Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001438
(load all 2 compositions to match this statistic)
Mapping
St001438: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => 0
[[2],[]]
 => 0
[[1,1],[]]
 => 0
[[2,1],[1]]
 => 1
[[3],[]]
 => 0
[[2,1],[]]
 => 0
[[3,1],[1]]
 => 1
[[2,2],[1]]
 => 1
[[3,2],[2]]
 => 2
[[1,1,1],[]]
 => 0
[[2,2,1],[1,1]]
 => 2
[[2,1,1],[1]]
 => 1
[[3,2,1],[2,1]]
 => 3
[[4],[]]
 => 0
[[3,1],[]]
 => 0
[[4,1],[1]]
 => 1
[[2,2],[]]
 => 0
[[3,2],[1]]
 => 1
[[4,2],[2]]
 => 2
[[2,1,1],[]]
 => 0
[[3,2,1],[1,1]]
 => 2
[[3,1,1],[1]]
 => 1
[[4,2,1],[2,1]]
 => 3
[[3,3],[2]]
 => 2
[[4,3],[3]]
 => 3
[[2,2,1],[1]]
 => 1
[[3,3,1],[2,1]]
 => 3
[[3,2,1],[2]]
 => 2
[[4,3,1],[3,1]]
 => 4
[[2,2,2],[1,1]]
 => 2
[[3,3,2],[2,2]]
 => 4
[[3,2,2],[2,1]]
 => 3
[[4,3,2],[3,2]]
 => 5
[[1,1,1,1],[]]
 => 0
[[2,2,2,1],[1,1,1]]
 => 3
[[2,2,1,1],[1,1]]
 => 2
[[3,3,2,1],[2,2,1]]
 => 5
[[2,1,1,1],[1]]
 => 1
[[3,2,2,1],[2,1,1]]
 => 4
[[3,2,1,1],[2,1]]
 => 3
[[4,3,2,1],[3,2,1]]
 => 6
[[5],[]]
 => 0
[[4,1],[]]
 => 0
[[5,1],[1]]
 => 1
[[3,2],[]]
 => 0
[[4,2],[1]]
 => 1
[[5,2],[2]]
 => 2
[[3,1,1],[]]
 => 0
[[4,2,1],[1,1]]
 => 2
[[4,1,1],[1]]
 => 1
Description
The number of missing boxes of a skew partition.
Matching statistic: St000228
(load all 20 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => 0
[[2],[]]
 => []
 => 0
[[1,1],[]]
 => []
 => 0
[[2,1],[1]]
 => [1]
 => 1
[[3],[]]
 => []
 => 0
[[2,1],[]]
 => []
 => 0
[[3,1],[1]]
 => [1]
 => 1
[[2,2],[1]]
 => [1]
 => 1
[[3,2],[2]]
 => [2]
 => 2
[[1,1,1],[]]
 => []
 => 0
[[2,2,1],[1,1]]
 => [1,1]
 => 2
[[2,1,1],[1]]
 => [1]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => 3
[[4],[]]
 => []
 => 0
[[3,1],[]]
 => []
 => 0
[[4,1],[1]]
 => [1]
 => 1
[[2,2],[]]
 => []
 => 0
[[3,2],[1]]
 => [1]
 => 1
[[4,2],[2]]
 => [2]
 => 2
[[2,1,1],[]]
 => []
 => 0
[[3,2,1],[1,1]]
 => [1,1]
 => 2
[[3,1,1],[1]]
 => [1]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => 3
[[3,3],[2]]
 => [2]
 => 2
[[4,3],[3]]
 => [3]
 => 3
[[2,2,1],[1]]
 => [1]
 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => 3
[[3,2,1],[2]]
 => [2]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => 4
[[2,2,2],[1,1]]
 => [1,1]
 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => 4
[[3,2,2],[2,1]]
 => [2,1]
 => 3
[[4,3,2],[3,2]]
 => [3,2]
 => 5
[[1,1,1,1],[]]
 => []
 => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 5
[[2,1,1,1],[1]]
 => [1]
 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 4
[[3,2,1,1],[2,1]]
 => [2,1]
 => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 6
[[5],[]]
 => []
 => 0
[[4,1],[]]
 => []
 => 0
[[5,1],[1]]
 => [1]
 => 1
[[3,2],[]]
 => []
 => 0
[[4,2],[1]]
 => [1]
 => 1
[[5,2],[2]]
 => [2]
 => 2
[[3,1,1],[]]
 => []
 => 0
[[4,2,1],[1,1]]
 => [1,1]
 => 2
[[4,1,1],[1]]
 => [1]
 => 1
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St001034
(load all 7 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => 0
[[2],[]]
 => []
 => []
 => 0
[[1,1],[]]
 => []
 => []
 => 0
[[2,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3],[]]
 => []
 => []
 => 0
[[2,1],[]]
 => []
 => []
 => 0
[[3,1],[1]]
 => [1]
 => [1,0]
 => 1
[[2,2],[1]]
 => [1]
 => [1,0]
 => 1
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[1,1,1],[]]
 => []
 => []
 => 0
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[2,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[4],[]]
 => []
 => []
 => 0
[[3,1],[]]
 => []
 => []
 => 0
[[4,1],[1]]
 => [1]
 => [1,0]
 => 1
[[2,2],[]]
 => []
 => []
 => 0
[[3,2],[1]]
 => [1]
 => [1,0]
 => 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,1,1],[]]
 => []
 => []
 => 0
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[3,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2,2,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
[[1,1,1,1],[]]
 => []
 => []
 => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 5
[[2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6
[[5],[]]
 => []
 => []
 => 0
[[4,1],[]]
 => []
 => []
 => 0
[[5,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3,2],[]]
 => []
 => []
 => 0
[[4,2],[1]]
 => [1]
 => [1,0]
 => 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[3,1,1],[]]
 => []
 => []
 => 0
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[4,1,1],[1]]
 => [1]
 => [1,0]
 => 1
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St000018
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => [] => 0
[[2],[]]
 => []
 => []
 => [] => 0
[[1,1],[]]
 => []
 => []
 => [] => 0
[[2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3],[]]
 => []
 => []
 => [] => 0
[[2,1],[]]
 => []
 => []
 => [] => 0
[[3,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[1,1,1],[]]
 => []
 => []
 => [] => 0
[[2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[4],[]]
 => []
 => []
 => [] => 0
[[3,1],[]]
 => []
 => []
 => [] => 0
[[4,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[2,2],[]]
 => []
 => []
 => [] => 0
[[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[4,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[2,1,1],[]]
 => []
 => []
 => [] => 0
[[3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[4,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[3,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 4
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 5
[[1,1,1,1],[]]
 => []
 => []
 => [] => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 5
[[2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 4
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 6
[[5],[]]
 => []
 => []
 => [] => 0
[[4,1],[]]
 => []
 => []
 => [] => 0
[[5,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,2],[]]
 => []
 => []
 => [] => 0
[[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[5,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[3,1,1],[]]
 => []
 => []
 => [] => 0
[[4,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[4,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000246
(load all 2 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => [] => 0
[[2],[]]
 => []
 => []
 => [] => 0
[[1,1],[]]
 => []
 => []
 => [] => 0
[[2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[3],[]]
 => []
 => []
 => [] => 0
[[2,1],[]]
 => []
 => []
 => [] => 0
[[3,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[[1,1,1],[]]
 => []
 => []
 => [] => 0
[[2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[[2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[[4],[]]
 => []
 => []
 => [] => 0
[[3,1],[]]
 => []
 => []
 => [] => 0
[[4,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[2,2],[]]
 => []
 => []
 => [] => 0
[[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[4,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[[2,1,1],[]]
 => []
 => []
 => [] => 0
[[3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[[3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[[4,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[[3,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 4
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 5
[[1,1,1,1],[]]
 => []
 => []
 => [] => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 5
[[2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 4
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 6
[[5],[]]
 => []
 => []
 => [] => 0
[[4,1],[]]
 => []
 => []
 => [] => 0
[[5,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[3,2],[]]
 => []
 => []
 => [] => 0
[[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[[5,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[[3,1,1],[]]
 => []
 => []
 => [] => 0
[[4,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[[4,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
Description
The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000293
(load all 6 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Values
[[1],[]]
 => []
 => => ? = 0
[[2],[]]
 => []
 => => ? = 0
[[1,1],[]]
 => []
 => => ? = 0
[[2,1],[1]]
 => [1]
 => 10 => 1
[[3],[]]
 => []
 => => ? = 0
[[2,1],[]]
 => []
 => => ? = 0
[[3,1],[1]]
 => [1]
 => 10 => 1
[[2,2],[1]]
 => [1]
 => 10 => 1
[[3,2],[2]]
 => [2]
 => 100 => 2
[[1,1,1],[]]
 => []
 => => ? = 0
[[2,2,1],[1,1]]
 => [1,1]
 => 110 => 2
[[2,1,1],[1]]
 => [1]
 => 10 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => 1010 => 3
[[4],[]]
 => []
 => => ? = 0
[[3,1],[]]
 => []
 => => ? = 0
[[4,1],[1]]
 => [1]
 => 10 => 1
[[2,2],[]]
 => []
 => => ? = 0
[[3,2],[1]]
 => [1]
 => 10 => 1
[[4,2],[2]]
 => [2]
 => 100 => 2
[[2,1,1],[]]
 => []
 => => ? = 0
[[3,2,1],[1,1]]
 => [1,1]
 => 110 => 2
[[3,1,1],[1]]
 => [1]
 => 10 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => 1010 => 3
[[3,3],[2]]
 => [2]
 => 100 => 2
[[4,3],[3]]
 => [3]
 => 1000 => 3
[[2,2,1],[1]]
 => [1]
 => 10 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => 1010 => 3
[[3,2,1],[2]]
 => [2]
 => 100 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => 10010 => 4
[[2,2,2],[1,1]]
 => [1,1]
 => 110 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => 1100 => 4
[[3,2,2],[2,1]]
 => [2,1]
 => 1010 => 3
[[4,3,2],[3,2]]
 => [3,2]
 => 10100 => 5
[[1,1,1,1],[]]
 => []
 => => ? = 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => 110 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 11010 => 5
[[2,1,1,1],[1]]
 => [1]
 => 10 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 10110 => 4
[[3,2,1,1],[2,1]]
 => [2,1]
 => 1010 => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 101010 => 6
[[5],[]]
 => []
 => => ? = 0
[[4,1],[]]
 => []
 => => ? = 0
[[5,1],[1]]
 => [1]
 => 10 => 1
[[3,2],[]]
 => []
 => => ? = 0
[[4,2],[1]]
 => [1]
 => 10 => 1
[[5,2],[2]]
 => [2]
 => 100 => 2
[[3,1,1],[]]
 => []
 => => ? = 0
[[4,2,1],[1,1]]
 => [1,1]
 => 110 => 2
[[4,1,1],[1]]
 => [1]
 => 10 => 1
[[5,2,1],[2,1]]
 => [2,1]
 => 1010 => 3
[[3,3],[1]]
 => [1]
 => 10 => 1
[[4,3],[2]]
 => [2]
 => 100 => 2
[[5,3],[3]]
 => [3]
 => 1000 => 3
[[2,2,1],[]]
 => []
 => => ? = 0
[[3,3,1],[1,1]]
 => [1,1]
 => 110 => 2
[[3,2,1],[1]]
 => [1]
 => 10 => 1
[[4,3,1],[2,1]]
 => [2,1]
 => 1010 => 3
[[4,2,1],[2]]
 => [2]
 => 100 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => 10010 => 4
[[3,2,2],[1,1]]
 => [1,1]
 => 110 => 2
[[4,3,2],[2,2]]
 => [2,2]
 => 1100 => 4
[[4,2,2],[2,1]]
 => [2,1]
 => 1010 => 3
[[5,3,2],[3,2]]
 => [3,2]
 => 10100 => 5
[[2,1,1,1],[]]
 => []
 => => ? = 0
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 3
[[3,2,1,1],[1,1]]
 => [1,1]
 => 110 => 2
[[1,1,1,1,1],[]]
 => []
 => => ? = 0
Description
The number of inversions of a binary word.
Matching statistic: St000290
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Values
[[1],[]]
 => []
 => => ? => ? = 0
[[2],[]]
 => []
 => => ? => ? = 0
[[1,1],[]]
 => []
 => => ? => ? = 0
[[2,1],[1]]
 => [1]
 => 10 => 10 => 1
[[3],[]]
 => []
 => => ? => ? = 0
[[2,1],[]]
 => []
 => => ? => ? = 0
[[3,1],[1]]
 => [1]
 => 10 => 10 => 1
[[2,2],[1]]
 => [1]
 => 10 => 10 => 1
[[3,2],[2]]
 => [2]
 => 100 => 010 => 2
[[1,1,1],[]]
 => []
 => => ? => ? = 0
[[2,2,1],[1,1]]
 => [1,1]
 => 110 => 110 => 2
[[2,1,1],[1]]
 => [1]
 => 10 => 10 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => 1010 => 0110 => 3
[[4],[]]
 => []
 => => ? => ? = 0
[[3,1],[]]
 => []
 => => ? => ? = 0
[[4,1],[1]]
 => [1]
 => 10 => 10 => 1
[[2,2],[]]
 => []
 => => ? => ? = 0
[[3,2],[1]]
 => [1]
 => 10 => 10 => 1
[[4,2],[2]]
 => [2]
 => 100 => 010 => 2
[[2,1,1],[]]
 => []
 => => ? => ? = 0
[[3,2,1],[1,1]]
 => [1,1]
 => 110 => 110 => 2
[[3,1,1],[1]]
 => [1]
 => 10 => 10 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => 1010 => 0110 => 3
[[3,3],[2]]
 => [2]
 => 100 => 010 => 2
[[4,3],[3]]
 => [3]
 => 1000 => 0010 => 3
[[2,2,1],[1]]
 => [1]
 => 10 => 10 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => 1010 => 0110 => 3
[[3,2,1],[2]]
 => [2]
 => 100 => 010 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => 10010 => 00110 => 4
[[2,2,2],[1,1]]
 => [1,1]
 => 110 => 110 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => 1100 => 1010 => 4
[[3,2,2],[2,1]]
 => [2,1]
 => 1010 => 0110 => 3
[[4,3,2],[3,2]]
 => [3,2]
 => 10100 => 10010 => 5
[[1,1,1,1],[]]
 => []
 => => ? => ? = 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 1110 => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => 110 => 110 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 11010 => 10110 => 5
[[2,1,1,1],[1]]
 => [1]
 => 10 => 10 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 10110 => 01110 => 4
[[3,2,1,1],[2,1]]
 => [2,1]
 => 1010 => 0110 => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 101010 => 100110 => 6
[[5],[]]
 => []
 => => ? => ? = 0
[[4,1],[]]
 => []
 => => ? => ? = 0
[[5,1],[1]]
 => [1]
 => 10 => 10 => 1
[[3,2],[]]
 => []
 => => ? => ? = 0
[[4,2],[1]]
 => [1]
 => 10 => 10 => 1
[[5,2],[2]]
 => [2]
 => 100 => 010 => 2
[[3,1,1],[]]
 => []
 => => ? => ? = 0
[[4,2,1],[1,1]]
 => [1,1]
 => 110 => 110 => 2
[[4,1,1],[1]]
 => [1]
 => 10 => 10 => 1
[[5,2,1],[2,1]]
 => [2,1]
 => 1010 => 0110 => 3
[[3,3],[1]]
 => [1]
 => 10 => 10 => 1
[[4,3],[2]]
 => [2]
 => 100 => 010 => 2
[[5,3],[3]]
 => [3]
 => 1000 => 0010 => 3
[[2,2,1],[]]
 => []
 => => ? => ? = 0
[[3,3,1],[1,1]]
 => [1,1]
 => 110 => 110 => 2
[[3,2,1],[1]]
 => [1]
 => 10 => 10 => 1
[[4,3,1],[2,1]]
 => [2,1]
 => 1010 => 0110 => 3
[[4,2,1],[2]]
 => [2]
 => 100 => 010 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => 10010 => 00110 => 4
[[3,2,2],[1,1]]
 => [1,1]
 => 110 => 110 => 2
[[4,3,2],[2,2]]
 => [2,2]
 => 1100 => 1010 => 4
[[4,2,2],[2,1]]
 => [2,1]
 => 1010 => 0110 => 3
[[5,3,2],[3,2]]
 => [3,2]
 => 10100 => 10010 => 5
[[2,1,1,1],[]]
 => []
 => => ? => ? = 0
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 1110 => 3
[[3,2,1,1],[1,1]]
 => [1,1]
 => 110 => 110 => 2
[[1,1,1,1,1],[]]
 => []
 => => ? => ? = 0
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000395
(load all 2 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Values
[[1],[]]
 => []
 => []
 => []
 => ? = 0
[[2],[]]
 => []
 => []
 => []
 => ? = 0
[[1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[2,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[3],[]]
 => []
 => []
 => []
 => ? = 0
[[2,1],[]]
 => []
 => []
 => []
 => ? = 0
[[3,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[2,2],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[1,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[2,1,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[[4],[]]
 => []
 => []
 => []
 => ? = 0
[[3,1],[]]
 => []
 => []
 => []
 => ? = 0
[[4,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[2,2],[]]
 => []
 => []
 => []
 => ? = 0
[[3,2],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[2,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[3,1,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[2,2,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 4
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 5
[[1,1,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 5
[[2,1,1,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 6
[[5],[]]
 => []
 => []
 => []
 => ? = 0
[[4,1],[]]
 => []
 => []
 => []
 => ? = 0
[[5,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[3,2],[]]
 => []
 => []
 => []
 => ? = 0
[[4,2],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[3,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[4,1,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[[3,3],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[2,2,1],[]]
 => []
 => []
 => []
 => ? = 0
[[3,3,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[3,2,1],[1]]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4
[[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[4,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 4
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 5
[[2,1,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3
[[3,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[1,1,1,1,1],[]]
 => []
 => []
 => []
 => ? = 0
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000719
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000719: Perfect matchings ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Values
[[1],[]]
 => []
 => []
 => []
 => ? = 0
[[2],[]]
 => []
 => []
 => []
 => ? = 0
[[1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[3],[]]
 => []
 => []
 => []
 => ? = 0
[[2,1],[]]
 => []
 => []
 => []
 => ? = 0
[[3,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[[1,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[[2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3
[[4],[]]
 => []
 => []
 => []
 => ? = 0
[[3,1],[]]
 => []
 => []
 => []
 => ? = 0
[[4,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[2,2],[]]
 => []
 => []
 => []
 => ? = 0
[[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[4,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[[2,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[[3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3
[[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[[4,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3
[[3,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 4
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 5
[[1,1,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 5
[[2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 4
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 6
[[5],[]]
 => []
 => []
 => []
 => ? = 0
[[4,1],[]]
 => []
 => []
 => []
 => ? = 0
[[5,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[3,2],[]]
 => []
 => []
 => []
 => ? = 0
[[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[5,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[[3,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[4,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[[4,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3
[[3,3],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[[5,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[[2,2,1],[]]
 => []
 => []
 => []
 => ? = 0
[[3,3,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3
[[4,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 4
[[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[[4,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 4
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3
[[5,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 5
[[2,1,1,1],[]]
 => []
 => []
 => []
 => ? = 0
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 3
[[3,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[[1,1,1,1,1],[]]
 => []
 => []
 => []
 => ? = 0
Description
The number of alignments in a perfect matching. An alignment is a pair of edges $(i,j)$, $(k,l)$ such that $i < j < k < l$. Since any two edges in a perfect matching are either nesting ([[St000041]]), crossing ([[St000042]]) or form an alignment, the sum of these numbers in a perfect matching with $n$ edges is $\binom{n}{2}$.
Matching statistic: St001759
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Values
[[1],[]]
 => []
 => []
 => [] => ? = 0
[[2],[]]
 => []
 => []
 => [] => ? = 0
[[1,1],[]]
 => []
 => []
 => [] => ? = 0
[[2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3],[]]
 => []
 => []
 => [] => ? = 0
[[2,1],[]]
 => []
 => []
 => [] => ? = 0
[[3,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[1,1,1],[]]
 => []
 => []
 => [] => ? = 0
[[2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[4],[]]
 => []
 => []
 => [] => ? = 0
[[3,1],[]]
 => []
 => []
 => [] => ? = 0
[[4,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[2,2],[]]
 => []
 => []
 => [] => ? = 0
[[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[4,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[2,1,1],[]]
 => []
 => []
 => [] => ? = 0
[[3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[4,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[3,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 4
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 5
[[1,1,1,1],[]]
 => []
 => []
 => [] => ? = 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 5
[[2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 4
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 6
[[5],[]]
 => []
 => []
 => [] => ? = 0
[[4,1],[]]
 => []
 => []
 => [] => ? = 0
[[5,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[3,2],[]]
 => []
 => []
 => [] => ? = 0
[[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[5,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[3,1,1],[]]
 => []
 => []
 => [] => ? = 0
[[4,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[4,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[3,3],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[5,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[[2,2,1],[]]
 => []
 => []
 => [] => ? = 0
[[3,3,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[4,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 4
[[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[4,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 4
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[[5,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 5
[[2,1,1,1],[]]
 => []
 => []
 => [] => ? = 0
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[[3,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[1,1,1,1,1],[]]
 => []
 => []
 => [] => ? = 0
Description
The Rajchgot index of a permutation. The '''Rajchgot index''' of a permutation $\sigma$ is the degree of the ''Grothendieck polynomial'' of $\sigma$. This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index [[St000004]] of the permutations smaller than or equal to $\sigma$ in the right ''weak Bruhat order''.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000189The number of elements in the poset. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.