Your data matches 70 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 13 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 9 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
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => 10 => 01 => 0
[[2],[]]
 => [2]
 => 100 => 011 => 0
[[1,1],[]]
 => [1,1]
 => 110 => 001 => 0
[[2,1],[1]]
 => [2,1]
 => 1010 => 0101 => 1
[[3],[]]
 => [3]
 => 1000 => 0111 => 0
[[2,1],[]]
 => [2,1]
 => 1010 => 0101 => 1
[[3,1],[1]]
 => [3,1]
 => 10010 => 01101 => 2
[[2,2],[1]]
 => [2,2]
 => 1100 => 0011 => 0
[[3,2],[2]]
 => [3,2]
 => 10100 => 01011 => 1
[[1,1,1],[]]
 => [1,1,1]
 => 1110 => 0001 => 0
[[2,2,1],[1,1]]
 => [2,2,1]
 => 11010 => 00101 => 1
[[2,1,1],[1]]
 => [2,1,1]
 => 10110 => 01001 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => 101010 => 010101 => 3
[[4],[]]
 => [4]
 => 10000 => 01111 => 0
[[3,1],[]]
 => [3,1]
 => 10010 => 01101 => 2
[[4,1],[1]]
 => [4,1]
 => 100010 => 011101 => 3
[[2,2],[]]
 => [2,2]
 => 1100 => 0011 => 0
[[3,2],[1]]
 => [3,2]
 => 10100 => 01011 => 1
[[4,2],[2]]
 => [4,2]
 => 100100 => 011011 => 2
[[2,1,1],[]]
 => [2,1,1]
 => 10110 => 01001 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => 101010 => 010101 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => 100110 => 011001 => 4
[[4,2,1],[2,1]]
 => [4,2,1]
 => 1001010 => 0110101 => 5
[[3,3],[2]]
 => [3,3]
 => 11000 => 00111 => 0
[[4,3],[3]]
 => [4,3]
 => 101000 => 010111 => 1
[[2,2,1],[1]]
 => [2,2,1]
 => 11010 => 00101 => 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => 110010 => 001101 => 2
[[3,2,1],[2]]
 => [3,2,1]
 => 101010 => 010101 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => 1010010 => 0101101 => 4
[[2,2,2],[1,1]]
 => [2,2,2]
 => 11100 => 00011 => 0
[[3,3,2],[2,2]]
 => [3,3,2]
 => 110100 => 001011 => 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => 101100 => 010011 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => 1010100 => 0101011 => 3
[[1,1,1,1],[]]
 => [1,1,1,1]
 => 11110 => 00001 => 0
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => 111010 => 000101 => 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => 110110 => 001001 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => 1101010 => 0010101 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => 101110 => 010001 => 3
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => 1011010 => 0100101 => 4
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => 1010110 => 0101001 => 5
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => 10101010 => 01010101 => 6
[[5],[]]
 => [5]
 => 100000 => 011111 => 0
[[4,1],[]]
 => [4,1]
 => 100010 => 011101 => 3
[[5,1],[1]]
 => [5,1]
 => 1000010 => 0111101 => 4
[[3,2],[]]
 => [3,2]
 => 10100 => 01011 => 1
[[4,2],[1]]
 => [4,2]
 => 100100 => 011011 => 2
[[5,2],[2]]
 => [5,2]
 => 1000100 => 0111011 => 3
[[3,1,1],[]]
 => [3,1,1]
 => 100110 => 011001 => 4
[[4,2,1],[1,1]]
 => [4,2,1]
 => 1001010 => 0110101 => 5
[[4,1,1],[1]]
 => [4,1,1]
 => 1000110 => 0111001 => 6
Description
The number of inversions of a binary word.
Matching statistic: St000394
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => [1,0]
 => 0
[[2],[]]
 => []
 => []
 => [1,0]
 => 0
[[1,1],[]]
 => []
 => []
 => [1,0]
 => 0
[[2,1],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[3],[]]
 => []
 => []
 => [1,0]
 => 0
[[2,1],[]]
 => []
 => []
 => [1,0]
 => 0
[[3,1],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[2,2],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[1,1,1],[]]
 => []
 => []
 => [1,0]
 => 0
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[[2,1,1],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[[4],[]]
 => []
 => []
 => [1,0]
 => 0
[[3,1],[]]
 => []
 => []
 => [1,0]
 => 0
[[4,1],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[2,2],[]]
 => []
 => []
 => [1,0]
 => 0
[[3,2],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[2,1,1],[]]
 => []
 => []
 => [1,0]
 => 0
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[[3,1,1],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[2,2,1],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[[1,1,1,1],[]]
 => []
 => []
 => [1,0]
 => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5
[[2,1,1,1],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 5
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 6
[[5],[]]
 => []
 => []
 => [1,0]
 => 0
[[4,1],[]]
 => []
 => []
 => [1,0]
 => 0
[[5,1],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[3,2],[]]
 => []
 => []
 => [1,0]
 => 0
[[4,2],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[3,1,1],[]]
 => []
 => []
 => [1,0]
 => 0
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[[4,1,1],[1]]
 => [1]
 => [1,0]
 => [1,1,0,0]
 => 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000029
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 95%●distinct values known / distinct values provided: 82%
Values
[[1],[]]
 => []
 => []
 => [1] => 0
[[2],[]]
 => []
 => []
 => [1] => 0
[[1,1],[]]
 => []
 => []
 => [1] => 0
[[2,1],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[3],[]]
 => []
 => []
 => [1] => 0
[[2,1],[]]
 => []
 => []
 => [1] => 0
[[3,1],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[2,2],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[[1,1,1],[]]
 => []
 => []
 => [1] => 0
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[[2,1,1],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[[4],[]]
 => []
 => []
 => [1] => 0
[[3,1],[]]
 => []
 => []
 => [1] => 0
[[4,1],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[2,2],[]]
 => []
 => []
 => [1] => 0
[[3,2],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[[2,1,1],[]]
 => []
 => []
 => [1] => 0
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[[3,1,1],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[[2,2,1],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 3
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 4
[[1,1,1,1],[]]
 => []
 => []
 => [1] => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 4
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 5
[[2,1,1,1],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 5
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => 6
[[5],[]]
 => []
 => []
 => [1] => 0
[[4,1],[]]
 => []
 => []
 => [1] => 0
[[5,1],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[3,2],[]]
 => []
 => []
 => [1] => 0
[[4,2],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[[3,1,1],[]]
 => []
 => []
 => [1] => 0
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[[4,1,1],[1]]
 => [1]
 => [1,0]
 => [2,1] => 1
[[5,4,2,1],[4,2,1]]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => ? ∊ {7,7,8,8,9,9,10}
[[5,4,3,1],[4,3,1]]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => ? ∊ {7,7,8,8,9,9,10}
[[5,4,3,2],[4,3,2]]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? ∊ {7,7,8,8,9,9,10}
[[4,4,3,2,1],[3,3,2,1]]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [7,3,4,1,6,2,5] => ? ∊ {7,7,8,8,9,9,10}
[[4,3,3,2,1],[3,2,2,1]]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? ∊ {7,7,8,8,9,9,10}
[[4,3,2,2,1],[3,2,1,1]]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => ? ∊ {7,7,8,8,9,9,10}
[[5,4,3,2,1],[4,3,2,1]]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [8,1,4,5,2,7,3,6] => ? ∊ {7,7,8,8,9,9,10}
Description
The depth of a permutation. This is given by $$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$. Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Matching statistic: St000395
(load all 5 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Values
[[1],[]]
 => []
 => []
 => ? = 0
[[2],[]]
 => []
 => []
 => ? ∊ {0,0}
[[1,1],[]]
 => []
 => []
 => ? ∊ {0,0}
[[2,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3],[]]
 => []
 => []
 => ? ∊ {0,0,0}
[[2,1],[]]
 => []
 => []
 => ? ∊ {0,0,0}
[[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],[]]
 => []
 => []
 => ? ∊ {0,0,0}
[[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],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0}
[[3,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0}
[[4,1],[1]]
 => [1]
 => [1,0]
 => 1
[[2,2],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0}
[[3,2],[1]]
 => [1]
 => [1,0]
 => 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0}
[[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]
 => 3
[[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]
 => 4
[[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0}
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 4
[[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]
 => 5
[[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],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[4,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[5,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3,2],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[4,2],[1]]
 => [1]
 => [1,0]
 => 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[3,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[4,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[3,3],[1]]
 => [1]
 => [1,0]
 => 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2,2,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[3,3,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[3,2,1],[1]]
 => [1]
 => [1,0]
 => 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
[[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[4,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 3
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 4
[[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 4
[[3,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0}
Description
The sum of the heights of the peaks of a Dyck path.
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],[]]
 => []
 => => ? => ? ∊ {0,0}
[[1,1],[]]
 => []
 => => ? => ? ∊ {0,0}
[[2,1],[1]]
 => [1]
 => 10 => 10 => 1
[[3],[]]
 => []
 => => ? => ? ∊ {0,0,0}
[[2,1],[]]
 => []
 => => ? => ? ∊ {0,0,0}
[[3,1],[1]]
 => [1]
 => 10 => 10 => 1
[[2,2],[1]]
 => [1]
 => 10 => 10 => 1
[[3,2],[2]]
 => [2]
 => 100 => 010 => 2
[[1,1,1],[]]
 => []
 => => ? => ? ∊ {0,0,0}
[[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],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0}
[[3,1],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0}
[[4,1],[1]]
 => [1]
 => 10 => 10 => 1
[[2,2],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0}
[[3,2],[1]]
 => [1]
 => 10 => 10 => 1
[[4,2],[2]]
 => [2]
 => 100 => 010 => 2
[[2,1,1],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0}
[[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],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0}
[[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],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0,0,0}
[[4,1],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0,0,0}
[[5,1],[1]]
 => [1]
 => 10 => 10 => 1
[[3,2],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0,0,0}
[[4,2],[1]]
 => [1]
 => 10 => 10 => 1
[[5,2],[2]]
 => [2]
 => 100 => 010 => 2
[[3,1,1],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0,0,0}
[[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],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0,0,0}
[[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],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0,0,0}
[[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],[]]
 => []
 => => ? => ? ∊ {0,0,0,0,0,0,0}
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 $\binom{n}{a}_q$.
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000719The number of alignments in a perfect matching. St001759The Rajchgot index of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000222The number of alignments in the permutation. 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. St000496The rcs statistic of a set partition. St001781The interlacing number of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000438The position of the last up step in a Dyck path. St000355The number of occurrences of the pattern 21-3. St000456The monochromatic index of a connected graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000454The largest eigenvalue of a graph if it is integral. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001875The number of simple modules with projective dimension at most 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000077The number of boxed and circled entries. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001330The hat guessing number of a graph. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.