Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000222
(load all 6 compositions to match this statistic)
Mapping
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 2
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 4
[1,2,5,4,3] => 3
[1,3,2,4,5] => 3
[1,3,2,5,4] => 4
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 4
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
Description
The number of alignments in the permutation.
Matching statistic: St001822
(load all 3 compositions to match this statistic)
Mapping
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] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 0
[3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 1
[2,3,4,1] => [2,3,4,1] => 0
[2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [2,4,3,1] => 1
[3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 1
[3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => ? = 3
[1,2,4,3,5] => [1,2,4,3,5] => ? = 3
[1,2,4,5,3] => [1,2,4,5,3] => ? = 2
[1,2,5,3,4] => [1,2,5,3,4] => ? = 4
[1,2,5,4,3] => [1,2,5,4,3] => ? = 3
[1,3,2,4,5] => [1,3,2,4,5] => ? = 3
[1,3,2,5,4] => [1,3,2,5,4] => ? = 4
[1,3,4,2,5] => [1,3,4,2,5] => ? = 2
[1,3,4,5,2] => [1,3,4,5,2] => ? = 1
[1,3,5,2,4] => [1,3,5,2,4] => ? = 3
[1,3,5,4,2] => [1,3,5,4,2] => ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => ? = 4
[1,4,2,5,3] => [1,4,2,5,3] => ? = 3
[1,4,3,2,5] => [1,4,3,2,5] => ? = 3
[1,4,3,5,2] => [1,4,3,5,2] => ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[1,4,5,3,2] => [1,4,5,3,2] => ? = 3
[1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[1,5,2,4,3] => [1,5,2,4,3] => ? = 4
[1,5,3,2,4] => [1,5,3,2,4] => ? = 4
[1,5,3,4,2] => [1,5,3,4,2] => ? = 3
[1,5,4,2,3] => [1,5,4,2,3] => ? = 3
[1,5,4,3,2] => [1,5,4,3,2] => ? = 4
[2,1,3,4,5] => [2,1,3,4,5] => ? = 3
[2,1,3,5,4] => [2,1,3,5,4] => ? = 4
[2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[2,1,5,3,4] => [2,1,5,3,4] => ? = 3
[2,1,5,4,3] => [2,1,5,4,3] => ? = 4
[2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[2,3,1,5,4] => [2,3,1,5,4] => ? = 3
[2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[2,4,1,3,5] => [2,4,1,3,5] => ? = 3
[2,4,1,5,3] => [2,4,1,5,3] => ? = 2
[2,4,3,1,5] => [2,4,3,1,5] => ? = 2
[2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[2,4,5,1,3] => [2,4,5,1,3] => ? = 1
[2,4,5,3,1] => [2,4,5,3,1] => ? = 2
[2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[2,5,1,4,3] => [2,5,1,4,3] => ? = 3
[2,5,3,1,4] => [2,5,3,1,4] => ? = 3
[2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[2,5,4,1,3] => [2,5,4,1,3] => ? = 2
[2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[3,1,2,4,5] => [3,1,2,4,5] => ? = 4
[3,1,2,5,4] => [3,1,2,5,4] => ? = 3
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, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.