Identifier
- St001727: Permutations ⟶ ℤ
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 2
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
[1,5,2,3,4] => 2
[1,5,2,4,3] => 2
[1,5,3,2,4] => 2
[1,5,3,4,2] => 2
[1,5,4,2,3] => 2
[1,5,4,3,2] => 2
[2,1,3,4,5] => 0
[2,1,3,5,4] => 0
[2,1,4,3,5] => 0
[2,1,4,5,3] => 0
[2,1,5,3,4] => 1
[2,1,5,4,3] => 1
[2,3,1,4,5] => 0
[2,3,1,5,4] => 0
[2,3,4,1,5] => 0
[2,3,4,5,1] => 0
[2,3,5,1,4] => 1
[2,3,5,4,1] => 1
[2,4,1,3,5] => 1
[2,4,1,5,3] => 1
[2,4,3,1,5] => 1
[2,4,3,5,1] => 1
[2,4,5,1,3] => 1
[2,4,5,3,1] => 1
[2,5,1,3,4] => 2
[2,5,1,4,3] => 2
[2,5,3,1,4] => 2
[2,5,3,4,1] => 2
[2,5,4,1,3] => 2
[2,5,4,3,1] => 2
[3,1,2,4,5] => 1
[3,1,2,5,4] => 1
[3,1,4,2,5] => 1
[3,1,4,5,2] => 1
[3,1,5,2,4] => 2
[3,1,5,4,2] => 2
[3,2,1,4,5] => 1
[3,2,1,5,4] => 1
[3,2,4,1,5] => 1
[3,2,4,5,1] => 1
[3,2,5,1,4] => 2
[3,2,5,4,1] => 2
[3,4,1,2,5] => 1
[3,4,1,5,2] => 1
[3,4,2,1,5] => 1
[3,4,2,5,1] => 1
[3,4,5,1,2] => 1
[3,4,5,2,1] => 1
[3,5,1,2,4] => 2
[3,5,1,4,2] => 2
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Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
References
[1] Coopman, M., Hamaker, Z. Involutions under Bruhat order and labeled Motzkin Paths arXiv:2106.06021
Code
def statistic(pi):
return sum(1 for i in range(len(pi)) for j in range(i+1, len(pi)) if pi[i] > pi[j] > i+1)
Created
Jun 14, 2021 at 15:37 by Martin Rubey
Updated
Jun 15, 2021 at 10:01 by Martin Rubey
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