Your data matches 48 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001925
Mapping
St001925: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => 0
[[1,0],[0,1]]
 => 1
[[0,1],[1,0]]
 => 1
[[1,0,0],[0,1,0],[0,0,1]]
 => 2
[[0,1,0],[1,0,0],[0,0,1]]
 => 2
[[1,0,0],[0,0,1],[0,1,0]]
 => 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => 0
[[0,0,1],[1,0,0],[0,1,0]]
 => 2
[[0,1,0],[0,0,1],[1,0,0]]
 => 2
[[0,0,1],[0,1,0],[1,0,0]]
 => 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 3
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 3
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 3
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => 3
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 3
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 3
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => 3
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 3
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => 3
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 3
Description
The minimal number of zeros in a row of an alternating sign matrix.
Matching statistic: St000159
(load all 2 compositions to match this statistic)
Mapping
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => 1 = 0 + 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => 2 = 1 + 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => 2 = 1 + 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => 3 = 2 + 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => 3 = 2 + 1
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => 3 = 2 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => 1 = 0 + 1
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => 3 = 2 + 1
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => 3 = 2 + 1
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => 3 = 2 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 2 = 1 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 2 = 1 + 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]
 => 2 = 1 + 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]
 => 2 = 1 + 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]
 => 2 = 1 + 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]
 => 2 = 1 + 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]
 => 4 = 3 + 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]
 => 2 = 1 + 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]
 => 2 = 1 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 2 = 1 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 2 = 1 + 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]
 => 4 = 3 + 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]
 => 2 = 1 + 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]
 => 2 = 1 + 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]
 => 4 = 3 + 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]
 => 2 = 1 + 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]
 => 2 = 1 + 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]
 => 4 = 3 + 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]
 => 2 = 1 + 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]
 => 2 = 1 + 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]
 => 2 = 1 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 4 = 3 + 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]
 => 2 = 1 + 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]
 => 4 = 3 + 1
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
(load all 2 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 = 0 + 2
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => 3 = 1 + 2
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => 3 = 1 + 2
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => 4 = 2 + 2
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => 4 = 2 + 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => 4 = 2 + 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => 2 = 0 + 2
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => 4 = 2 + 2
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => 4 = 2 + 2
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => 4 = 2 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 1 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 1 + 2
[[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 = 1 + 2
[[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 = 1 + 2
[[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 = 1 + 2
[[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 + 2
[[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 = 3 + 2
[[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 = 1 + 2
[[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 = 1 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 1 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 1 + 2
[[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 = 3 + 2
[[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 = 1 + 2
[[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 + 2
[[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 = 3 + 2
[[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 = 1 + 2
[[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 = 1 + 2
[[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 = 3 + 2
[[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 = 1 + 2
[[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 = 1 + 2
[[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 = 1 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 3 + 2
[[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 = 1 + 2
[[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 = 3 + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000292
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
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => 10 => 0
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => 1010 => 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => 1010 => 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => 101010 => 2
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => 101010 => 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => 101010 => 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => 11100 => 0
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => 101010 => 2
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => 101010 => 2
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => 101010 => 2
[[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 => 3
[[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 => 3
[[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 => 3
[[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 => 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 => 3
[[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 => 3
[[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 => 3
[[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 => 3
[[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 => 3
[[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 => 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 => 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 => 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 => 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 => 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 => 3
[[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 => 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 => 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 => 3
[[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 => 3
[[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 => 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 => 3
[[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 => 3
[[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 => 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 => 3
[[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 => 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 => 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 => 3
[[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 => 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 => 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 => 3
[[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 => 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 => 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 => 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 => 3
[[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 => 3
[[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 => 3
[[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 => 3
[[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 => 3
[[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 => 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 => 3
Description
The number of ascents of a binary word.
Matching statistic: St001032
Mapping
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1,0]
 => [1,0]
 => [1,1,0,0]
 => 0
[[1,0],[0,1]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[[0,1],[1,0]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[0,0,1],[1,0,0],[0,1,0]]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[0,1,0],[0,0,1],[1,0,0]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[0,0,1],[0,1,0],[1,0,0]]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
Description
The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. In other words, this is the number of valleys and peaks whose first step is in odd position, the initial position equal to 1. The generating function is given in [1].
Matching statistic: St001036
Mapping
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => [1,0]
 => 0
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001037
Mapping
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1]
 => [1,0]
 => 0
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[[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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 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]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
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 = 0 + 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 1 + 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 = 2 + 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 = 2 + 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 = 2 + 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 = 0 + 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 = 2 + 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 = 2 + 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 = 2 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 0 + 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [2,2,2]
 => 11100 => 1 = 0 + 1
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [3,2,1]
 => 101010 => 3 = 2 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 1
Description
The number of descents of a binary word.
Matching statistic: St000306
(load all 2 compositions to match this statistic)
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 = 0 + 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2 = 1 + 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 = 2 + 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 = 2 + 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 = 2 + 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 = 0 + 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 = 2 + 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 = 2 + 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 = 2 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 1 + 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 = 1 + 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 = 1 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 3 + 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 = 1 + 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 = 3 + 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 $D$.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000390The number of runs of ones in a binary word. 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. St000015The number of peaks of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We 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: St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000385The number of vertices with out-degree 1 in a binary tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001875The number of simple modules with projective dimension at most 1. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. St001645The pebbling number of a connected graph. St000080The rank of the poset. St000663The number of right floats of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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 $eA$. St001812The biclique partition number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000264The girth of a graph, which is not a tree. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.