- FindStat

Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001925
(load all 8 compositions to match this statistic)

Mapping

St001925: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => 0 = 2 - 2
[[1,0],[0,1]]
 => 1 = 3 - 2
[[0,1],[1,0]]
 => 1 = 3 - 2
[[1,0,0],[0,1,0],[0,0,1]]
 => 2 = 4 - 2
[[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 4 - 2
[[1,0,0],[0,0,1],[0,1,0]]
 => 2 = 4 - 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => 0 = 2 - 2
[[0,0,1],[1,0,0],[0,1,0]]
 => 2 = 4 - 2
[[0,1,0],[0,0,1],[1,0,0]]
 => 2 = 4 - 2
[[0,0,1],[0,1,0],[1,0,0]]
 => 2 = 4 - 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 3 = 5 - 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 3 = 5 - 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 3 = 5 - 2
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 1 = 3 - 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 3 = 5 - 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => 3 = 5 - 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 3 = 5 - 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 3 = 5 - 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 3 = 5 - 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 1 = 3 - 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 1 = 3 - 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => 1 = 3 - 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => 1 = 3 - 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => 1 = 3 - 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 3 = 5 - 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => 1 = 3 - 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => 1 = 3 - 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3 = 5 - 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => 3 = 5 - 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => 1 = 3 - 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 3 = 5 - 2
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => 3 = 5 - 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => 1 = 3 - 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 3 = 5 - 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => 1 = 3 - 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => 1 = 3 - 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 3 = 5 - 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => 1 = 3 - 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => 1 = 3 - 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 3 = 5 - 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => 1 = 3 - 2
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => 1 = 3 - 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => 1 = 3 - 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => 3 = 5 - 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 3 = 5 - 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 3 = 5 - 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 3 = 5 - 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => 3 = 5 - 2
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => 1 = 3 - 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 3 = 5 - 2

Description

The minimal number of zeros in a row of an alternating sign matrix.

Matching statistic: St000318
(load all 4 compositions to match this statistic)

Mapping

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1]
 => 2
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => 3
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => 3
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => 4
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => 4
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => 4
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => 2
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => 5
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => 3
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => 3
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 5
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => 3
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 5

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000015

Mapping

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1]
 => [1,0,1,0]
 => 2
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5

Description

The number of peaks of a Dyck path.

Matching statistic: St001068

Mapping

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1]
 => [1,0,1,0]
 => 2
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St001203

Mapping

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1]
 => [1,0,1,0]
 => 2
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5

Description

We associate to 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 Dyck path as follows: In the list

$L$ delete the first entry

$c_0$ and substract from all other entries

$n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.

Matching statistic: St000053

Mapping

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1]
 => [1,0,1,0]
 => 1 = 2 - 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1

Description

The number of valleys of the Dyck path.

Matching statistic: St000291

Mapping

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1]
 => 10 => 1 = 2 - 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => 1010 => 2 = 3 - 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => 1010 => 2 = 3 - 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => 11100 => 1 = 2 - 1
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => 1110010 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => 10011100 => 2 = 3 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => 10011100 => 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => 1110010 => 2 = 3 - 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => 10011100 => 2 = 3 - 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => 1110010 => 2 = 3 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => 10011100 => 2 = 3 - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => 1110010 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1

Description

The number of descents of a binary word.

Matching statistic: St000306

Mapping

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1]
 => [1,0,1,0]
 => 1 = 2 - 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

Matching statistic: St000390

Mapping

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1]
 => 10 => 1 = 2 - 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => 1010 => 2 = 3 - 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => 1010 => 2 = 3 - 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => 11100 => 1 = 2 - 1
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 4 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => 1110010 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => 10011100 => 2 = 3 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => 10011100 => 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => 1110010 => 2 = 3 - 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => 10011100 => 2 = 3 - 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => 1110010 => 2 = 3 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => 1101100 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => 10011100 => 2 = 3 - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => 1110010 => 2 = 3 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => 10101010 => 4 = 5 - 1

Description

The number of runs of ones in a binary word.

Matching statistic: St001028

Mapping

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1]
 => [1,0,1,0]
 => 3 = 2 + 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1

Description

Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for 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

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000292The number of ascents of a binary word. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000159The number of distinct parts of the integer partition. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths.