Your data matches 75 different statistics following compositions of up to 3 maps.
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Matching statistic: St000102
(load all 3 compositions to match this statistic)
Mapping
St000102: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 1
[[2,2]]
 => 0
[[1],[2]]
 => 0
[[1,1,2]]
 => 1
[[1,2,2]]
 => 1
[[2,2,2]]
 => 0
[[1,1],[2]]
 => 0
[[1,2],[2]]
 => 0
[[1,1,3]]
 => 1
[[1,2,3]]
 => 3
[[1,3,3]]
 => 1
[[2,2,3]]
 => 1
[[2,3,3]]
 => 1
[[3,3,3]]
 => 0
[[1,1],[3]]
 => 0
[[1,2],[3]]
 => 2
[[1,3],[2]]
 => 1
[[1,3],[3]]
 => 0
[[2,2],[3]]
 => 0
[[2,3],[3]]
 => 0
[[1],[2],[3]]
 => 0
[[1,1,1,2]]
 => 1
[[1,1,2,2]]
 => 2
[[1,2,2,2]]
 => 1
[[2,2,2,2]]
 => 0
[[1,1,1],[2]]
 => 0
[[1,1,2],[2]]
 => 1
[[1,2,2],[2]]
 => 0
[[1,1],[2,2]]
 => 0
[[1,1,1,3]]
 => 1
[[1,1,2,3]]
 => 3
[[1,1,3,3]]
 => 2
[[1,2,2,3]]
 => 3
[[1,2,3,3]]
 => 3
[[1,3,3,3]]
 => 1
[[2,2,2,3]]
 => 1
[[2,2,3,3]]
 => 2
[[2,3,3,3]]
 => 1
[[3,3,3,3]]
 => 0
[[1,1,1],[3]]
 => 0
[[1,1,2],[3]]
 => 2
[[1,1,3],[2]]
 => 1
[[1,1,3],[3]]
 => 1
[[1,2,2],[3]]
 => 2
[[1,2,3],[2]]
 => 1
[[1,2,3],[3]]
 => 2
[[1,3,3],[2]]
 => 1
[[1,3,3],[3]]
 => 0
[[2,2,2],[3]]
 => 0
[[2,2,3],[3]]
 => 1
Description
The charge of a semistandard tableau.
Matching statistic: St000259
(load all 3 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 60%●distinct values known / distinct values provided: 36%
Values
[[1,2]]
 => [1,2] => ([],2)
 => ([],1)
 => 0
[[2,2]]
 => [1,2] => ([],2)
 => ([],1)
 => 0
[[1],[2]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[[1,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[[2,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[[1,1],[2]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[[1,2],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1
[[1,1,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[[1,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[[1,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[[2,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[[2,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[[3,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[[1,1],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[[1,3],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,2,3}
[[1,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,2,3}
[[2,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[[2,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,2,3}
[[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[1,1,1],[2]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,1,2],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,2}
[[1,2,2],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,2}
[[1,1],[2,2]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
[[1,1,1,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[1,1,2,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[1,1,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[1,2,2,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[1,2,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[1,3,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[2,2,2,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[2,2,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[2,3,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[3,3,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[[1,1,1],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,1,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,1,3],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[1,1,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[1,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[[1,2,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[1,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[1,3,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[1,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[2,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[[2,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[2,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[1,1],[2,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
[[1,1],[3,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
[[1,2],[2,3]]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[[1,2],[3,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
[[2,2],[3,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
[[1,1],[2],[3]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,2],[2],[3]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,3],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[1,1,1,1,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[[1,1,1,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[[1,1,2,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[[1,2,2,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[[2,2,2,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[[1,1,1,1],[2]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[[1,1,1,2],[2]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,2,2}
[[1,1,2,2],[2]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,2,2}
[[1,2,2,2],[2]]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,2,2}
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,2,2}
[[1,1,4],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,3,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,4,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,3,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,4,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,4,4],[4]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[2,3,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[2,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[2,4,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[2,4,4],[4]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[3,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[3,4,4],[4]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,4],[2],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,4],[3],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[2,4],[3],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,1,3],[2]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[[1,1,1,3],[3]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[[1,1,2,3],[2]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[[1,1,2,3],[3]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[[1,1,3,3],[2]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[[1,1,3,3],[3]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[[1,2,2,3],[2]]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000777
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 60%●distinct values known / distinct values provided: 36%
Values
[[1,2]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[[2,2]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[[1,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[[2,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[2]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,2],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1 + 1
[[1,1,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[[1,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[[2,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[[2,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[[3,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,3],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,2,3} + 1
[[1,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,2,3} + 1
[[2,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[2,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,2,3} + 1
[[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,2],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,2} + 1
[[1,2,2],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,2} + 1
[[1,1],[2,2]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,1,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[1,1,2,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[1,1,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[1,2,2,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[1,2,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[1,3,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[2,2,2,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[2,2,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[2,3,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[3,3,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,3],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,3} + 1
[[1,1,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,3} + 1
[[1,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,2,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,3} + 1
[[1,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,3} + 1
[[1,3,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,3} + 1
[[1,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,3} + 1
[[2,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[2,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,3} + 1
[[2,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,3} + 1
[[1,1],[2,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1],[3,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,2],[2,3]]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 4 = 3 + 1
[[1,2],[3,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[2,2],[3,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1],[2],[3]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[2],[3]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[1,3],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,2,3} + 1
[[1,1,1,1,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[[1,1,2,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[[1,2,2,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[[2,2,2,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1],[2]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,1,2],[2]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,2,2} + 1
[[1,1,2,2],[2]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,2,2} + 1
[[1,2,2,2],[2]]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,2,2} + 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,2,2} + 1
[[1,1,4],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,1,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,1,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,2,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,3,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,4,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,3,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,4,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,4,4],[4]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[2,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[2,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[2,3,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[2,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[2,4,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[2,4,4],[4]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[3,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[3,4,4],[4]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,4],[2],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,4],[3],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[2,4],[3],[4]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,5,6} + 1
[[1,1,1,3],[2]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4} + 1
[[1,1,1,3],[3]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4} + 1
[[1,1,2,3],[2]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4} + 1
[[1,1,2,3],[3]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4} + 1
[[1,1,3,3],[2]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4} + 1
[[1,1,3,3],[3]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4} + 1
[[1,2,2,3],[2]]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4} + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000668
(load all 3 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 49%●distinct values known / distinct values provided: 27%
Values
[[1,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[2,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[1],[2]]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,1}
[[1,1,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[2,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,1],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,2],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,1,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[3,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[2,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[3,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,1],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,1,2]]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2,2}
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 2
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 3 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 49%●distinct values known / distinct values provided: 27%
Values
[[1,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[2,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[1],[2]]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,1}
[[1,1,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[2,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,1],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,2],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,1,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[3,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[2,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[3,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,1],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,1,2]]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2,2}
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 2
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
Description
The product of the parts of an integer partition.
Matching statistic: St000770
(load all 3 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 49%●distinct values known / distinct values provided: 45%
Values
[[1,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[2,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[1],[2]]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,1}
[[1,1,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[2,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,1],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,2],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,1,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[3,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[2,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[3,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,1],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,1,2]]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2,2}
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 4
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
Description
The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St000815
(load all 4 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 49%●distinct values known / distinct values provided: 36%
Values
[[1,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[2,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[1],[2]]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,1}
[[1,1,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[2,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,1],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,2],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,1,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[3,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[2,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[3,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,1],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,1,2]]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2,2}
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [3]
 => 3
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
Description
The number of semistandard Young tableaux of partition weight of given shape. The weight of a semistandard Young tableaux is the sequence $(m_1, m_2,\dots)$, where $m_i$ is the number of occurrences of the number $i$ in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence. Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.
Matching statistic: St000933
(load all 3 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 49%●distinct values known / distinct values provided: 27%
Values
[[1,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[2,2]]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[[1],[2]]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,1}
[[1,1,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[2,2,2]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[[1,1],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,2],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[[1,1,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[3,3,3]]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[2,2,2,2]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[3,3,3,3]]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,1],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[2]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,2],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,2,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[2,3,3],[3]]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3}
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1,1,2]]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2,2}
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 2
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,2],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,2,3],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
Description
The number of multipartitions of sizes given by an integer partition. This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
Matching statistic: St000454
(load all 12 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 49%●distinct values known / distinct values provided: 45%
Values
[[1,2]]
 => [1,2] => ([],2)
 => ([],2)
 => 0
[[2,2]]
 => [1,2] => ([],2)
 => ([],2)
 => 0
[[1],[2]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,2]]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[[1,2,2]]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[[2,2,2]]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[[1,1],[2]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? = 1
[[1,2],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[[1,1,3]]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[[1,2,3]]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[[1,3,3]]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[[2,2,3]]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[[2,3,3]]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[[3,3,3]]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[[1,1],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,3}
[[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,3}
[[1,3],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[[1,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[[2,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,3}
[[2,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[1,1,1],[2]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1}
[[1,1,2],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1}
[[1,2,2],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[[1,1],[2,2]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[1,1,1,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[1,1,2,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[1,1,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[1,2,2,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[1,2,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[1,3,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[2,2,2,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[2,2,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[2,3,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[3,3,3,3]]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[[1,1,1],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[1,1,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[1,1,3],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[1,1,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[1,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[1,2,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[[1,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[1,3,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[[1,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[[2,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[2,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[2,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[[1,1],[2,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[1,1],[3,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[1,2],[2,3]]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[1,2],[3,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[2,2],[3,3]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[1,1],[2],[3]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[1,2],[2],[3]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,3}
[[1,3],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[[1,1,1,1,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[[1,1,1,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[[1,1,2,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[[1,2,2,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[[2,2,2,2,2]]
 => [1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[[1,1,1,1],[2]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,1,1,2],[2]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1}
[[1,1,2,2],[2]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1}
[[1,2,2,2],[2]]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1}
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[1,1,1],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,4],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[3,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[3,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2],[2,4]]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3],[2,4]]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3],[3,4]]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,3],[3,4]]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1],[2],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2],[2],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3],[2],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3],[3],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000456
(load all 7 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 45%
Values
[[1,2]]
 => [1,2] => [1] => ([],1)
 => ? ∊ {0,0,1}
[[2,2]]
 => [1,2] => [1] => ([],1)
 => ? ∊ {0,0,1}
[[1],[2]]
 => [2,1] => [1] => ([],1)
 => ? ∊ {0,0,1}
[[1,1,2]]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {0,0,0,1}
[[1,2,2]]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {0,0,0,1}
[[2,2,2]]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {0,0,0,1}
[[1,1],[2]]
 => [3,1,2] => [1,2] => ([],2)
 => ? ∊ {0,0,0,1}
[[1,2],[2]]
 => [2,1,3] => [2,1] => ([(0,1)],2)
 => 1
[[1,1,3]]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,3}
[[1,2,3]]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,3}
[[1,3,3]]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,3}
[[2,2,3]]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,3}
[[2,3,3]]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,3}
[[3,3,3]]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,3}
[[1,1],[3]]
 => [3,1,2] => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,3}
[[1,2],[3]]
 => [3,1,2] => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,3}
[[1,3],[2]]
 => [2,1,3] => [2,1] => ([(0,1)],2)
 => 1
[[1,3],[3]]
 => [2,1,3] => [2,1] => ([(0,1)],2)
 => 1
[[2,2],[3]]
 => [3,1,2] => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,3}
[[2,3],[3]]
 => [2,1,3] => [2,1] => ([(0,1)],2)
 => 1
[[1],[2],[3]]
 => [3,2,1] => [2,1] => ([(0,1)],2)
 => 1
[[1,1,1,2]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,1,2}
[[1,1,2,2]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,1,2}
[[1,2,2,2]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,1,2}
[[2,2,2,2]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,1,2}
[[1,1,1],[2]]
 => [4,1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,1,2}
[[1,1,2],[2]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,2,2],[2]]
 => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,2}
[[1,1],[2,2]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,1,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,1,2,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,1,3,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,2,2,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,2,3,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,3,3,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[2,2,2,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[2,2,3,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[2,3,3,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[3,3,3,3]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,1,1],[3]]
 => [4,1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,1,2],[3]]
 => [4,1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,1,3],[2]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,3],[3]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,2,2],[3]]
 => [4,1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,2,3],[2]]
 => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,2,3],[3]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,3,3],[2]]
 => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,3,3],[3]]
 => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[2,2,2],[3]]
 => [4,1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[2,2,3],[3]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,3,3],[3]]
 => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,1],[2,3]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1],[3,3]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,2],[2,3]]
 => [2,4,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,2],[3,3]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,2],[3,3]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3}
[[1,3],[2],[3]]
 => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,1,1,2]]
 => [1,2,3,4,5] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,2}
[[1,1,1,2,2]]
 => [1,2,3,4,5] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,2}
[[1,1,2,2,2]]
 => [1,2,3,4,5] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,2}
[[1,2,2,2,2]]
 => [1,2,3,4,5] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,2}
[[2,2,2,2,2]]
 => [1,2,3,4,5] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,2}
[[1,1,1,1],[2]]
 => [5,1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,2}
[[1,1,1,2],[2]]
 => [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[1,1,2,2],[2]]
 => [3,1,2,4,5] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,2}
[[1,2,2,2],[2]]
 => [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,2}
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[1,1,4],[2]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,4],[3]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,4],[4]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,2,4],[3]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,2,4],[4]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,3,4],[4]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,2,4],[3]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,2,4],[4]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,3,4],[4]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[3,3,4],[4]]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1],[2,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1],[3,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1],[4,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,2],[4,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,3],[4,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,2],[3,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,2],[4,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,3],[4,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[3,3],[4,4]]
 => [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1],[3],[4]]
 => [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4],[2],[4]]
 => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4],[3],[4]]
 => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[2,2],[3],[4]]
 => [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,4],[3],[4]]
 => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,1,1,3],[2]]
 => [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
The following 65 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St001280The number of parts of an integer partition that are at least two. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001118The acyclic chromatic index of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001330The hat guessing number of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001060The distinguishing index of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001644The dimension of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. 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. 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. 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. St001128The exponens consonantiae of a partition. St000379The number of Hamiltonian cycles in a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001098The 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 vertex labelled trees. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.