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Statistic identifier: St001927
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Collection: Signed permutations
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Description: Sparre Andersen's number of positives of a signed permutation.
For $\pi$ a signed permutation of length $n$, first create the tuple $x = (x_1, \dots, x_n)$, where $x_i = c_{|\pi_1|} \operatorname{sgn}(\pi_{|\pi_1|}) + \cdots + c_{|\pi_i|} \operatorname{sgn}(\pi_{|\pi_i|})$ and $(c_1, \dots ,c_n) = (1, 2, \dots, 2^{n-1})$. The actual value of the c-tuple for Andersen'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 number of strictly positive values 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 $\binom{2k}{k} \binom{2n-2k}{n-k} \frac{n!}{2^n}$. This statistic is equidistributed with Sparre Andersen's `Position of Maximum' statistic.
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References: [1] Andersen, E. S. On the fluctuations of sums of random variables [[DOI:10.7146/math.scand.a-10385]]
[2] Andersen, E. S. On the fluctuations of sums of random variables II [[DOI:10.7146/math.scand.a-10407]]
[3] Jacobs, K. Discrete Stochastics [[DOI:10.1007/978-3-0348-8645-1]]
[4] Triangle T(m,s), m >= 0, 0 <= s <= m, arising in the computation of certain integrals. [[OEIS:A059366]]
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Code:
-----------------------------------------------------------------------------
Statistic values:
[1,2] => 2
[1,-2] => 1
[-1,2] => 1
[-1,-2] => 0
[2,1] => 2
[2,-1] => 0
[-2,1] => 2
[-2,-1] => 0
[1,2,3] => 3
[1,2,-3] => 2
[1,-2,3] => 2
[1,-2,-3] => 1
[-1,2,3] => 2
[-1,2,-3] => 1
[-1,-2,3] => 1
[-1,-2,-3] => 0
[1,3,2] => 3
[1,3,-2] => 1
[1,-3,2] => 3
[1,-3,-2] => 1
[-1,3,2] => 2
[-1,3,-2] => 0
[-1,-3,2] => 2
[-1,-3,-2] => 0
[2,1,3] => 3
[2,1,-3] => 2
[2,-1,3] => 1
[2,-1,-3] => 0
[-2,1,3] => 3
[-2,1,-3] => 2
[-2,-1,3] => 1
[-2,-1,-3] => 0
[2,3,1] => 3
[2,3,-1] => 1
[2,-3,1] => 2
[2,-3,-1] => 0
[-2,3,1] => 3
[-2,3,-1] => 1
[-2,-3,1] => 2
[-2,-3,-1] => 0
[3,1,2] => 3
[3,1,-2] => 0
[3,-1,2] => 3
[3,-1,-2] => 0
[-3,1,2] => 3
[-3,1,-2] => 0
[-3,-1,2] => 3
[-3,-1,-2] => 0
[3,2,1] => 3
[3,2,-1] => 0
[3,-2,1] => 3
[3,-2,-1] => 0
[-3,2,1] => 3
[-3,2,-1] => 0
[-3,-2,1] => 3
[-3,-2,-1] => 0
[1,2,3,4] => 4
[1,2,3,-4] => 3
[1,2,-3,4] => 3
[1,2,-3,-4] => 2
[1,-2,3,4] => 3
[1,-2,3,-4] => 2
[1,-2,-3,4] => 2
[1,-2,-3,-4] => 1
[-1,2,3,4] => 3
[-1,2,3,-4] => 2
[-1,2,-3,4] => 2
[-1,2,-3,-4] => 1
[-1,-2,3,4] => 2
[-1,-2,3,-4] => 1
[-1,-2,-3,4] => 1
[-1,-2,-3,-4] => 0
[1,2,4,3] => 4
[1,2,4,-3] => 2
[1,2,-4,3] => 4
[1,2,-4,-3] => 2
[1,-2,4,3] => 3
[1,-2,4,-3] => 1
[1,-2,-4,3] => 3
[1,-2,-4,-3] => 1
[-1,2,4,3] => 3
[-1,2,4,-3] => 1
[-1,2,-4,3] => 3
[-1,2,-4,-3] => 1
[-1,-2,4,3] => 2
[-1,-2,4,-3] => 0
[-1,-2,-4,3] => 2
[-1,-2,-4,-3] => 0
[1,3,2,4] => 4
[1,3,2,-4] => 3
[1,3,-2,4] => 2
[1,3,-2,-4] => 1
[1,-3,2,4] => 4
[1,-3,2,-4] => 3
[1,-3,-2,4] => 2
[1,-3,-2,-4] => 1
[-1,3,2,4] => 3
[-1,3,2,-4] => 2
[-1,3,-2,4] => 1
[-1,3,-2,-4] => 0
[-1,-3,2,4] => 3
[-1,-3,2,-4] => 2
[-1,-3,-2,4] => 1
[-1,-3,-2,-4] => 0
[1,3,4,2] => 4
[1,3,4,-2] => 2
[1,3,-4,2] => 3
[1,3,-4,-2] => 1
[1,-3,4,2] => 4
[1,-3,4,-2] => 2
[1,-3,-4,2] => 3
[1,-3,-4,-2] => 1
[-1,3,4,2] => 3
[-1,3,4,-2] => 1
[-1,3,-4,2] => 2
[-1,3,-4,-2] => 0
[-1,-3,4,2] => 3
[-1,-3,4,-2] => 1
[-1,-3,-4,2] => 2
[-1,-3,-4,-2] => 0
[1,4,2,3] => 4
[1,4,2,-3] => 1
[1,4,-2,3] => 4
[1,4,-2,-3] => 1
[1,-4,2,3] => 4
[1,-4,2,-3] => 1
[1,-4,-2,3] => 4
[1,-4,-2,-3] => 1
[-1,4,2,3] => 3
[-1,4,2,-3] => 0
[-1,4,-2,3] => 3
[-1,4,-2,-3] => 0
[-1,-4,2,3] => 3
[-1,-4,2,-3] => 0
[-1,-4,-2,3] => 3
[-1,-4,-2,-3] => 0
[1,4,3,2] => 4
[1,4,3,-2] => 1
[1,4,-3,2] => 4
[1,4,-3,-2] => 1
[1,-4,3,2] => 4
[1,-4,3,-2] => 1
[1,-4,-3,2] => 4
[1,-4,-3,-2] => 1
[-1,4,3,2] => 3
[-1,4,3,-2] => 0
[-1,4,-3,2] => 3
[-1,4,-3,-2] => 0
[-1,-4,3,2] => 3
[-1,-4,3,-2] => 0
[-1,-4,-3,2] => 3
[-1,-4,-3,-2] => 0
[2,1,3,4] => 4
[2,1,3,-4] => 3
[2,1,-3,4] => 3
[2,1,-3,-4] => 2
[2,-1,3,4] => 2
[2,-1,3,-4] => 1
[2,-1,-3,4] => 1
[2,-1,-3,-4] => 0
[-2,1,3,4] => 4
[-2,1,3,-4] => 3
[-2,1,-3,4] => 3
[-2,1,-3,-4] => 2
[-2,-1,3,4] => 2
[-2,-1,3,-4] => 1
[-2,-1,-3,4] => 1
[-2,-1,-3,-4] => 0
[2,1,4,3] => 4
[2,1,4,-3] => 2
[2,1,-4,3] => 4
[2,1,-4,-3] => 2
[2,-1,4,3] => 2
[2,-1,4,-3] => 0
[2,-1,-4,3] => 2
[2,-1,-4,-3] => 0
[-2,1,4,3] => 4
[-2,1,4,-3] => 2
[-2,1,-4,3] => 4
[-2,1,-4,-3] => 2
[-2,-1,4,3] => 2
[-2,-1,4,-3] => 0
[-2,-1,-4,3] => 2
[-2,-1,-4,-3] => 0
[2,3,1,4] => 4
[2,3,1,-4] => 3
[2,3,-1,4] => 2
[2,3,-1,-4] => 1
[2,-3,1,4] => 3
[2,-3,1,-4] => 2
[2,-3,-1,4] => 1
[2,-3,-1,-4] => 0
[-2,3,1,4] => 4
[-2,3,1,-4] => 3
[-2,3,-1,4] => 2
[-2,3,-1,-4] => 1
[-2,-3,1,4] => 3
[-2,-3,1,-4] => 2
[-2,-3,-1,4] => 1
[-2,-3,-1,-4] => 0
[2,3,4,1] => 4
[2,3,4,-1] => 2
[2,3,-4,1] => 3
[2,3,-4,-1] => 1
[2,-3,4,1] => 3
[2,-3,4,-1] => 1
[2,-3,-4,1] => 2
[2,-3,-4,-1] => 0
[-2,3,4,1] => 4
[-2,3,4,-1] => 2
[-2,3,-4,1] => 3
[-2,3,-4,-1] => 1
[-2,-3,4,1] => 3
[-2,-3,4,-1] => 1
[-2,-3,-4,1] => 2
[-2,-3,-4,-1] => 0
[2,4,1,3] => 4
[2,4,1,-3] => 1
[2,4,-1,3] => 4
[2,4,-1,-3] => 1
[2,-4,1,3] => 3
[2,-4,1,-3] => 0
[2,-4,-1,3] => 3
[2,-4,-1,-3] => 0
[-2,4,1,3] => 4
[-2,4,1,-3] => 1
[-2,4,-1,3] => 4
[-2,4,-1,-3] => 1
[-2,-4,1,3] => 3
[-2,-4,1,-3] => 0
[-2,-4,-1,3] => 3
[-2,-4,-1,-3] => 0
[2,4,3,1] => 4
[2,4,3,-1] => 1
[2,4,-3,1] => 4
[2,4,-3,-1] => 1
[2,-4,3,1] => 3
[2,-4,3,-1] => 0
[2,-4,-3,1] => 3
[2,-4,-3,-1] => 0
[-2,4,3,1] => 4
[-2,4,3,-1] => 1
[-2,4,-3,1] => 4
[-2,4,-3,-1] => 1
[-2,-4,3,1] => 3
[-2,-4,3,-1] => 0
[-2,-4,-3,1] => 3
[-2,-4,-3,-1] => 0
[3,1,2,4] => 4
[3,1,2,-4] => 3
[3,1,-2,4] => 1
[3,1,-2,-4] => 0
[3,-1,2,4] => 4
[3,-1,2,-4] => 3
[3,-1,-2,4] => 1
[3,-1,-2,-4] => 0
[-3,1,2,4] => 4
[-3,1,2,-4] => 3
[-3,1,-2,4] => 1
[-3,1,-2,-4] => 0
[-3,-1,2,4] => 4
[-3,-1,2,-4] => 3
[-3,-1,-2,4] => 1
[-3,-1,-2,-4] => 0
[3,1,4,2] => 4
[3,1,4,-2] => 2
[3,1,-4,2] => 2
[3,1,-4,-2] => 0
[3,-1,4,2] => 4
[3,-1,4,-2] => 2
[3,-1,-4,2] => 2
[3,-1,-4,-2] => 0
[-3,1,4,2] => 4
[-3,1,4,-2] => 2
[-3,1,-4,2] => 2
[-3,1,-4,-2] => 0
[-3,-1,4,2] => 4
[-3,-1,4,-2] => 2
[-3,-1,-4,2] => 2
[-3,-1,-4,-2] => 0
[3,2,1,4] => 4
[3,2,1,-4] => 3
[3,2,-1,4] => 1
[3,2,-1,-4] => 0
[3,-2,1,4] => 4
[3,-2,1,-4] => 3
[3,-2,-1,4] => 1
[3,-2,-1,-4] => 0
[-3,2,1,4] => 4
[-3,2,1,-4] => 3
[-3,2,-1,4] => 1
[-3,2,-1,-4] => 0
[-3,-2,1,4] => 4
[-3,-2,1,-4] => 3
[-3,-2,-1,4] => 1
[-3,-2,-1,-4] => 0
[3,2,4,1] => 4
[3,2,4,-1] => 2
[3,2,-4,1] => 2
[3,2,-4,-1] => 0
[3,-2,4,1] => 4
[3,-2,4,-1] => 2
[3,-2,-4,1] => 2
[3,-2,-4,-1] => 0
[-3,2,4,1] => 4
[-3,2,4,-1] => 2
[-3,2,-4,1] => 2
[-3,2,-4,-1] => 0
[-3,-2,4,1] => 4
[-3,-2,4,-1] => 2
[-3,-2,-4,1] => 2
[-3,-2,-4,-1] => 0
[3,4,1,2] => 4
[3,4,1,-2] => 1
[3,4,-1,2] => 3
[3,4,-1,-2] => 0
[3,-4,1,2] => 4
[3,-4,1,-2] => 1
[3,-4,-1,2] => 3
[3,-4,-1,-2] => 0
[-3,4,1,2] => 4
[-3,4,1,-2] => 1
[-3,4,-1,2] => 3
[-3,4,-1,-2] => 0
[-3,-4,1,2] => 4
[-3,-4,1,-2] => 1
[-3,-4,-1,2] => 3
[-3,-4,-1,-2] => 0
[3,4,2,1] => 4
[3,4,2,-1] => 1
[3,4,-2,1] => 3
[3,4,-2,-1] => 0
[3,-4,2,1] => 4
[3,-4,2,-1] => 1
[3,-4,-2,1] => 3
[3,-4,-2,-1] => 0
[-3,4,2,1] => 4
[-3,4,2,-1] => 1
[-3,4,-2,1] => 3
[-3,4,-2,-1] => 0
[-3,-4,2,1] => 4
[-3,-4,2,-1] => 1
[-3,-4,-2,1] => 3
[-3,-4,-2,-1] => 0
[4,1,2,3] => 4
[4,1,2,-3] => 0
[4,1,-2,3] => 4
[4,1,-2,-3] => 0
[4,-1,2,3] => 4
[4,-1,2,-3] => 0
[4,-1,-2,3] => 4
[4,-1,-2,-3] => 0
[-4,1,2,3] => 4
[-4,1,2,-3] => 0
[-4,1,-2,3] => 4
[-4,1,-2,-3] => 0
[-4,-1,2,3] => 4
[-4,-1,2,-3] => 0
[-4,-1,-2,3] => 4
[-4,-1,-2,-3] => 0
[4,1,3,2] => 4
[4,1,3,-2] => 0
[4,1,-3,2] => 4
[4,1,-3,-2] => 0
[4,-1,3,2] => 4
[4,-1,3,-2] => 0
[4,-1,-3,2] => 4
[4,-1,-3,-2] => 0
[-4,1,3,2] => 4
[-4,1,3,-2] => 0
[-4,1,-3,2] => 4
[-4,1,-3,-2] => 0
[-4,-1,3,2] => 4
[-4,-1,3,-2] => 0
[-4,-1,-3,2] => 4
[-4,-1,-3,-2] => 0
[4,2,1,3] => 4
[4,2,1,-3] => 0
[4,2,-1,3] => 4
[4,2,-1,-3] => 0
[4,-2,1,3] => 4
[4,-2,1,-3] => 0
[4,-2,-1,3] => 4
[4,-2,-1,-3] => 0
[-4,2,1,3] => 4
[-4,2,1,-3] => 0
[-4,2,-1,3] => 4
[-4,2,-1,-3] => 0
[-4,-2,1,3] => 4
[-4,-2,1,-3] => 0
[-4,-2,-1,3] => 4
[-4,-2,-1,-3] => 0
[4,2,3,1] => 4
[4,2,3,-1] => 0
[4,2,-3,1] => 4
[4,2,-3,-1] => 0
[4,-2,3,1] => 4
[4,-2,3,-1] => 0
[4,-2,-3,1] => 4
[4,-2,-3,-1] => 0
[-4,2,3,1] => 4
[-4,2,3,-1] => 0
[-4,2,-3,1] => 4
[-4,2,-3,-1] => 0
[-4,-2,3,1] => 4
[-4,-2,3,-1] => 0
[-4,-2,-3,1] => 4
[-4,-2,-3,-1] => 0
[4,3,1,2] => 4
[4,3,1,-2] => 0
[4,3,-1,2] => 4
[4,3,-1,-2] => 0
[4,-3,1,2] => 4
[4,-3,1,-2] => 0
[4,-3,-1,2] => 4
[4,-3,-1,-2] => 0
[-4,3,1,2] => 4
[-4,3,1,-2] => 0
[-4,3,-1,2] => 4
[-4,3,-1,-2] => 0
[-4,-3,1,2] => 4
[-4,-3,1,-2] => 0
[-4,-3,-1,2] => 4
[-4,-3,-1,-2] => 0
[4,3,2,1] => 4
[4,3,2,-1] => 0
[4,3,-2,1] => 4
[4,3,-2,-1] => 0
[4,-3,2,1] => 4
[4,-3,2,-1] => 0
[4,-3,-2,1] => 4
[4,-3,-2,-1] => 0
[-4,3,2,1] => 4
[-4,3,2,-1] => 0
[-4,3,-2,1] => 4
[-4,3,-2,-1] => 0
[-4,-3,2,1] => 4
[-4,-3,2,-1] => 0
[-4,-3,-2,1] => 4
[-4,-3,-2,-1] => 0
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Created: Sep 12, 2023 at 11:02 by Arvind Ayyer
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Last Updated: Aug 14, 2024 at 10:48 by Arvind Ayyer