Your data matches 371 different statistics following compositions of up to 3 maps.
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Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [3]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [2,1]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [2,1]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [2,1]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,1,1]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [4]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,1,1]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,1,1,1]
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [5]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[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]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,0,1,1,0,0]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,0,1,1,0,0]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[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]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[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]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[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]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[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]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
Description
The sum of the heights of the peaks of a Dyck path.
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [3]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [2,1]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [2,1]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [2,1]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,1,1]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [4]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,1,1]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,1,1,1]
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [5]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[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]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
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.
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [3]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [2,1]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [2,1]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [2,1]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,1,1]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [4]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,1,1]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,1,1,1]
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [5]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[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]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
Description
The hook length of the last cell along the main diagonal of an integer partition.
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00277: Permutations catalanizationPermutations
St000725: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,2,3] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,3,2] => 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,3,2] => 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [2,3,1] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [3,2,1] => 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3,4] => 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,4,3] => 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,4,3] => 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,4,3] => 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,3,4,2] => 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,3,4,2] => 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,3,4,2] => 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [2,3,4,1] => 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,4,3,2] => 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,4,3,2] => 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,4,3,2] => 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,4,3,1] => 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,4,3,2] => 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,4,3,2] => 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,4,3,1] => 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [3,4,2,1] => 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [4,3,2,1] => 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 5
[[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]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 5
[[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]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 5
[[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]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,4,5,3] => 5
[[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]]
=> [1,2,5,3,4] => [1,2,4,5,3] => 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,4,5,3] => 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,4,5,3] => 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,4,5,3] => 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,4,5,3] => 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,3,4,5,2] => 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,3,4,5,2] => 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,3,4,5,2] => 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,3,4,5,2] => 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,3,5,4,2] => 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,3,5,4,2] => 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,3,5,4,2] => 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,3,5,4,2] => 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [2,3,5,4,1] => 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 5
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [3]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [2,1]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [2,1]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [2,1]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,1,1]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [4]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,1]
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [3,1]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,1,1]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [2,1,1]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,1,1,1]
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [5]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[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]]
=> [1,2,3,5,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[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]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [4,1]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [4,1]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [3,1,1]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [3,1,1]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [3,1,1]
=> 5
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.
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
St001004: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,2,3] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,3,2] => 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,3,2] => 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [3,1,2] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [2,3,1] => 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3,4] => 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,4,3] => 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,4,3] => 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,4,3] => 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,4,2,3] => 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,4,2,3] => 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,4,2,3] => 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [4,1,2,3] => 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,3,4,2] => 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,3,4,2] => 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,3,4,2] => 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [3,4,1,2] => 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,3,4,2] => 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,3,4,2] => 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [3,4,1,2] => 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [3,1,4,2] => 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [2,3,4,1] => 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 5
[[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]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 5
[[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]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 5
[[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]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,5,3,4] => 5
[[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]]
=> [1,2,5,3,4] => [1,2,5,3,4] => 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,5,3,4] => 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,5,3,4] => 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,5,3,4] => 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,5,3,4] => 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,5,2,3,4] => 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,5,2,3,4] => 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,5,2,3,4] => 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,5,2,3,4] => 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [5,1,2,3,4] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,5,3] => 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,5,3] => 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,5,3] => 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,5,3] => 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,5,3] => 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,5,3] => 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,4,5,2,3] => 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,4,5,2,3] => 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,4,5,2,3] => 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,4,5,2,3] => 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [4,1,5,2,3] => 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,4,5,3] => 5
Description
The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,0,1,1,0,0]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,0,1,1,0,0]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[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]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[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]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[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]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[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]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00072: Permutations binary search tree: left to rightBinary trees
St001554: Binary trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [.,[[.,.],.]]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [.,[[.,.],.]]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [[.,[.,.]],.]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [[[.,.],.],.]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [[.,[[.,.],.]],.]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [[.,[[.,.],.]],.]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> 4
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 5
[[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]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 5
[[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]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 5
[[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]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 5
[[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]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> 5
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> 5
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> 5
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [[.,[.,[[.,.],.]]],.]
=> 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 5
Description
The number of distinct nonempty subtrees of a binary tree.
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00072: Permutations binary search tree: left to rightBinary trees
St000385: Binary trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 2 = 3 - 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [.,[[.,.],.]]
=> 2 = 3 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [.,[[.,.],.]]
=> 2 = 3 - 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [[.,[.,.]],.]
=> 2 = 3 - 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [[[.,.],.],.]
=> 2 = 3 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 3 = 4 - 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 3 = 4 - 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 3 = 4 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 3 = 4 - 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 3 = 4 - 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 3 = 4 - 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 3 = 4 - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 3 = 4 - 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 3 = 4 - 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 3 = 4 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [[.,[[.,.],.]],.]
=> 3 = 4 - 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 3 = 4 - 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 3 = 4 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [[.,[[.,.],.]],.]
=> 3 = 4 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> 3 = 4 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 3 = 4 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 5 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 4 = 5 - 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]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 4 = 5 - 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]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 4 = 5 - 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]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 4 = 5 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 4 = 5 - 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]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 4 = 5 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 4 = 5 - 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 4 = 5 - 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 4 = 5 - 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 4 = 5 - 1
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 4 = 5 - 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 4 = 5 - 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 4 = 5 - 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 4 = 5 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 4 = 5 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 4 = 5 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 4 = 5 - 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 4 = 5 - 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 4 = 5 - 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 4 = 5 - 1
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> 4 = 5 - 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> 4 = 5 - 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> 4 = 5 - 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> 4 = 5 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [[.,[.,[[.,.],.]]],.]
=> 4 = 5 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 4 = 5 - 1
Description
The number of vertices with out-degree 1 in a binary tree. See the references for several connections of this statistic. In particular, the number $T(n,k)$ of binary trees with $n$ vertices and $k$ out-degree $1$ vertices is given by $T(n,k) = 0$ for $n-k$ odd and $$T(n,k)=\frac{2^k}{n+1}\binom{n+1}{k}\binom{n+1-k}{(n-k)/2}$$ for $n-k$ is even.
The following 361 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000393The number of strictly increasing runs in a binary word. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. 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. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001267The length of the Lyndon factorization of the binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001437The flex of a binary word. St000022The number of fixed points of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000203The number of external nodes of a binary tree. St000293The number of inversions of a binary word. St000294The number of distinct factors of a binary word. St000308The height of the tree associated to a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000518The number of distinct subsequences in a binary word. St000528The height of a poset. St000548The number of different non-empty partial sums of an integer partition. St000625The sum of the minimal distances to a greater element. St000654The first descent of a permutation. St000657The smallest part of an integer composition. St000702The number of weak deficiencies of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000808The number of up steps of the associated bargraph. St000863The length of the first row of the shifted shape of a permutation. St000873The aix statistic of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001034The area of the parallelogram polyomino associated with the Dyck path. St001074The number of inversions of the cyclic embedding of a permutation. St001093The detour number of a graph. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001497The position of the largest weak excedence of a permutation. St001523The degree of symmetry of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000050The depth or height of a binary tree. St000070The number of antichains in a poset. St000081The number of edges of a graph. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000259The diameter of a connected graph. St000296The length of the symmetric border of a binary word. St000441The number of successions of a permutation. St000520The number of patterns in a permutation. St000553The number of blocks of a graph. St000627The exponent of a binary word. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000672The number of minimal elements in Bruhat order not less than the permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000806The semiperimeter of the associated bargraph. St000921The number of internal inversions of a binary word. St000922The minimal number such that all substrings of this length are unique. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000982The length of the longest constant subword. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001405The number of bonds in a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001430The number of positive entries in a signed permutation. St001479The number of bridges of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. 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. St001884The number of borders of a binary word. St001917The order of toric promotion on the set of labellings of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St000060The greater neighbor of the maximum. St000295The length of the border of a binary word. St000313The number of degree 2 vertices of a graph. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000519The largest length of a factor maximising the subword complexity. St000552The number of cut vertices of a graph. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001130The number of two successive successions in a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001308The number of induced paths on three vertices in a graph. St001368The number of vertices of maximal degree in a graph. St001521Half the total irregularity of a graph. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St001958The degree of the polynomial interpolating the values of a permutation. St000447The number of pairs of vertices of a graph with distance 3. St001306The number of induced paths on four vertices in a graph. St000019The cardinality of the support of a permutation. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St001925The minimal number of zeros in a row of an alternating sign matrix. St001622The number of join-irreducible elements of a lattice. St000026The position of the first return of a Dyck path. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St001279The sum of the parts of an integer partition that are at least two. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. St000505The biggest entry in the block containing the 1. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000010The length of the partition. St000028The number of stack-sorts needed to sort a permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000064The number of one-box pattern of a permutation. St000147The largest part of an integer partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000221The number of strong fixed points of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000501The size of the first part in the decomposition of a permutation. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000717The number of ordinal summands of a poset. St000733The row containing the largest entry of a standard tableau. St000844The size of the largest block in the direct sum decomposition of a permutation. St000906The length of the shortest maximal chain in a poset. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. St001252Half the sum of the even parts of a partition. St001717The largest size of an interval in a poset. St000058The order of a permutation. St000080The rank of the poset. St000094The depth of an ordered tree. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000167The number of leaves of an ordered tree. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000653The last descent of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001391The disjunction number of a graph. St001480The number of simple summands of the module J^2/J^3. St001516The number of cyclic bonds of a permutation. St001649The length of a longest trail in a graph. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000021The number of descents of a permutation. St000309The number of vertices with even degree. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000836The number of descents of distance 2 of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000171The degree of the graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000924The number of topologically connected components of a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000018The number of inversions of a permutation. St001268The size of the largest ordinal summand in the poset. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001725The harmonious chromatic number of a graph. St000673The number of non-fixed points of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001468The smallest fixpoint of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001429The number of negative entries in a signed permutation. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000327The number of cover relations in a poset. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001927Sparre Andersen's number of positives of a signed permutation. St001948The number of augmented double ascents of a permutation. St001557The number of inversions of the second entry of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St000784The maximum of the length and the largest part of the integer partition. St000422The energy of a graph, if it is integral. St000157The number of descents of a standard tableau. St000507The number of ascents of a standard tableau. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000035The number of left outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001645The pebbling number of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001861The number of Bruhat lower covers of a permutation. St001434The number of negative sum pairs of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000135The number of lucky cars of the parking function. St000744The length of the path to the largest entry in a standard Young tableau. St001115The number of even descents of a permutation. St000145The Dyson rank of a partition. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000691The number of changes of a binary word. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000017The number of inversions of a standard tableau. St000159The number of distinct parts of the integer partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000628The balance of a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000753The Grundy value for the game of Kayles on a binary word. St000759The smallest missing part in an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000820The number of compositions obtained by rotating the composition. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001432The order dimension of the partition. St001484The number of singletons of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001926Sparre Andersen's position of the maximum of a signed permutation. St000183The side length of the Durfee square of an integer partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000781The number of proper colouring schemes of a Ferrers diagram. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001280The number of parts of an integer partition that are at least two. St000142The number of even parts of a partition. St000146The Andrews-Garvan crank of a partition. St000389The number of runs of ones of odd length in a binary word. St000474Dyson's crank of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001092The number of distinct even parts of a partition. St001175The size of a partition minus the hook length of the base cell. St001525The number of symmetric hooks on the diagonal of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001675The number of parts equal to the part in the reversed composition. St001845The number of join irreducibles minus the rank of a lattice.