Your data matches 26 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000186
(load all 82 compositions to match this statistic)
Mapping
St000186: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => 1
[[1,0],[1]]
 => 1
[[2,0],[0]]
 => 2
[[2,0],[1]]
 => 2
[[2,0],[2]]
 => 2
[[1,1],[1]]
 => 2
[[1,0,0],[0,0],[0]]
 => 1
[[1,0,0],[1,0],[0]]
 => 1
[[1,0,0],[1,0],[1]]
 => 1
[[3,0],[0]]
 => 3
[[3,0],[1]]
 => 3
[[3,0],[2]]
 => 3
[[3,0],[3]]
 => 3
[[2,1],[1]]
 => 3
[[2,1],[2]]
 => 3
[[2,0,0],[0,0],[0]]
 => 2
[[2,0,0],[1,0],[0]]
 => 2
[[2,0,0],[1,0],[1]]
 => 2
[[2,0,0],[2,0],[0]]
 => 2
[[2,0,0],[2,0],[1]]
 => 2
[[2,0,0],[2,0],[2]]
 => 2
[[1,1,0],[1,0],[0]]
 => 2
[[1,1,0],[1,0],[1]]
 => 2
[[1,1,0],[1,1],[1]]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => 1
[[4,0],[0]]
 => 4
[[4,0],[1]]
 => 4
[[4,0],[2]]
 => 4
[[4,0],[3]]
 => 4
[[4,0],[4]]
 => 4
[[3,1],[1]]
 => 4
[[3,1],[2]]
 => 4
[[3,1],[3]]
 => 4
[[2,2],[2]]
 => 4
[[3,0,0],[0,0],[0]]
 => 3
[[3,0,0],[1,0],[0]]
 => 3
[[3,0,0],[1,0],[1]]
 => 3
[[3,0,0],[2,0],[0]]
 => 3
[[3,0,0],[2,0],[1]]
 => 3
[[3,0,0],[2,0],[2]]
 => 3
[[3,0,0],[3,0],[0]]
 => 3
[[3,0,0],[3,0],[1]]
 => 3
[[3,0,0],[3,0],[2]]
 => 3
[[3,0,0],[3,0],[3]]
 => 3
[[2,1,0],[1,0],[0]]
 => 3
[[2,1,0],[1,0],[1]]
 => 3
[[2,1,0],[1,1],[1]]
 => 3
Description
The sum of the first row in a Gelfand-Tsetlin pattern.
Matching statistic: St000228
(load all 11 compositions to match this statistic)
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => 1
[[1,0],[1]]
 => [[1]]
 => [1]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => 3
[[4,1,1,0],[4,1,0],[4,1],[4]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[4]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[3]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[3]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[3]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[3]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[3]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[3]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[3]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[2]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[1,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[1,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,1],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,0],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,0],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,0],[1]]
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,0],[1]]
 => ?
 => ?
 => ? = 6
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => 10 => 1
[[1,0],[1]]
 => [[1]]
 => [1]
 => 10 => 1
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => 100 => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,1]
 => 110 => 2
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => 100 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => 110 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => 10 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => 10 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => 10 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => 1000 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [2,1]
 => 1010 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [2,1]
 => 1010 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => 1000 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => 1010 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => 1010 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => 100 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,1]
 => 110 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,1]
 => 110 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => 100 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,1]
 => 110 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => 100 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => 110 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => 110 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => 110 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => 10 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => 10 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => 10 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => 10 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => 10000 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [3,1]
 => 10010 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [2,2]
 => 1100 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [3,1]
 => 10010 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => 10000 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => 10010 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [2,2]
 => 1100 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => 10010 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => 1100 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => 1000 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,1,1]
 => 1110 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => 1000 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => 1000 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => 1010 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => 1010 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [1,1,1]
 => 1110 => 3
[[4,1,1,0],[4,1,0],[4,1],[4]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[4]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[3]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[3]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[3]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[3]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[3]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[3]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[3]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[2]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[1,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,1],[1,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[1,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[2,1,1],[1,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[2,1,0],[1,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[1,1,0],[1,1],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[4,1,0],[1,0],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[3,1,0],[1,0],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[2,1,0],[1,0],[1]]
 => ?
 => ?
 => ? => ? = 6
[[4,1,1,0],[1,1,0],[1,0],[1]]
 => ?
 => ?
 => ? => ? = 6
Description
The number of inversions of a binary word.
Matching statistic: St000459
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 1
[[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,1]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,1]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,1]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,1,1]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => [2,1]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => [2,1]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,1]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => [2]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => [2]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => [1]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => [2,1,1]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => [2,1,1]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => [2,1,1]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => [2,1,1]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,1,1]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => [2,1]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => [2,1]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => [2,1]
 => 3
[[4,1,1,0],[4,1,0],[4,1],[4]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[4]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 1
[[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,1]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,1]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,1]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,1,1]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => [2,1]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => [2,1]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,1]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => [2]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => [2]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => [1]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => [2,1,1]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => [2,1,1]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => [2,1,1]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => [2,1,1]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,1,1]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => [2,1]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => [2,1]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => [2,1]
 => 3
[[4,1,1,0],[4,1,0],[4,1],[4]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[4]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 1
[[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,1]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,1]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,1]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,1,1]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => [2,1]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => [2,1]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,1]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,1]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => [2]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => [2]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => [1]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => [2,1,1]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => [2,1,1]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => [2,1,1]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => [2,1,1]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,1,1]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,1,1]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => [2,1]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => [2,1]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => [2,1]
 => 3
[[4,1,1,0],[4,1,0],[4,1],[4]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[4]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
Description
The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001020
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => [1,0]
 => 1
[[1,0],[1]]
 => [[1]]
 => [1] => [1,0]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,0,1,0]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,0,1,0]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,0,1,0]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [1,1,0,0]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1,0]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1,0]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1,0]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => [1,0,1,0]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => [1,0,1,0]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => [1,0,1,0]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,0,1,0]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,0,1,0]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,0,1,0]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => [1,1,0,0]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => [1,1,0,0]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [1,1,0,0]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => [1,0]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1,0]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1,0]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1,0]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 3
[[4,1,1,0],[4,1,0],[4,1],[4]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[4]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001034
(load all 2 compositions to match this statistic)
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => [1,0]
 => 1
[[1,0],[1]]
 => [[1]]
 => [1]
 => [1,0]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => [1,0,1,0]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => [1,0]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => [1,0]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => [1,0]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => [1,0,1,0]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => [1,0]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => [1,0]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => [1,0]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => [1,0]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[4,1,1,0],[4,1,0],[4,1],[4]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[4]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[3]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[3]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[3]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[3]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[3]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[3]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[3]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[4,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[3,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[3,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[2,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[3,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[2,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[2,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,1],[1,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[2,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,1],[1,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[2,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,1],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[4,1,0],[1,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[3,1,0],[1,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[2,1,0],[1,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,1,1,0],[1,1,0],[1,0],[1]]
 => ?
 => ?
 => ?
 => ? = 6
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St001332
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001332: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => [1] => 0 = 1 - 1
[[1,0],[1]]
 => [[1]]
 => [1] => [1] => 0 = 1 - 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,2] => 1 = 2 - 1
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,2] => 1 = 2 - 1
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,2] => 1 = 2 - 1
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [1,2] => 1 = 2 - 1
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1] => 0 = 1 - 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1] => 0 = 1 - 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1] => 0 = 1 - 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => [1,3,2] => 2 = 3 - 1
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => [1,2,3] => 2 = 3 - 1
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => [1,2] => 1 = 2 - 1
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => [1,2] => 1 = 2 - 1
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => [1,2] => 1 = 2 - 1
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,2] => 1 = 2 - 1
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,2] => 1 = 2 - 1
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,2] => 1 = 2 - 1
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => [1,2] => 1 = 2 - 1
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => [1,2] => 1 = 2 - 1
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [1,2] => 1 = 2 - 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => [1] => 0 = 1 - 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1] => 0 = 1 - 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1] => 0 = 1 - 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1] => 0 = 1 - 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => [1,3,4,2] => 3 = 4 - 1
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => [1,2,4,3] => 3 = 4 - 1
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => [1,2,3,4] => 3 = 4 - 1
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => [1,2,3,4] => 3 = 4 - 1
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => [1,3,2] => 2 = 3 - 1
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => [1,3,2] => 2 = 3 - 1
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => [1,3,2] => 2 = 3 - 1
[[4,1,1,0],[4,1,0],[4,1],[4]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,0],[4]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,1],[3]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,0],[3]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,1],[3]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,1],[3,1],[3]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,1],[3]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,0],[3]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,0],[3]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,1],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,0],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,1],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,1],[3,1],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,1],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,0],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[2,1],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,1],[2,1],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,0],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[2,1],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[2,1,1],[2,1],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[2,1,0],[2,1],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[2,0],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[2,0],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[2,1,0],[2,0],[2]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,1],[3,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[2,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,1],[2,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[2,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[2,1,1],[2,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[2,1,0],[2,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[2,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[1,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,1],[1,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[2,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[1,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[2,1,1],[1,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[2,1,0],[2,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[2,1,0],[1,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[1,1,0],[1,1],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[4,1,0],[1,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[3,1,0],[1,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[2,1,0],[1,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
[[4,1,1,0],[1,1,0],[1,0],[1]]
 => ?
 => ? => ? => ? = 6 - 1
Description
The number of steps on the non-negative side of the walk associated with the permutation. Consider the walk taking an up step for each ascent, and a down step for each descent of the permutation. Then this statistic is the number of steps that begin and end at non-negative height.
Matching statistic: St001382
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 0 = 1 - 1
[[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 0 = 1 - 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,1]
 => 1 = 2 - 1
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,1]
 => 1 = 2 - 1
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,1]
 => 1 = 2 - 1
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2]
 => 1 = 2 - 1
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1]
 => 0 = 1 - 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 0 = 1 - 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 0 = 1 - 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => [2,1]
 => 2 = 3 - 1
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => [2,1]
 => 2 = 3 - 1
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => [1,1]
 => 1 = 2 - 1
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => [1,1]
 => 1 = 2 - 1
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => [1,1]
 => 1 = 2 - 1
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,1]
 => 1 = 2 - 1
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,1]
 => 1 = 2 - 1
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,1]
 => 1 = 2 - 1
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => [2]
 => 1 = 2 - 1
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => [2]
 => 1 = 2 - 1
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2]
 => 1 = 2 - 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => [1]
 => 0 = 1 - 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1]
 => 0 = 1 - 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1]
 => 0 = 1 - 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1]
 => 0 = 1 - 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 3 = 4 - 1
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 3 = 4 - 1
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 3 = 4 - 1
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => [1,1,1,1]
 => 3 = 4 - 1
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => [1,1,1,1]
 => 3 = 4 - 1
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => [2,1,1]
 => 3 = 4 - 1
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => [2,1,1]
 => 3 = 4 - 1
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => [2,1,1]
 => 3 = 4 - 1
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => [2,1,1]
 => 3 = 4 - 1
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,1,1]
 => 2 = 3 - 1
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => [2,1]
 => 2 = 3 - 1
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => [2,1]
 => 2 = 3 - 1
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => [2,1]
 => 2 = 3 - 1
[[4,1,1,0],[4,1,0],[4,1],[4]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,0],[4]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,1],[3]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,0],[3]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,1],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,1],[3]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,0],[3]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,1],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,0],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,1],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,1],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,0],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[2,1,1],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[2,1,0],[2,1],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[2,1,0],[2,0],[2]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[4,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,1],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[3,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[2,1,1],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[2,1,0],[2,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[2,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[2,1,0],[2,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[2,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[1,1,0],[1,1],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[4,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[3,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[2,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
[[4,1,1,0],[1,1,0],[1,0],[1]]
 => ?
 => ? => ?
 => ? = 6 - 1
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000189The number of elements in the poset. St001430The number of positive entries in a signed permutation. St001622The number of join-irreducible elements of a lattice. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001925The minimal number of zeros in a row of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St000197The number of entries equal to positive one in the alternating sign matrix. St000135The number of lucky cars of the parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001621The number of atoms of a lattice. St001926Sparre Andersen's position of the maximum of a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice.