Your data matches 275 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
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
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St001034
(load all 10 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
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
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: 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: St000246
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
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
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
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000222
(load all 2 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => [1,0]
 => [1] => 0
[[2],[]]
 => [2]
 => [1,0,1,0]
 => [2,1] => 0
[[1,1],[]]
 => [1,1]
 => [1,1,0,0]
 => [1,2] => 0
[[2,1],[1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 1
[[3],[]]
 => [3]
 => [1,0,1,0,1,0]
 => [2,3,1] => 0
[[2,1],[]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 1
[[3,1],[1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[[2,2],[1]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[[3,2],[2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 2
[[1,1,1],[]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [3,1,2] => 0
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 2
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 3
[[4],[]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 0
[[3,1],[]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[[4,1],[1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 1
[[2,2],[]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[[3,2],[1]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 2
[[4,2],[2]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 2
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 3
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 2
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,6,4,5] => 5
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 0
[[4,3],[3]]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 2
[[2,2,1],[1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 2
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => 1
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,5,1,6,3,4] => 3
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => 3
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 3
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => 5
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 0
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => 4
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,2,6,3,5] => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,1,3,6,4,5] => 6
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,1,5,6,3,4] => 4
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [2,5,1,3,7,4,6] => ? = 4
Description
The number of alignments in the permutation.
Matching statistic: St001384
(load all 3 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => [1,0]
 => []
 => ? = 0
[[2],[]]
 => [2]
 => [1,0,1,0]
 => [1]
 => 0
[[1,1],[]]
 => [1,1]
 => [1,1,0,0]
 => []
 => ? = 0
[[2,1],[1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[[3],[]]
 => [3]
 => [1,0,1,0,1,0]
 => [2,1]
 => 0
[[2,1],[]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[[3,1],[1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[[2,2],[1]]
 => [2,2]
 => [1,1,1,0,0,0]
 => []
 => ? = 0
[[3,2],[2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 2
[[1,1,1],[]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1]
 => 0
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[[4],[]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[[3,1],[]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[[4,1],[1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 3
[[2,2],[]]
 => [2,2]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,0}
[[3,2],[1]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 2
[[4,2],[2]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 4
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 2
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => 5
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[[4,3],[3]]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 2
[[2,2,1],[1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 1
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1,1]
 => 3
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0}
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 3
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => 4
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 0
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 2
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1,1]
 => 5
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1,1]
 => 4
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,2,1,1,1]
 => 6
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
The following 265 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000189The number of elements in the poset. St000197The number of entries equal to positive one in the alternating sign matrix. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000290The major index of a binary word. St000494The number of inversions of distance at most 3 of a permutation. St000631The number of distinct palindromic decompositions of a binary word. St000719The number of alignments in a perfect matching. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001485The modular major index of a binary word. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001759The Rajchgot index of a permutation. St000355The number of occurrences of the pattern 21-3. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000668The least common multiple of the parts of the partition. St000693The modular (standard) major index of a standard tableau. St000708The product of the parts of an integer partition. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000216The absolute length of a permutation. St000248The number of anti-singletons of a set partition. St000461The rix statistic of a permutation. St000472The sum of the ascent bottoms of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000493The los statistic of a set partition. St000495The number of inversions of distance at most 2 of a permutation. St000503The maximal difference between two elements in a common block. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000653The last descent of a permutation. St000673The number of non-fixed points of a permutation. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000730The maximal arc length of a set partition. St000792The Grundy value for the game of ruler on a binary word. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000933The number of multipartitions of sizes given by an integer partition. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001077The prefix exchange distance of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001500The global dimension of magnitude 1 Nakayama algebras. St001556The number of inversions of the third entry of a permutation. St001721The degree of a binary word. St001811The Castelnuovo-Mumford regularity of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001498The normalised height of a Nakayama algebra with magnitude 1. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. 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. St000077The number of boxed and circled entries. St000456The monochromatic index of a connected graph. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000045The number of linear extensions of a binary tree. St000219The number of occurrences of the pattern 231 in a permutation. St000699The toughness times the least common multiple of 1,. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000567The sum of the products of all pairs of parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000707The product of the factorials of the parts. St000929The constant term of the character polynomial of 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. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St001378The product of the cohook lengths of the integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001330The hat guessing number of a graph. St000478Another weight of a partition according to Alladi. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000706The product of the factorials of the multiplicities of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000934The 2-degree of an integer partition. 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. St001568The smallest positive integer that does not appear twice in the partition. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000420The number of Dyck paths that are weakly above a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000438The position of the last up step in a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000976The sum of the positions of double up-steps of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. 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. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001959The product of the heights of the peaks of a Dyck path. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001060The distinguishing index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000944The 3-degree of an integer partition. St000981The length of the longest zigzag subpath. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001541The Gini index of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000264The girth of a graph, which is not a tree. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.