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Your data matches 85 different statistics following compositions of up to 3 maps.
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Matching statistic: St000170
St000170: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 1
[[2,2]]
=> 2
[[1],[2]]
=> 1
[[1,3]]
=> 1
[[2,3]]
=> 2
[[3,3]]
=> 3
[[1],[3]]
=> 1
[[2],[3]]
=> 2
[[1,1,2]]
=> 1
[[1,2,2]]
=> 1
[[2,2,2]]
=> 2
[[1,1],[2]]
=> 1
[[1,2],[2]]
=> 1
[[1,4]]
=> 1
[[2,4]]
=> 2
[[3,4]]
=> 3
[[4,4]]
=> 4
[[1],[4]]
=> 1
[[2],[4]]
=> 2
[[3],[4]]
=> 3
[[1,1,3]]
=> 1
[[1,2,3]]
=> 1
[[1,3,3]]
=> 1
[[2,2,3]]
=> 2
[[2,3,3]]
=> 2
[[3,3,3]]
=> 3
[[1,1],[3]]
=> 1
[[1,2],[3]]
=> 1
[[1,3],[2]]
=> 1
[[1,3],[3]]
=> 1
[[2,2],[3]]
=> 2
[[2,3],[3]]
=> 2
[[1],[2],[3]]
=> 1
[[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> 1
[[1,2,2,2]]
=> 1
[[2,2,2,2]]
=> 2
[[1,1,1],[2]]
=> 1
[[1,1,2],[2]]
=> 1
[[1,2,2],[2]]
=> 1
[[1,1],[2,2]]
=> 3
[[1,5]]
=> 1
[[2,5]]
=> 2
[[3,5]]
=> 3
[[4,5]]
=> 4
[[5,5]]
=> 5
[[1],[5]]
=> 1
[[2],[5]]
=> 2
[[3],[5]]
=> 3
[[4],[5]]
=> 4
Description
The trace of a semistandard tableau.
This is the sum of the entries on the diagonal.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 57%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 57%
Values
[[1,2]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2}
[[2,2]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2}
[[1],[2]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2}
[[1,3]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2}
[[1,2,2]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2}
[[2,2,2]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2}
[[1,1],[2]]
=> [3,1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2}
[[1,2],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,4]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [3,1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [3,1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2,2,3}
[[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)
=> ? ∊ {1,1,1,1,2,2,2,2,3}
[[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)
=> ? ∊ {1,1,1,1,2,3}
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,3}
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,3}
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,3}
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,3}
[[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)
=> ? ∊ {1,1,1,1,2,3}
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,3,4]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4,4]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[2,2,4]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[2,3,4]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[2,4,4]]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,4],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,4],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,4],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,4],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,4],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[3,4],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[4]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[3],[4]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3],[4]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[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
[[1,2,3],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2,2,3],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[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],[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
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[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
[[1,5],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,5],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,5],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,5],[5]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,5],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,5],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,5],[5]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[3,5],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[3,5],[5]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[4,5],[5]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[3],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[4],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[4],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[4],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[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
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.
Matching statistic: St000284
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[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}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[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]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[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],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 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
Description
The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000704
(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
St000704: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[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}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[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]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[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],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 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
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000706
(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
St000706: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 43%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[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}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[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]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[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],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[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]
=> 6
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The product of the factorials of the multiplicities of an integer partition.
Matching statistic: St000939
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 43%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[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}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[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]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[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],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[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
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000993
(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
St000993: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 43%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[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}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[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]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[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],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[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
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001128
(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
St001128: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[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}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[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]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[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],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 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
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: St001568
(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
St001568: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[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}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[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]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[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],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[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
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000668
(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
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 43%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[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]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[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]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,3}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,3}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[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],[2,2]]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[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
Description
The least common multiple of the parts of the partition.
The following 75 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000675The number of centered multitunnels of a Dyck path. St000708The product of the parts of an integer partition. 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. 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. St000933The number of multipartitions of sizes given by an integer partition. 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. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001943The sum of the squares of the hook lengths of an integer partition. St001877Number of indecomposable injective modules with projective dimension 2. St001118The acyclic chromatic index of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000464The Schultz index of a connected graph. 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. St001281The normalized isoperimetric number of a graph. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001545The second Elser number of a connected graph. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001592The maximal number of simple paths between any two different vertices of a graph. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. 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.
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