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Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St000356
Mp00223: Permutations —runsort⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => 1
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,2,3] => 2
[2,1,3,4] => [1,3,4,2] => 1
[2,1,4,3] => [1,4,2,3] => 2
[2,3,1,4] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => 2
[3,2,1,4] => [1,4,2,3] => 2
[3,2,4,1] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,5,3,4] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => 2
[1,3,5,4,2] => [1,3,5,2,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => 3
[1,4,3,2,5] => [1,4,2,5,3] => 3
[1,4,3,5,2] => [1,4,2,3,5] => 2
[1,4,5,2,3] => [1,4,5,2,3] => 2
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000358
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [1,3,2] => [3,1,2] => 1
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 2
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 1
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => 2
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 1
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => 2
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 2
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => 2
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001727
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [3,1,2] => [3,1,2] => 1
[2,3,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2
[2,3,1,4] => [4,1,3,2] => [4,3,1,2] => 2
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => 0
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 2
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 2
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,3,5,4,1] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [2,4,5,3,1] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,4,3,5,1] => 1
[1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [2,5,3,4,1] => 2
[1,3,2,4,5] => [5,4,2,3,1] => [3,4,2,5,1] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [3,5,2,4,1] => 2
[1,3,4,2,5] => [5,2,4,3,1] => [3,4,5,1,2] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [3,2,4,5,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,1,2] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [3,2,5,4,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [4,5,2,3,1] => 3
[1,4,3,5,2] => [2,5,3,4,1] => [4,2,5,3,1] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,2,5,1] => 2
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 $\pi(i) > \pi(j) > i$.
Matching statistic: St000223
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [1,3,2] => [2,3,1] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => [3,2,4,1] => 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => [3,2,4,1] => 1
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,3,5,4,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [2,4,5,3,1] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [2,4,3,5,1] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [3,4,2,5,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [3,5,2,4,1] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [3,4,5,2,1] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [3,2,4,5,1] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,2,1] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,2,1] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => 3
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [4,3,2,5,1] => 2
Description
The number of nestings in the permutation.
Matching statistic: St000809
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000809: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000809: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,4,2,3] => 3
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => 3
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,5,2,4,3] => 2
Description
The reduced reflection length of the permutation.
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].
Matching statistic: St000307
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 29%
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[1,3,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,3,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[3,2,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[1,3,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[1,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 + 1
[1,4,3,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 + 1
[1,4,3,5,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[1,4,5,3,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 4 + 1
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 4 + 1
[1,5,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[1,5,4,2,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,1,3,5,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[2,1,4,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[2,1,4,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[2,1,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,1,5,4,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,3,1,4,5] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[2,3,1,5,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,3,4,1,5] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,4,1,3,5] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 4 + 1
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 4 + 1
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St001632
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 29%
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => ([],1)
=> ? = 0 + 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[1,3,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,3,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[3,2,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[1,3,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[1,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 + 1
[1,4,3,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 + 1
[1,4,3,5,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[1,4,5,3,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 4 + 1
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 4 + 1
[1,5,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[1,5,4,2,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,1,3,5,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[2,1,4,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[2,1,4,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[2,1,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,1,5,4,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,3,1,4,5] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[2,3,1,5,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,3,4,1,5] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,4,1,3,5] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 4 + 1
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 4 + 1
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001905
Mp00223: Permutations —runsort⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 3
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 3
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[1,5,2,4,3] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 4
[1,5,3,2,4] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 4
[1,5,3,4,2] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[1,5,4,2,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
[2,1,3,5,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[2,1,5,4,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[2,3,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[2,3,4,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[2,4,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 4
[2,4,3,1,5] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 4
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[2,4,5,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[2,4,5,3,1] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
[2,5,1,3,4] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
[2,5,1,4,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 3
[2,5,3,1,4] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 3
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
[2,5,4,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
[3,1,2,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
[3,1,2,5,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
Description
The number of preferred parking spots in a parking function less than the index of the car.
Let $(a_1,\dots,a_n)$ be a parking function. Then this statistic returns the number of indices $1\leq i\leq n$ such that $a_i < i$.
Matching statistic: St001822
Mp00223: Permutations —runsort⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001822: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001822: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[2,1,3] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[3,1,2] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 1
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 1
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 1
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 3
[1,4,3,2,5] => [1,4,2,5,3] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 3
[1,4,3,5,2] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[1,5,2,4,3] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[1,5,3,2,4] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[1,5,3,4,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[1,5,4,2,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[1,5,4,3,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[2,1,3,4,5] => [1,3,4,5,2] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1
[2,1,3,5,4] => [1,3,5,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 2
[2,1,4,3,5] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[2,1,5,4,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[2,3,1,4,5] => [1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[2,3,4,1,5] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[2,3,5,1,4] => [1,4,2,3,5] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[2,4,1,3,5] => [1,3,5,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[2,4,3,1,5] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[2,4,3,5,1] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[2,4,5,1,3] => [1,3,2,4,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[2,4,5,3,1] => [1,2,4,5,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1
[2,5,1,3,4] => [1,3,4,2,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1
[2,5,1,4,3] => [1,4,2,5,3] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 3
[2,5,3,1,4] => [1,4,2,5,3] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 3
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 2
[2,5,4,1,3] => [1,3,2,5,4] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 2
[3,1,2,4,5] => [1,2,4,5,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1
[3,1,2,5,4] => [1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 2
Description
The number of alignments of a signed permutation.
An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, [[St001866]], an alignment of type EN, [[St001867]], or an alignment of type NE, [[St001868]].
Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, [[St001862]], let \operatorname{wex} be the number of weak excedances, [[St001863]], and let \operatorname{neg} be the number of negative entries, [[St001429]]. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
Matching statistic: St001823
Mp00223: Permutations —runsort⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001823: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001823: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 3
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 3
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 3
[1,5,2,4,3] => [1,5,2,4,3] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 4
[1,5,3,2,4] => [1,5,2,4,3] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 4
[1,5,3,4,2] => [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 3
[1,5,4,2,3] => [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 3
[1,5,4,3,2] => [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 3
[2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[2,1,3,5,4] => [1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 2
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 3
[2,1,5,4,3] => [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 3
[2,3,1,4,5] => [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 3
[2,3,4,1,5] => [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[2,4,1,3,5] => [1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 4
[2,4,3,1,5] => [1,5,2,4,3] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 4
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[2,4,5,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[2,5,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
[2,5,1,4,3] => [1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 3
[2,5,3,1,4] => [1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 3
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2
[2,5,4,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2
[3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[3,1,2,5,4] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2
Description
The Stasinski-Voll length of a signed permutation.
The Stasinski-Voll length of a signed permutation $\sigma$ is
$$
L(\sigma) = \frac{1}{2} \#\{(i,j) ~\mid -n \leq i < j \leq n,~ i \not\equiv j \operatorname{mod} 2,~ \sigma(i) > \sigma(j)\},
$$
where $n$ is the size of $\sigma$.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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