Your data matches 91 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001435
(load all 2 compositions to match this statistic)
Mapping
St001435: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => 0 = 1 - 1
[[2],[]]
 => 0 = 1 - 1
[[1,1],[]]
 => 0 = 1 - 1
[[2,1],[1]]
 => 1 = 2 - 1
[[3],[]]
 => 0 = 1 - 1
[[2,1],[]]
 => 0 = 1 - 1
[[3,1],[1]]
 => 1 = 2 - 1
[[2,2],[1]]
 => 1 = 2 - 1
[[3,2],[2]]
 => 2 = 3 - 1
[[1,1,1],[]]
 => 0 = 1 - 1
[[2,2,1],[1,1]]
 => 1 = 2 - 1
[[2,1,1],[1]]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => 2 = 3 - 1
[[4],[]]
 => 0 = 1 - 1
[[3,1],[]]
 => 0 = 1 - 1
[[4,1],[1]]
 => 1 = 2 - 1
[[2,2],[]]
 => 0 = 1 - 1
[[3,2],[1]]
 => 1 = 2 - 1
[[4,2],[2]]
 => 2 = 3 - 1
[[2,1,1],[]]
 => 0 = 1 - 1
[[3,2,1],[1,1]]
 => 1 = 2 - 1
[[3,1,1],[1]]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => 2 = 3 - 1
[[3,3],[2]]
 => 2 = 3 - 1
[[4,3],[3]]
 => 3 = 4 - 1
[[2,2,1],[1]]
 => 1 = 2 - 1
[[3,3,1],[2,1]]
 => 2 = 3 - 1
[[3,2,1],[2]]
 => 2 = 3 - 1
[[4,3,1],[3,1]]
 => 3 = 4 - 1
[[2,2,2],[1,1]]
 => 1 = 2 - 1
[[3,3,2],[2,2]]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => 2 = 3 - 1
[[4,3,2],[3,2]]
 => 3 = 4 - 1
[[1,1,1,1],[]]
 => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => 1 = 2 - 1
[[2,2,1,1],[1,1]]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => 2 = 3 - 1
[[2,1,1,1],[1]]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => 3 = 4 - 1
[[5],[]]
 => 0 = 1 - 1
[[4,1],[]]
 => 0 = 1 - 1
[[5,1],[1]]
 => 1 = 2 - 1
[[3,2],[]]
 => 0 = 1 - 1
[[4,2],[1]]
 => 1 = 2 - 1
[[5,2],[2]]
 => 2 = 3 - 1
[[3,1,1],[]]
 => 0 = 1 - 1
[[4,2,1],[1,1]]
 => 1 = 2 - 1
[[4,1,1],[1]]
 => 1 = 2 - 1
Description
The number of missing boxes in the first row.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => 0 = 1 - 1
[[2],[]]
 => []
 => 0 = 1 - 1
[[1,1],[]]
 => []
 => 0 = 1 - 1
[[2,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3],[]]
 => []
 => 0 = 1 - 1
[[2,1],[]]
 => []
 => 0 = 1 - 1
[[3,1],[1]]
 => [1]
 => 1 = 2 - 1
[[2,2],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2],[2]]
 => [2]
 => 1 = 2 - 1
[[1,1,1],[]]
 => []
 => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
[[2,1,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[4],[]]
 => []
 => 0 = 1 - 1
[[3,1],[]]
 => []
 => 0 = 1 - 1
[[4,1],[1]]
 => [1]
 => 1 = 2 - 1
[[2,2],[]]
 => []
 => 0 = 1 - 1
[[3,2],[1]]
 => [1]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => 1 = 2 - 1
[[2,1,1],[]]
 => []
 => 0 = 1 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
[[3,1,1],[1]]
 => [1]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[3,3],[2]]
 => [2]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => 1 = 2 - 1
[[2,2,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[3,2,1],[2]]
 => [2]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => 2 = 3 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => 2 = 3 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => 2 = 3 - 1
[[1,1,1,1],[]]
 => []
 => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 3 = 4 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 3 = 4 - 1
[[2,1,1,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 3 = 4 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 3 = 4 - 1
[[5],[]]
 => []
 => 0 = 1 - 1
[[4,1],[]]
 => []
 => 0 = 1 - 1
[[5,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2],[]]
 => []
 => 0 = 1 - 1
[[4,2],[1]]
 => [1]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => 1 = 2 - 1
[[3,1,1],[]]
 => []
 => 0 = 1 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
[[4,1,1],[1]]
 => [1]
 => 1 = 2 - 1
Description
The length of the partition.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => 0 = 1 - 1
[[2],[]]
 => []
 => 0 = 1 - 1
[[1,1],[]]
 => []
 => 0 = 1 - 1
[[2,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3],[]]
 => []
 => 0 = 1 - 1
[[2,1],[]]
 => []
 => 0 = 1 - 1
[[3,1],[1]]
 => [1]
 => 1 = 2 - 1
[[2,2],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2],[2]]
 => [2]
 => 2 = 3 - 1
[[1,1,1],[]]
 => []
 => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[2,1,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[4],[]]
 => []
 => 0 = 1 - 1
[[3,1],[]]
 => []
 => 0 = 1 - 1
[[4,1],[1]]
 => [1]
 => 1 = 2 - 1
[[2,2],[]]
 => []
 => 0 = 1 - 1
[[3,2],[1]]
 => [1]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => 2 = 3 - 1
[[2,1,1],[]]
 => []
 => 0 = 1 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[3,1,1],[1]]
 => [1]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[3,3],[2]]
 => [2]
 => 2 = 3 - 1
[[4,3],[3]]
 => [3]
 => 3 = 4 - 1
[[2,2,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[3,2,1],[2]]
 => [2]
 => 2 = 3 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => 3 = 4 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => 3 = 4 - 1
[[1,1,1,1],[]]
 => []
 => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1 = 2 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 2 = 3 - 1
[[2,1,1,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 3 = 4 - 1
[[5],[]]
 => []
 => 0 = 1 - 1
[[4,1],[]]
 => []
 => 0 = 1 - 1
[[5,1],[1]]
 => [1]
 => 1 = 2 - 1
[[3,2],[]]
 => []
 => 0 = 1 - 1
[[4,2],[1]]
 => [1]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => 2 = 3 - 1
[[3,1,1],[]]
 => []
 => 0 = 1 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => 1 = 2 - 1
[[4,1,1],[1]]
 => [1]
 => 1 = 2 - 1
Description
The largest part of an integer partition.
Matching statistic: St000225
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => 0 = 1 - 1
[[2],[]]
 => [2]
 => 0 = 1 - 1
[[1,1],[]]
 => [1,1]
 => 0 = 1 - 1
[[2,1],[1]]
 => [2,1]
 => 1 = 2 - 1
[[3],[]]
 => [3]
 => 0 = 1 - 1
[[2,1],[]]
 => [2,1]
 => 1 = 2 - 1
[[3,1],[1]]
 => [3,1]
 => 2 = 3 - 1
[[2,2],[1]]
 => [2,2]
 => 0 = 1 - 1
[[3,2],[2]]
 => [3,2]
 => 1 = 2 - 1
[[1,1,1],[]]
 => [1,1,1]
 => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => 1 = 2 - 1
[[2,1,1],[1]]
 => [2,1,1]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [3,2,1]
 => 2 = 3 - 1
[[4],[]]
 => [4]
 => 0 = 1 - 1
[[3,1],[]]
 => [3,1]
 => 2 = 3 - 1
[[4,1],[1]]
 => [4,1]
 => 3 = 4 - 1
[[2,2],[]]
 => [2,2]
 => 0 = 1 - 1
[[3,2],[1]]
 => [3,2]
 => 1 = 2 - 1
[[4,2],[2]]
 => [4,2]
 => 2 = 3 - 1
[[2,1,1],[]]
 => [2,1,1]
 => 1 = 2 - 1
[[3,2,1],[1,1]]
 => [3,2,1]
 => 2 = 3 - 1
[[3,1,1],[1]]
 => [3,1,1]
 => 2 = 3 - 1
[[4,2,1],[2,1]]
 => [4,2,1]
 => 3 = 4 - 1
[[3,3],[2]]
 => [3,3]
 => 0 = 1 - 1
[[4,3],[3]]
 => [4,3]
 => 1 = 2 - 1
[[2,2,1],[1]]
 => [2,2,1]
 => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => 2 = 3 - 1
[[3,2,1],[2]]
 => [3,2,1]
 => 2 = 3 - 1
[[4,3,1],[3,1]]
 => [4,3,1]
 => 3 = 4 - 1
[[2,2,2],[1,1]]
 => [2,2,2]
 => 0 = 1 - 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => 1 = 2 - 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => 1 = 2 - 1
[[4,3,2],[3,2]]
 => [4,3,2]
 => 2 = 3 - 1
[[1,1,1,1],[]]
 => [1,1,1,1]
 => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => 1 = 2 - 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => 2 = 3 - 1
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => 3 = 4 - 1
[[5],[]]
 => [5]
 => 0 = 1 - 1
[[4,1],[]]
 => [4,1]
 => 3 = 4 - 1
[[5,1],[1]]
 => [5,1]
 => 4 = 5 - 1
[[3,2],[]]
 => [3,2]
 => 1 = 2 - 1
[[4,2],[1]]
 => [4,2]
 => 2 = 3 - 1
[[5,2],[2]]
 => [5,2]
 => 3 = 4 - 1
[[3,1,1],[]]
 => [3,1,1]
 => 2 = 3 - 1
[[4,2,1],[1,1]]
 => [4,2,1]
 => 3 = 4 - 1
[[4,1,1],[1]]
 => [4,1,1]
 => 3 = 4 - 1
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000013
(load all 5 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2],[]]
 => []
 => []
 => 0 = 1 - 1
[[1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[2,2],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[1,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2 = 3 - 1
[[2,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[4],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[2,2],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,2],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[2,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2 = 3 - 1
[[3,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[[2,2,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2 = 3 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[[1,1,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2 = 3 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[[2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[[5],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[5,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,2],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,2],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[3,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2 = 3 - 1
[[4,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000378
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2],[]]
 => []
 => []
 => 0 = 1 - 1
[[1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,1],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[3],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,1],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[2,2],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[3,2],[2]]
 => [2]
 => [1,1]
 => 1 = 2 - 1
[[1,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2 = 3 - 1
[[2,1,1],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [3]
 => 2 = 3 - 1
[[4],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,1],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[2,2],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,2],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,1]
 => 1 = 2 - 1
[[2,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2 = 3 - 1
[[3,1,1],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [3]
 => 2 = 3 - 1
[[3,3],[2]]
 => [2]
 => [1,1]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
[[2,2,1],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [3]
 => 2 = 3 - 1
[[3,2,1],[2]]
 => [2]
 => [1,1]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => [2]
 => 2 = 3 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => [4]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [3]
 => 2 = 3 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [5]
 => 2 = 3 - 1
[[1,1,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [2,1]
 => 3 = 4 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => [2]
 => 2 = 3 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,2,1]
 => 3 = 4 - 1
[[2,1,1,1],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [2,2]
 => 3 = 4 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [3]
 => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [5,1]
 => 3 = 4 - 1
[[5],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[5,1],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[3,2],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,2],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,1]
 => 1 = 2 - 1
[[3,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2 = 3 - 1
[[4,1,1],[1]]
 => [1]
 => [1]
 => 1 = 2 - 1
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000676
(load all 7 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2],[]]
 => []
 => []
 => 0 = 1 - 1
[[1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[2,2],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[[1,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[[2,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[4],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[2,2],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,2],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[[2,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[[3,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[[2,2,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[[1,1,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[[5],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[5,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[3,2],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,2],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[[3,1,1],[]]
 => []
 => []
 => 0 = 1 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[[4,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 2 - 1
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000740
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => [1] => 1
[[2],[]]
 => []
 => []
 => [1] => 1
[[1,1],[]]
 => []
 => []
 => [1] => 1
[[2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[3],[]]
 => []
 => []
 => [1] => 1
[[2,1],[]]
 => []
 => []
 => [1] => 1
[[3,1],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3
[[1,1,1],[]]
 => []
 => []
 => [1] => 1
[[2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[[2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[[4],[]]
 => []
 => []
 => [1] => 1
[[3,1],[]]
 => []
 => []
 => [1] => 1
[[4,1],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[2,2],[]]
 => []
 => []
 => [1] => 1
[[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[4,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3
[[2,1,1],[]]
 => []
 => []
 => [1] => 1
[[3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[[3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3
[[4,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 4
[[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[[3,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3
[[4,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 3
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 4
[[1,1,1,1],[]]
 => []
 => []
 => [1] => 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 2
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[[2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[[5],[]]
 => []
 => []
 => [1] => 1
[[4,1],[]]
 => []
 => []
 => [1] => 1
[[5,1],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[3,2],[]]
 => []
 => []
 => [1] => 1
[[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
[[5,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3
[[3,1,1],[]]
 => []
 => []
 => [1] => 1
[[4,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[[4,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 2
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Matching statistic: St001291
(load all 5 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => [1,0]
 => 1
[[2],[]]
 => []
 => []
 => [1,0]
 => 1
[[1,1],[]]
 => []
 => []
 => [1,0]
 => 1
[[2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[3],[]]
 => []
 => []
 => [1,0]
 => 1
[[2,1],[]]
 => []
 => []
 => [1,0]
 => 1
[[3,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[[1,1,1],[]]
 => []
 => []
 => [1,0]
 => 1
[[2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[4],[]]
 => []
 => []
 => [1,0]
 => 1
[[3,1],[]]
 => []
 => []
 => [1,0]
 => 1
[[4,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[2,2],[]]
 => []
 => []
 => [1,0]
 => 1
[[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[[2,1,1],[]]
 => []
 => []
 => [1,0]
 => 1
[[3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[[4,3],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[3,2,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[[4,3,1],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4
[[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[4,3,2],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4
[[1,1,1,1],[]]
 => []
 => []
 => [1,0]
 => 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[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]
 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[[5],[]]
 => []
 => []
 => [1,0]
 => 1
[[4,1],[]]
 => []
 => []
 => [1,0]
 => 1
[[5,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[3,2],[]]
 => []
 => []
 => [1,0]
 => 1
[[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[5,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[[3,1,1],[]]
 => []
 => []
 => [1,0]
 => 1
[[4,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000007
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[3],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[3,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[2,2],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[3,2],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[[1,1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[[2,1,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 3 - 1
[[4],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[3,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[4,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[2,2],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[3,2],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[[2,1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[[3,1,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 3 - 1
[[3,3],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 2 - 1
[[2,2,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 3 - 1
[[3,2,1],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2 = 3 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 3 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 2 = 3 - 1
[[1,1,1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3 = 4 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 3 = 4 - 1
[[2,1,1,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3 = 4 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 3 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => 3 = 4 - 1
[[5],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[4,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[5,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[3,2],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[4,2],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[[3,1,1],[]]
 => []
 => []
 => [] => 0 = 1 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[[4,1,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
The following 81 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000025The number of initial rises of a Dyck path. St000507The number of ascents of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000144The pyramid weight of the Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000054The first entry of the permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000288The number of ones in a binary word. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001497The position of the largest weak excedence of a permutation. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000094The depth of an ordered tree. St001180Number of indecomposable injective modules with projective dimension at most 1. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St001839The number of excedances of a set partition. 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 $eA$. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St000075The orbit size of a standard tableau under promotion. 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. St001060The distinguishing index of a graph. St001645The pebbling number of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. 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. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001330The hat guessing number of a graph. 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. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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.