Identifier
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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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.
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.
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
[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
Code
def CumulativeSums(pi):
r"""
For pi a signed permutation of length n, returns
the tuple (x_1, ..., x_n),
where x_i = c_{|pi_1|} sgn(pi_{|pi_1|}) + ... + c_{|pi_i|} sgn(pi_{|pi_i|})
and (c_1, ... ,c_n) = (1, 2, ..., 2^{n-1}).
The actual value of the c-tuple for these statistics
does not matter so long as no sums or differences
of any subset of the c_i's is zero.
"""
n = len(pi)
c = [2**i for i in range(n)]
return [sum(c[abs(pi[i]) - 1]*sgn(pi[abs(pi[i]) - 1]) for i in range(j)) for j in range(n+1)]
def statistic(pi):
r"""
For pi a signed permutation of length n, returns the number
of strictly positive terms in CumulativeSums(pi)
"""
cumul = CumulativeSums(pi)
return sum(1 for i in range(len(pi)+1) if cumul[i] > 0)
Created
Sep 12, 2023 at 11:02 by Arvind Ayyer
Updated
Aug 14, 2024 at 10:48 by Arvind Ayyer
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