- FindStat

Your data matches 20 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001925
(load all 4 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
[[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: 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: 21% ●values known / values provided: 21%●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
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 0 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 0 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 0 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,3,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,3,4],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,5],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1

Description

The number of valleys of the Dyck path.

Matching statistic: St001499

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
St001499: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●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
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 0 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 0 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 0 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,3,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,3,4],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,5],[3,3,5],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1

Description

The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.

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: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St001505

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
St001505: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of elements generated by the Dyck path as a map in the full transformation monoid. We view the resolution quiver of a Dyck path (corresponding to an LNakayamaalgebra) as a transformation and associate to it the submonoid generated by this map in the full transformation monoid.

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: 21% ●values known / values provided: 21%●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
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
 => [5,3,3,3,1]
 => 1001110010 => ? = 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
 => [5,4,2,2,2]
 => 1010011100 => ? = 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
 => [5,4,2,2,2]
 => 1010011100 => ? = 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,5],[4,5],[5]]
 => [5,3,3,3,1]
 => 1001110010 => ? = 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 0
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 0

Description

The number of ascents of a binary word.

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: 21% ●values known / values provided: 21%●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
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
 => [5,3,3,3,1]
 => 1001110010 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
 => [5,4,2,2,2]
 => 1010011100 => ? = 2 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
 => [5,4,2,2,2]
 => 1010011100 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,5],[4,5],[5]]
 => [5,3,3,3,1]
 => 1001110010 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 0 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 0 + 1

Description

The number of descents of a binary word.

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: 21% ●values known / values provided: 21%●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
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
 => [5,3,3,3,1]
 => 1001110010 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ? => ? = 2 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
 => [5,4,2,2,2]
 => 1010011100 => ? = 2 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
 => [5,4,2,2,2]
 => 1010011100 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,5],[4,5],[5]]
 => [5,3,3,3,1]
 => 1001110010 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 0 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
 => ?
 => ? => ? = 0 + 1

Description

The number of runs of ones in a binary word.

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: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 83%

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

Description

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

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: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 83%

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
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [4,4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
 => [5,3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
 => [4,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
 => [4,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]]
 => [4,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,4],[5]]
 => [4,4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,4],[5]]
 => [4,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]]
 => [4,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => [4,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
 => [4,4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
 => [5,4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 2 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
 => [5,4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,5],[5]]
 => [5,3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,5],[5]]
 => [4,4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,5],[5]]
 => [4,4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
 => [4,4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,5],[5]]
 => [5,3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,5],[5]]
 => [4,4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,5],[5]]
 => [4,3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
 => [4,4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,5],[5]]
 => [5,3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => [4,3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => [4,3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
 => ?
 => ?
 => ? = 2 + 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 10 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. 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 $eA$. St000015The number of peaks of a Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a 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. St000159The number of distinct parts of the integer partition. 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.