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Your data matches 85 different statistics following compositions of up to 3 maps.
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Matching statistic: St000737
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000737: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000737: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [[1,2]]
=> 1
[[2,2]]
=> [[2,2]]
=> 2
[[1],[2]]
=> [[1,2]]
=> 1
[[1,3]]
=> [[1,3]]
=> 1
[[2,3]]
=> [[2,3]]
=> 2
[[3,3]]
=> [[3,3]]
=> 3
[[1],[3]]
=> [[1,3]]
=> 1
[[2],[3]]
=> [[2,3]]
=> 2
[[1,1,2]]
=> [[1,1,2]]
=> 1
[[1,2,2]]
=> [[1,2,2]]
=> 1
[[2,2,2]]
=> [[2,2,2]]
=> 2
[[1,1],[2]]
=> [[1,1,2]]
=> 1
[[1,2],[2]]
=> [[1,2,2]]
=> 1
[[1,4]]
=> [[1,4]]
=> 1
[[2,4]]
=> [[2,4]]
=> 2
[[3,4]]
=> [[3,4]]
=> 3
[[4,4]]
=> [[4,4]]
=> 4
[[1],[4]]
=> [[1,4]]
=> 1
[[2],[4]]
=> [[2,4]]
=> 2
[[3],[4]]
=> [[3,4]]
=> 3
[[1,1,3]]
=> [[1,1,3]]
=> 1
[[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,3,3]]
=> [[1,3,3]]
=> 1
[[2,2,3]]
=> [[2,2,3]]
=> 2
[[2,3,3]]
=> [[2,3,3]]
=> 2
[[3,3,3]]
=> [[3,3,3]]
=> 3
[[1,1],[3]]
=> [[1,1,3]]
=> 1
[[1,2],[3]]
=> [[1,2,3]]
=> 1
[[1,3],[2]]
=> [[1,2],[3]]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> 1
[[2,2],[3]]
=> [[2,2,3]]
=> 2
[[2,3],[3]]
=> [[2,3,3]]
=> 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> [[1,1,2,2]]
=> 1
[[1,2,2,2]]
=> [[1,2,2,2]]
=> 1
[[2,2,2,2]]
=> [[2,2,2,2]]
=> 2
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> 1
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> 1
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> 1
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> 1
[[1,5]]
=> [[1,5]]
=> 1
[[2,5]]
=> [[2,5]]
=> 2
[[3,5]]
=> [[3,5]]
=> 3
[[4,5]]
=> [[4,5]]
=> 4
[[5,5]]
=> [[5,5]]
=> 5
[[1],[5]]
=> [[1,5]]
=> 1
[[2],[5]]
=> [[2,5]]
=> 2
[[3],[5]]
=> [[3,5]]
=> 3
[[4],[5]]
=> [[4,5]]
=> 4
Description
The last entry on the main diagonal of a semistandard tableau.
Matching statistic: St001128
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 55%●distinct values known / distinct values provided: 33%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 55%●distinct values known / distinct values provided: 33%
Values
[[1,2]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[2,2]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1],[2]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,2,2]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,2,2]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,1],[2]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,2}
[[1,2],[2]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,2}
[[1,4]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,3,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,2,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,3,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[3,3,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,1],[3]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,1,2,2,2,2,3}
[[1,2],[3]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,1,2,2,2,2,3}
[[1,3],[2]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,3}
[[1,3],[3]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,3}
[[2,2],[3]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,1,2,2,2,2,3}
[[2,3],[3]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,3}
[[1,1,1,2]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,2}
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,2}
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[1,5]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,2,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,3,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,4,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,2,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,3,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[2,4,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[3,3,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[3,4,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[4,4,4]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[[1,1],[4]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2],[4]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,3],[4]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[3]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[4]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[2,2],[4]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[2,3],[4]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[2,4],[3]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[2,4],[4]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[3,3],[4]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[3,4],[4]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[3],[4]]
=> [3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[2],[3],[4]]
=> [3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,1,1,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,2,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,2,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,3,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[2,2,2,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[2,2,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[2,3,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[3,3,3,3]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
Description
The exponens consonantiae of a partition.
This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
Matching statistic: St000207
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,5],[2]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[2]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[3]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4,4],[3]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,5],[2]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[2]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[3]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4,4],[3]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000460
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,5],[2]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[2]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[3]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4,4],[3]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000618
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 45%●distinct values known / distinct values provided: 17%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 45%●distinct values known / distinct values provided: 17%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,5],[2]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[2]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[3]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4,4],[3]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
Description
The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St000667
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,5],[2]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[2]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[3]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4,4],[3]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000755
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,5],[2]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[2]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[3]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4,4],[3]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St000781
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 45%●distinct values known / distinct values provided: 17%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 45%●distinct values known / distinct values provided: 17%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,5],[2]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[2]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[3]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4,4],[3]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St000870
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> [1]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1
[[1,5],[2]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> [1]
=> 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[2]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4,4],[3]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4,4],[3]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4]]
=> [[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [3,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [[2,3,4],[4]]
=> [3,1]
=> [1]
=> 1
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.
The following 75 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000675The number of centered multitunnels of a Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000937The number of positive values of the symmetric group character corresponding to the partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000170The trace of a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001118The acyclic chromatic index of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001281The normalized isoperimetric number of a graph. St001545The second Elser number of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001060The distinguishing index of a graph. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees.
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