Identifier
- St000375: 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] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 1
[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] => 0
[1,2,5,4,3] => 0
[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] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
[1,4,5,3,2] => 1
[1,5,2,3,4] => 0
[1,5,2,4,3] => 0
[1,5,3,2,4] => 0
[1,5,3,4,2] => 0
[1,5,4,2,3] => 0
[1,5,4,3,2] => 1
[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] => 0
[2,1,5,4,3] => 0
[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] => 0
[2,3,5,4,1] => 0
[2,4,1,3,5] => 0
[2,4,1,5,3] => 0
[2,4,3,1,5] => 0
[2,4,3,5,1] => 0
[2,4,5,1,3] => 0
[2,4,5,3,1] => 1
[2,5,1,3,4] => 0
[2,5,1,4,3] => 0
[2,5,3,1,4] => 0
[2,5,3,4,1] => 0
[2,5,4,1,3] => 0
[2,5,4,3,1] => 1
[3,1,2,4,5] => 0
[3,1,2,5,4] => 0
[3,1,4,2,5] => 0
[3,1,4,5,2] => 0
[3,1,5,2,4] => 0
[3,1,5,4,2] => 0
[3,2,1,4,5] => 0
[3,2,1,5,4] => 0
[3,2,4,1,5] => 0
[3,2,4,5,1] => 0
[3,2,5,1,4] => 0
[3,2,5,4,1] => 0
[3,4,1,2,5] => 0
[3,4,1,5,2] => 0
[3,4,2,1,5] => 1
[3,4,2,5,1] => 1
[3,4,5,1,2] => 0
[3,4,5,2,1] => 1
[3,5,1,2,4] => 0
[3,5,1,4,2] => 0
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Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..
References
[1] Chen, J. N., Zhou, R. D. P. On the Sign-imbalance of Permutation Tableaux arXiv:1602.00105
Code
def statistic(pi):
patterns = pi.pattern_positions([3,2,1])
return sum( 1 for i in range(len(pi)) if pi[i] not in pi.weak_excedences() and any( i == pattern[1] for pattern in patterns ) )
Created
Feb 02, 2016 at 20:19 by Christian Stump
Updated
May 10, 2019 at 17:29 by Henning Ulfarsson
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