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Your data matches 102 different statistics following compositions of up to 3 maps.
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Matching statistic: St000739
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000739: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000739: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [[1,2]]
=> [[1,2]]
=> [[1,2]]
=> 1
[[2,2]]
=> [[2,2]]
=> [[2,2]]
=> [[2,2]]
=> 2
[[1],[2]]
=> [[1,2]]
=> [[1,2]]
=> [[1,2]]
=> 1
[[1,1,2]]
=> [[1,1,2]]
=> [[1,1,2]]
=> [[1,1,2]]
=> 1
[[1,2,2]]
=> [[1,2,2]]
=> [[1,2,2]]
=> [[1,2,2]]
=> 1
[[2,2,2]]
=> [[2,2,2]]
=> [[2,2,2]]
=> [[2,2,2]]
=> 2
[[1,1],[2]]
=> [[1,1,2]]
=> [[1,1,2]]
=> [[1,1,2]]
=> 1
[[1,2],[2]]
=> [[1,2,2]]
=> [[1,2,2]]
=> [[1,2,2]]
=> 1
[[1,1,3]]
=> [[1,1,3]]
=> [[1,1,3]]
=> [[1,1,3]]
=> 1
[[1,2,3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,3,3]]
=> [[1,3,3]]
=> [[1,3,3]]
=> [[1,3,3]]
=> 1
[[2,2,3]]
=> [[2,2,3]]
=> [[2,2,3]]
=> [[2,2,3]]
=> 2
[[2,3,3]]
=> [[2,3,3]]
=> [[2,3,3]]
=> [[2,3,3]]
=> 2
[[3,3,3]]
=> [[3,3,3]]
=> [[3,3,3]]
=> [[3,3,3]]
=> 3
[[1,1],[3]]
=> [[1,1,3]]
=> [[1,1,3]]
=> [[1,1,3]]
=> 1
[[1,2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,3],[2]]
=> [[1,2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [[1,3,3]]
=> [[1,3,3]]
=> 1
[[2,2],[3]]
=> [[2,2,3]]
=> [[2,2,3]]
=> [[2,2,3]]
=> 2
[[2,3],[3]]
=> [[2,3,3]]
=> [[2,3,3]]
=> [[2,3,3]]
=> 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [[1,1,1,2]]
=> [[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> 1
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [[1,2,2,2]]
=> [[1,2,2,2]]
=> 1
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [[2,2,2,2]]
=> [[2,2,2,2]]
=> 2
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [[1,1,1,2]]
=> [[1,1,1,2]]
=> 1
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> 1
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [[1,2,2,2]]
=> [[1,2,2,2]]
=> 1
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> 1
[[1,1,1,3]]
=> [[1,1,1,3]]
=> [[1,1,1,3]]
=> [[1,1,1,3]]
=> 1
[[1,1,2,3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> 1
[[1,1,3,3]]
=> [[1,1,3,3]]
=> [[1,1,3,3]]
=> [[1,1,3,3]]
=> 1
[[1,2,2,3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> 1
[[1,2,3,3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> 1
[[1,3,3,3]]
=> [[1,3,3,3]]
=> [[1,3,3,3]]
=> [[1,3,3,3]]
=> 1
[[2,2,2,3]]
=> [[2,2,2,3]]
=> [[2,2,2,3]]
=> [[2,2,2,3]]
=> 2
[[2,2,3,3]]
=> [[2,2,3,3]]
=> [[2,2,3,3]]
=> [[2,2,3,3]]
=> 2
[[2,3,3,3]]
=> [[2,3,3,3]]
=> [[2,3,3,3]]
=> [[2,3,3,3]]
=> 2
[[3,3,3,3]]
=> [[3,3,3,3]]
=> [[3,3,3,3]]
=> [[3,3,3,3]]
=> 3
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> [[1,1,1,3]]
=> [[1,1,1,3]]
=> 1
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> 1
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> [[1,1,3,3]]
=> [[1,1,3,3]]
=> 1
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> 1
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> 1
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> [[1,3,3,3]]
=> [[1,3,3,3]]
=> 1
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> [[2,2,2,3]]
=> [[2,2,2,3]]
=> 2
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> [[2,2,3,3]]
=> [[2,2,3,3]]
=> 2
Description
The first entry in the last row of a semistandard tableau.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 80%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[1,1,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[1,1,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[1,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[1,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[1,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[2,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[2,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[2,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,1,1,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1,1,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1,1,1],[2]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,1,2],[2]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,2,2],[2]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2,2,2],[2]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1,1,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,1,2,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,1,3,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,1,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,2,2,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,2,3,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,2,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,3,3,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,3,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,4,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[2,2,2,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[2,2,3,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[2,2,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[2,3,3,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[2,3,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[2,4,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[3,3,3,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[3,3,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[3,4,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[4,4,4,4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,1,1],[4]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,2],[4]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,4],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,3],[4]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,4],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,4],[4]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2,2],[4]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2,4],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001199
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 67%●distinct values known / distinct values provided: 40%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 67%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[[2,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[[1],[2]]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 2
[[1,1,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,2}
[[1,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,2}
[[2,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,2}
[[1,1],[2]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,2],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,2}
[[1,1,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,3}
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,3}
[[1,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,3}
[[2,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,3}
[[2,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,3}
[[3,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,3}
[[1,1],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,3}
[[1,3],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,3}
[[2,2],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[2,3],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,3}
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1}
[[1,2,2,2]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1}
[[2,2,2,2]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1}
[[1,1,1],[2]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1,2],[2]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1}
[[1,1],[2,2]]
=> [3,4,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[[1,1,1,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,1,2,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,1,3,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,2,2,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,2,3,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,3,3,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[2,2,2,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[2,2,3,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[2,3,3,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[3,3,3,3]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,1,1],[3]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1,2],[3]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1,3],[2]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[1,1,3],[3]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[1,2,2],[3]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,2,3],[2]]
=> [2,1,3,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,2,3],[3]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[1,3,3],[2]]
=> [2,1,3,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,3,3],[3]]
=> [2,1,3,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[2,2,2],[3]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[2,2,3],[3]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[2,3,3],[3]]
=> [2,1,3,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,1],[2,3]]
=> [3,4,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[[1,1],[3,3]]
=> [3,4,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[[1,2],[2,3]]
=> [2,4,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[[1,2],[3,3]]
=> [3,4,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[[2,2],[3,3]]
=> [3,4,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2}
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2}
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2}
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2}
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2}
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2}
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[[1,1,1,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,1,2,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,1,3,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,1,4,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,2,2,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,2,3,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,2,4,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,3,3,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,3,4,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,4,4,4]]
=> [1,2,3,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4}
[[1,1,1],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1,2],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1,4],[2]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[1,1,3],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1,4],[3]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[1,1,4],[4]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[1,2,2],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,2,3],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[1,2,4],[4]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[1,3,3],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[1,3,4],[4]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[2,2,2],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[2,2,3],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[2,2,4],[3]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[2,2,4],[4]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[2,3,3],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[2,3,4],[4]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[[3,3,3],[4]]
=> [4,1,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[3,3,4],[4]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
Description
The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
Matching statistic: St001491
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 80%
Values
[[1,2]]
=> [2]
=> []
=> ? => ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> ? => ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> 1 => 1
[[1,1,2]]
=> [3]
=> []
=> ? => ? ∊ {1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ? => ? ∊ {1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ? => ? ∊ {1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> 1 => 1
[[1,2],[2]]
=> [2,1]
=> [1]
=> 1 => 1
[[1,1,3]]
=> [3]
=> []
=> ? => ? ∊ {1,1,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ? => ? ∊ {1,1,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ? => ? ∊ {1,1,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ? => ? ∊ {1,1,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ? => ? ∊ {1,1,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ? => ? ∊ {1,1,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> 1 => 1
[[1,2],[3]]
=> [2,1]
=> [1]
=> 1 => 1
[[1,3],[2]]
=> [2,1]
=> [1]
=> 1 => 1
[[1,3],[3]]
=> [2,1]
=> [1]
=> 1 => 1
[[2,2],[3]]
=> [2,1]
=> [1]
=> 1 => 1
[[2,3],[3]]
=> [2,1]
=> [1]
=> 1 => 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 11 => 2
[[1,1,1,2]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,2}
[[1,1,1,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,2,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,2,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,2,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3,3]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 11 => 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 11 => 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 11 => 2
[[1,1,1,1,2]]
=> [5]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,2}
[[1,1,1,2,2]]
=> [5]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,2}
[[1,1,2,2,2]]
=> [5]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,2}
[[1,2,2,2,2]]
=> [5]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,2}
[[2,2,2,2,2]]
=> [5]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,2}
[[1,1,1,1],[2]]
=> [4,1]
=> [1]
=> 1 => 1
[[1,1,1,2],[2]]
=> [4,1]
=> [1]
=> 1 => 1
[[1,1,2,2],[2]]
=> [4,1]
=> [1]
=> 1 => 1
[[1,2,2,2],[2]]
=> [4,1]
=> [1]
=> 1 => 1
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,1,2}
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,1,2}
[[1,1,1,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,1,2,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,1,3,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,1,4,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,2,2,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,2,3,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,2,4,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,3,3,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,3,4,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,4,4,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[2,2,2,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[2,2,3,4]]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[[1,1,1],[4]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1,2],[4]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1,4],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1,3],[4]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1,4],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,1,4],[4]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,2,2],[4]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,2,4],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,2,4],[4]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,4,4],[2]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,3,3],[4]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,3,4],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,3,4],[4]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,4,4],[3]]
=> [3,1]
=> [1]
=> 1 => 1
[[1,4,4],[4]]
=> [3,1]
=> [1]
=> 1 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let An=K[x]/(xn).
We associate to a nonempty subset S of an (n-1)-set the module MS, which is the direct sum of An-modules with indecomposable non-projective direct summands of dimension i when i is in S (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of MS. We decode the subset as a binary word so that for example the subset S={1,3} of {1,2,3} is decoded as 101.
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: 40% ●values known / values provided: 58%●distinct values known / distinct values provided: 40%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 58%●distinct values known / distinct values provided: 40%
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}
[[1,1,2]]
=> [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}
[[1,1,3]]
=> [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
[[1,1,1,3]]
=> [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
[[1,1,1],[3]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[1,1,2],[3]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[1,1,3],[2]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[1,1,3],[3]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[1,2,2],[3]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[[1,2,3],[3]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[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
[[2,2,2],[3]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[2,2,3],[3]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[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],[2,3]]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[1,2],[3,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,3}
[[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
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,1}
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1}
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5]
=> []
=> ? ∊ {1,1,1}
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 2
[[1,1,1,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,2,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,4,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,2,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2,4,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,3,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,1,2],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,1,4],[2]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,1,3],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,1,4],[3]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,1,4],[4]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,2,2],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,2,3],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,2,4],[4]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,3,3],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,3,4],[4]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[2,2,2],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[2,2,3],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[2,2,4],[3]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[2,2,4],[4]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[2,3,3],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[2,3,4],[4]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[3,3,3],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[3,3,4],[4]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,2],[2,4]]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[[1,3],[2,4]]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
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: St000704
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 49%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 49%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[2,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[3,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,1,1,2]]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,2],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,2],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,3],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,3],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition λ=(λ1>⋯>λk>0), the number of [[SemistandardTableaux|semistandard tableaux]] of shape λ with maximal entry k.
Equivalently, this is the evaluation sλ(1,…,1) of the Schur function sλ in k variables, or, explicitly,
∏(i,j)∈Lk+j−ihook(i,j)
where the product is over all cells (i,j)∈L and hook(i,j) is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000939
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 80%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1,1,1,2]]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,2],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,2],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,3],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,3],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000993
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 80%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1,1,1,2]]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,2],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,2],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,3],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,3],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001568
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 49%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 49%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1,1,1,2]]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,3],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,2],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,2],[3,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,3],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,2,3],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000675
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2}
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,2}
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,1,1,1,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,2],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,3],[2,2]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,2],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,3],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,3],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2,2],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2,2],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2,3],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2,3],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations D=ABC of a Dyck path, such that B is a Dyck path and A and B have the same length.
The following 92 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000260The radius of a connected graph. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000706The product of the factorials of the multiplicities of an integer partition. St000170The trace of a semistandard tableau. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given 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. St001609The number of coloured trees such that the multiplicities of colours 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. 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. St001410The minimal entry of a semistandard tableau. St000259The diameter of a connected graph. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001330The hat guessing number of a graph. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000456The monochromatic index of a connected graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St001877Number of indecomposable injective modules with projective dimension 2. 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. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. 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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