
[2,-1]     => 0
[-2,1]     => 1
[-2,-1]    => 0
[1,2,3]    => 3
[1,2,-3]   => 2
[1,-2,3]   => 3
[1,-2,-3]  => 1
[-1,2,3]   => 3
[-1,2,-3]  => 2
[-1,-2,3]  => 0
[-1,-2,-3] => 0
[1,3,2]    => 3
[1,3,-2]   => 1
[1,-3,2]   => 2
[1,-3,-2]  => 1
[-1,3,2]   => 3
[-1,3,-2]  => 0
[-1,-3,2]  => 2
[-1,-3,-2] => 0
[2,1,3]    => 3
[2,1,-3]   => 2
[2,-1,3]   => 3
[2,-1,-3]  => 0
[-2,1,3]   => 3
[-2,1,-3]  => 1
[-2,-1,3]  => 0
[-2,-1,-3] => 0
[2,3,1]    => 3
[2,3,-1]   => 1
[2,-3,1]   => 3
[2,-3,-1]  => 0
[-2,3,1]   => 2
[-2,3,-1]  => 1
[-2,-3,1]  => 2
[-2,-3,-1] => 0
[3,1,2]    => 3
[3,1,-2]   => 0
[3,-1,2]   => 2
[3,-1,-2]  => 0
[-3,1,2]   => 3
[-3,1,-2]  => 0
[-3,-1,2]  => 1
[-3,-1,-2] => 0
[3,2,1]    => 3
[3,2,-1]   => 0
[3,-2,1]   => 1
[3,-2,-1]  => 0
[-3,2,1]   => 2
[-3,2,-1]  => 0
[-3,-2,1]  => 1
[-3,-2,-1] => 0

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Created: Sep 11, 2023 at 07:41 by Arvind Ayyer

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Last Updated: Sep 12, 2023 at 11:05 by Arvind Ayyer