Identifier
Values
[1] => 0
[-1] => 0
[1,2] => 0
[1,-2] => 1
[-1,2] => 0
[-1,-2] => 0
[2,1] => 0
[2,-1] => 1
[-2,1] => 0
[-2,-1] => 0
[1,2,3] => 0
[1,2,-3] => 2
[1,-2,3] => 1
[1,-2,-3] => 1
[-1,2,3] => 0
[-1,2,-3] => 2
[-1,-2,3] => 0
[-1,-2,-3] => 0
[1,3,2] => 0
[1,3,-2] => 2
[1,-3,2] => 1
[1,-3,-2] => 1
[-1,3,2] => 0
[-1,3,-2] => 1
[-1,-3,2] => 1
[-1,-3,-2] => 0
[2,1,3] => 0
[2,1,-3] => 2
[2,-1,3] => 1
[2,-1,-3] => 3
[-2,1,3] => 0
[-2,1,-3] => 1
[-2,-1,3] => 0
[-2,-1,-3] => 1
[2,3,1] => 1
[2,3,-1] => 2
[2,-3,1] => 1
[2,-3,-1] => 2
[-2,3,1] => 1
[-2,3,-1] => 1
[-2,-3,1] => 0
[-2,-3,-1] => 0
[3,1,2] => 0
[3,1,-2] => 3
[3,-1,2] => 1
[3,-1,-2] => 1
[-3,1,2] => 0
[-3,1,-2] => 1
[-3,-1,2] => 0
[-3,-1,-2] => 0
[3,2,1] => 0
[3,2,-1] => 2
[3,-2,1] => 2
[3,-2,-1] => 1
[-3,2,1] => 0
[-3,2,-1] => 1
[-3,-2,1] => 0
[-3,-2,-1] => 0
[1,2,3,4] => 0
[1,2,3,-4] => 3
[1,2,-3,4] => 2
[1,2,-3,-4] => 2
[1,-2,3,4] => 1
[1,-2,3,-4] => 4
[1,-2,-3,4] => 1
[1,-2,-3,-4] => 1
[-1,2,3,4] => 0
[-1,2,3,-4] => 3
[-1,2,-3,4] => 2
[-1,2,-3,-4] => 2
[-1,-2,3,4] => 0
[-1,-2,3,-4] => 3
[-1,-2,-3,4] => 0
[-1,-2,-3,-4] => 0
[1,2,4,3] => 0
[1,2,4,-3] => 3
[1,2,-4,3] => 2
[1,2,-4,-3] => 2
[1,-2,4,3] => 1
[1,-2,4,-3] => 2
[1,-2,-4,3] => 3
[1,-2,-4,-3] => 1
[-1,2,4,3] => 0
[-1,2,4,-3] => 3
[-1,2,-4,3] => 2
[-1,2,-4,-3] => 2
[-1,-2,4,3] => 0
[-1,-2,4,-3] => 1
[-1,-2,-4,3] => 2
[-1,-2,-4,-3] => 0
[1,3,2,4] => 0
[1,3,2,-4] => 3
[1,3,-2,4] => 2
[1,3,-2,-4] => 5
[1,-3,2,4] => 1
[1,-3,2,-4] => 2
[1,-3,-2,4] => 1
[1,-3,-2,-4] => 2
[-1,3,2,4] => 0
[-1,3,2,-4] => 3
[-1,3,-2,4] => 1
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Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
References
[1] Fang, W., Mühle, H., Novelli, J.-C. Parabolic Tamari Lattices in Linear Type B arXiv:2112.13400
Code
def statistic(pi):
n = len(pi)
return sum(1 for i in range(-n, n+1)
for j in range(max(1, i+1), n+1)
for k in range(max(1, j+1), n+1)
if i != 0 and j != 0 and k != 0
and pi(j) > pi(i)
and pi(i) == (1 if pi(k) == -1 else pi(k)+1))
Created
Dec 09, 2022 at 15:15 by Martin Rubey
Updated
Dec 09, 2022 at 15:15 by Martin Rubey
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