Identifier
- St000957: Permutations ⟶ ℤ
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 3
[3,1,2,4] => 2
[3,1,4,2] => 3
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 4
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 4
[4,3,1,2] => 3
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 4
[1,4,5,3,2] => 3
[1,5,2,3,4] => 3
[1,5,2,4,3] => 3
[1,5,3,2,4] => 3
[1,5,3,4,2] => 4
[1,5,4,2,3] => 3
[1,5,4,3,2] => 3
[2,1,3,4,5] => 1
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 3
[2,1,5,3,4] => 3
[2,1,5,4,3] => 3
[2,3,1,4,5] => 2
[2,3,1,5,4] => 3
[2,3,4,1,5] => 3
[2,3,4,5,1] => 4
[2,3,5,1,4] => 4
[2,3,5,4,1] => 4
[2,4,1,3,5] => 3
[2,4,1,5,3] => 4
[2,4,3,1,5] => 3
[2,4,3,5,1] => 4
[2,4,5,1,3] => 5
[2,4,5,3,1] => 4
[2,5,1,3,4] => 4
[2,5,1,4,3] => 4
[2,5,3,1,4] => 4
[2,5,3,4,1] => 5
[2,5,4,1,3] => 4
[2,5,4,3,1] => 4
[3,1,2,4,5] => 2
[3,1,2,5,4] => 3
[3,1,4,2,5] => 3
[3,1,4,5,2] => 4
[3,1,5,2,4] => 4
[3,1,5,4,2] => 4
[3,2,1,4,5] => 2
[3,2,1,5,4] => 3
[3,2,4,1,5] => 3
[3,2,4,5,1] => 4
[3,2,5,1,4] => 4
[3,2,5,4,1] => 4
[3,4,1,2,5] => 4
[3,4,1,5,2] => 5
[3,4,2,1,5] => 3
[3,4,2,5,1] => 4
[3,4,5,1,2] => 6
[3,4,5,2,1] => 4
[3,5,1,2,4] => 5
[3,5,1,4,2] => 5
[3,5,2,1,4] => 4
>>> Load all 1200 entries. <<<
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Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Code
def statistic(pi):
return len(PermutationGroupElement(pi).bruhat_lower_covers())
Created
Aug 27, 2017 at 22:50 by Martin Rubey
Updated
Jan 15, 2018 at 08:16 by Martin Rubey
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