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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2] => 2
[[2,2]]
=> [1,2] => [2] => 2
[[1],[2]]
=> [2,1] => [2] => 2
[[1,3]]
=> [1,2] => [2] => 2
[[2,3]]
=> [1,2] => [2] => 2
[[3,3]]
=> [1,2] => [2] => 2
[[1],[3]]
=> [2,1] => [2] => 2
[[2],[3]]
=> [2,1] => [2] => 2
[[1,1,2]]
=> [1,2,3] => [3] => 2
[[1,2,2]]
=> [1,2,3] => [3] => 2
[[2,2,2]]
=> [1,2,3] => [3] => 2
[[1,1],[2]]
=> [3,1,2] => [3] => 2
[[1,2],[2]]
=> [2,1,3] => [3] => 2
[[1,4]]
=> [1,2] => [2] => 2
[[2,4]]
=> [1,2] => [2] => 2
[[3,4]]
=> [1,2] => [2] => 2
[[4,4]]
=> [1,2] => [2] => 2
[[1],[4]]
=> [2,1] => [2] => 2
[[2],[4]]
=> [2,1] => [2] => 2
[[3],[4]]
=> [2,1] => [2] => 2
[[1,1,3]]
=> [1,2,3] => [3] => 2
[[1,2,3]]
=> [1,2,3] => [3] => 2
[[1,3,3]]
=> [1,2,3] => [3] => 2
[[2,2,3]]
=> [1,2,3] => [3] => 2
[[2,3,3]]
=> [1,2,3] => [3] => 2
[[3,3,3]]
=> [1,2,3] => [3] => 2
[[1,1],[3]]
=> [3,1,2] => [3] => 2
[[1,2],[3]]
=> [3,1,2] => [3] => 2
[[1,3],[2]]
=> [2,1,3] => [3] => 2
[[1,3],[3]]
=> [2,1,3] => [3] => 2
[[2,2],[3]]
=> [3,1,2] => [3] => 2
[[2,3],[3]]
=> [2,1,3] => [3] => 2
[[1],[2],[3]]
=> [3,2,1] => [2,1] => 3
[[1,1,1,2]]
=> [1,2,3,4] => [4] => 2
[[1,1,2,2]]
=> [1,2,3,4] => [4] => 2
[[1,2,2,2]]
=> [1,2,3,4] => [4] => 2
[[2,2,2,2]]
=> [1,2,3,4] => [4] => 2
[[1,1,1],[2]]
=> [4,1,2,3] => [4] => 2
[[1,1,2],[2]]
=> [3,1,2,4] => [4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => [4] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [4] => 2
[[1,5]]
=> [1,2] => [2] => 2
[[2,5]]
=> [1,2] => [2] => 2
[[3,5]]
=> [1,2] => [2] => 2
[[4,5]]
=> [1,2] => [2] => 2
[[5,5]]
=> [1,2] => [2] => 2
[[1],[5]]
=> [2,1] => [2] => 2
[[2],[5]]
=> [2,1] => [2] => 2
[[3],[5]]
=> [2,1] => [2] => 2
[[4],[5]]
=> [2,1] => [2] => 2
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St001352
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[2,2]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[1],[2]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[2,3]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[3,3]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[1],[3]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[[1,2,2]]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[[2,2,2]]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[[1,1],[2]]
=> [3,1,2] => [3] => ([],3)
=> 1 = 2 - 1
[[1,2],[2]]
=> [2,1,3] => [3] => ([],3)
=> 1 = 2 - 1
[[1,4]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[2,4]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[3,4]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[4,4]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[1],[4]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[[1,2,3]]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[[1,3,3]]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[[2,2,3]]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[[2,3,3]]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[[3,3,3]]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[[1,1],[3]]
=> [3,1,2] => [3] => ([],3)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [3] => ([],3)
=> 1 = 2 - 1
[[1,3],[2]]
=> [2,1,3] => [3] => ([],3)
=> 1 = 2 - 1
[[1,3],[3]]
=> [2,1,3] => [3] => ([],3)
=> 1 = 2 - 1
[[2,2],[3]]
=> [3,1,2] => [3] => ([],3)
=> 1 = 2 - 1
[[2,3],[3]]
=> [2,1,3] => [3] => ([],3)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [4] => ([],4)
=> 1 = 2 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [4] => ([],4)
=> 1 = 2 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [4] => ([],4)
=> 1 = 2 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [4] => ([],4)
=> 1 = 2 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4] => ([],4)
=> 1 = 2 - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [4] => ([],4)
=> 1 = 2 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [4] => ([],4)
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4] => ([],4)
=> 1 = 2 - 1
[[1,5]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[2,5]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[3,5]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[4,5]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[5,5]]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[[1],[5]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[[1,1,1,1,1,1,1,2]]
=> [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,1,1,2,2]]
=> [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,1,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[2,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,1,1,1],[2]]
=> [8,1,2,3,4,5,6,7] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,1,1,2],[2]]
=> [7,1,2,3,4,5,6,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,1,2,2],[2]]
=> [6,1,2,3,4,5,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,2,2,2],[2]]
=> [5,1,2,3,4,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,2,2,2,2],[2]]
=> [4,1,2,3,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,2,2,2,2,2],[2]]
=> [3,1,2,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,2,2,2,2,2,2],[2]]
=> [2,1,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,1,1],[2,2]]
=> [7,8,1,2,3,4,5,6] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,1,2],[2,2]]
=> [6,7,1,2,3,4,5,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,2,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,1],[2,2,2]]
=> [6,7,8,1,2,3,4,5] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1,2],[2,2,2]]
=> [5,6,7,1,2,3,4,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,2,2],[2,2,2]]
=> [4,5,6,1,2,3,7,8] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1],[2,2,2,2]]
=> [5,6,7,8,1,2,3,4] => [8] => ([],8)
=> ? = 2 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [1,2,7] => ([(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [1,5,4] => ([(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [1,2,7] => ([(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [1,2,7] => ([(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [1,5,4] => ([(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [1,2,7] => ([(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [1,2,7] => ([(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [1,4,5] => ([(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [1,2,7] => ([(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => [1,3,6] => ([(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => [1,3,6] => ([(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => [1,3,6] => ([(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => [1,3,6] => ([(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => [1,2,7] => ([(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => [1,3,6] => ([(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
Description
The number of internal nodes in the modular decomposition of a graph.
Matching statistic: St001488
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[2,2]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[1],[2]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[1,3]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[2,3]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[3,3]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[1],[3]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[2],[3]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[1,1,2]]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[[1,2,2]]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[[2,2,2]]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[[1,1],[2]]
=> [3,1,2] => [3] => [[3],[]]
=> 2
[[1,2],[2]]
=> [2,1,3] => [3] => [[3],[]]
=> 2
[[1,4]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[2,4]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[3,4]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[4,4]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[1],[4]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[2],[4]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[3],[4]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[1,1,3]]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[[1,2,3]]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[[1,3,3]]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[[2,2,3]]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[[2,3,3]]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[[3,3,3]]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[[1,1],[3]]
=> [3,1,2] => [3] => [[3],[]]
=> 2
[[1,2],[3]]
=> [3,1,2] => [3] => [[3],[]]
=> 2
[[1,3],[2]]
=> [2,1,3] => [3] => [[3],[]]
=> 2
[[1,3],[3]]
=> [2,1,3] => [3] => [[3],[]]
=> 2
[[2,2],[3]]
=> [3,1,2] => [3] => [[3],[]]
=> 2
[[2,3],[3]]
=> [2,1,3] => [3] => [[3],[]]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [2,1] => [[2,2],[1]]
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [4] => [[4],[]]
=> 2
[[1,1,2,2]]
=> [1,2,3,4] => [4] => [[4],[]]
=> 2
[[1,2,2,2]]
=> [1,2,3,4] => [4] => [[4],[]]
=> 2
[[2,2,2,2]]
=> [1,2,3,4] => [4] => [[4],[]]
=> 2
[[1,1,1],[2]]
=> [4,1,2,3] => [4] => [[4],[]]
=> 2
[[1,1,2],[2]]
=> [3,1,2,4] => [4] => [[4],[]]
=> 2
[[1,2,2],[2]]
=> [2,1,3,4] => [4] => [[4],[]]
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [4] => [[4],[]]
=> 2
[[1,5]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[2,5]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[3,5]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[4,5]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[5,5]]
=> [1,2] => [2] => [[2],[]]
=> 2
[[1],[5]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[2],[5]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[3],[5]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[4],[5]]
=> [2,1] => [2] => [[2],[]]
=> 2
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,2],[2]]
=> [5,1,2,3,4,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,1,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,2,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,2,2,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,2,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,2,2,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,2,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,3,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,2,2,2,2,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,2,2,2,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,2,2,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,2,3,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[2,2,2,2,2,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[2,2,2,2,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[2,2,2,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[2,2,3,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[2,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[3,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,1],[3]]
=> [6,1,2,3,4,5] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,2],[3]]
=> [6,1,2,3,4,5] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,3],[2]]
=> [5,1,2,3,4,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,1,3],[3]]
=> [5,1,2,3,4,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,2,3],[2]]
=> [4,1,2,3,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,2,3],[3]]
=> [5,1,2,3,4,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,3,3],[2]]
=> [4,1,2,3,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,1,3,3],[3]]
=> [4,1,2,3,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,2,3],[2]]
=> [3,1,2,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,2,3],[3]]
=> [5,1,2,3,4,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,3,3],[2]]
=> [3,1,2,4,5,6] => [6] => [[6],[]]
=> ? = 2
[[1,1,2,3,3],[3]]
=> [4,1,2,3,5,6] => [6] => [[6],[]]
=> ? = 2
Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
Matching statistic: St001491
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[2,2]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[1],[2]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[1,3]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[2,3]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[3,3]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[1],[3]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[2],[3]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[1,1,2]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[1,2,2]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[2,2,2]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[1,1],[2]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,2],[2]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,4]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[2,4]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[3,4]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[4,4]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[1],[4]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[2],[4]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[3],[4]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[1,1,3]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[1,2,3]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[1,3,3]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[2,2,3]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[2,3,3]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[3,3,3]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[1,1],[3]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,2],[3]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,3],[2]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,3],[3]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[2,2],[3]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[2,3],[3]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1],[2],[3]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[1,1,1,2]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[1,1,2,2]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[1,2,2,2]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[2,2,2,2]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[1,1,1],[2]]
=> [3,1]
=> 10010 => 00010 => ? = 2 - 1
[[1,1,2],[2]]
=> [3,1]
=> 10010 => 00010 => ? = 2 - 1
[[1,2,2],[2]]
=> [3,1]
=> 10010 => 00010 => ? = 2 - 1
[[1,1],[2,2]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[1,5]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[2,5]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[3,5]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[4,5]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[5,5]]
=> [2]
=> 100 => 000 => ? = 2 - 1
[[1],[5]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[2],[5]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[3],[5]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[4],[5]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[1,1,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[1,2,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[1,3,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[1,4,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[2,2,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[2,3,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[2,4,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[3,3,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[3,4,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[4,4,4]]
=> [3]
=> 1000 => 0000 => ? = 2 - 1
[[1,1],[4]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,2],[4]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,4],[2]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,3],[4]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,4],[3]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,4],[4]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[2,2],[4]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[2,3],[4]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[2,4],[3]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[2,4],[4]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[3,3],[4]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[3,4],[4]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1],[2],[4]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[1],[3],[4]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[2],[3],[4]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[1,1,1,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[1,1,2,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[1,1,3,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[1,2,2,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[1,2,3,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[1,3,3,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[2,2,2,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[2,2,3,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[2,3,3,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[3,3,3,3]]
=> [4]
=> 10000 => 00000 => ? = 2 - 1
[[1,1],[2,3]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[1,1],[3,3]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[1,2],[2,3]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[1,2],[3,3]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[2,2],[3,3]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[1],[6]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[2],[6]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[3],[6]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[4],[6]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[5],[6]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[1,1],[5]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,2],[5]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,5],[2]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,3],[5]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[1,5],[3]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000782
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 20%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 20%
Values
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[1,3]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[2,3]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[1,1,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[1,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[2,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[3,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[1,1,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[1,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[2,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[2,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[[1,1,1,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[1,1,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[1,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,1,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2 - 1
[[1,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[2,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[3,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[4,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[[1,1,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[1,2,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[1,3,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[1,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[2,2,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[2,3,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[2,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[3,3,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[3,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[4,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[[1,1],[4]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,2],[4]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,4],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,3],[4]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,4],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1,4],[4]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[2,2],[4]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[2,3],[4]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[2,4],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[2,4],[4]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[3,3],[4]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[3,4],[4]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[[1,1,1,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[1,1,2,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[1,1,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[1,2,2,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[1,2,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[1,3,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[2,2,2,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[2,2,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[2,3,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[3,3,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 - 1
[[1,1,1],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,1,2],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,1,3],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,1,3],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,2,2],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,2,3],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,2,3],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,3,3],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,3,3],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[[1,6]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[2,6]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[3,6]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[4,6]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[5,6]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[[6,6]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St000739
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000739: Semistandard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000739: Semistandard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[2]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,1,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,1],[2]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,2],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[4,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[3],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,1,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[3,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,1],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,3],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1,3],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[2,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[2,3],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1],[2],[3]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 3
[[1,1,1,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[[1,1,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[[1,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[[2,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[[1,1,1],[2]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ? = 2
[[1,1,2],[2]]
=> [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ? = 2
[[1,2,2],[2]]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 2
[[1,1],[2,2]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 2
[[1,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[4,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[5,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[3],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[4],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,1,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[3,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[3,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[4,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,1],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,4],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,4],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[2,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[2,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[2,4],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[2,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[3,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[3,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1],[2],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 3
[[1],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 3
[[1,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[4,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[5,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[6,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[3],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[4],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[5],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[4,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[5,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[6,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[7,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[3],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[4],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[5],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[6],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
Description
The first entry in the last row of a semistandard tableau.
Matching statistic: St001401
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St001401: Semistandard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St001401: Semistandard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[2]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,1,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,1],[2]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,2],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[4,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[3],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,1,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[3,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,1],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,3],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1,3],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[2,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[2,3],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1],[2],[3]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 3
[[1,1,1,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[[1,1,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[[1,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[[2,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2
[[1,1,1],[2]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ? = 2
[[1,1,2],[2]]
=> [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ? = 2
[[1,2,2],[2]]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 2
[[1,1],[2,2]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 2
[[1,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[4,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[5,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[3],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[4],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,1,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[2,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[3,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[3,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[4,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
[[1,1],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,4],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,4],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[2,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[2,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[2,4],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[2,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[3,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[3,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
[[1],[2],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 3
[[1],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 3
[[1,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[4,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[5,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[6,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[3],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[4],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[5],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[3,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[4,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[5,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[6,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[7,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[1],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[2],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[3],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[4],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[5],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[6],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
[[1,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
[[2,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 2
Description
The number of distinct entries in a semistandard tableau.
Matching statistic: St001637
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,2]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[2]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,3]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,3]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,3]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1,2]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,2,2]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,2,2]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,1],[2]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,2],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,2,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,3,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,2,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,3,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,1],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,3],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,3],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,3],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[1,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,2,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,3,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,2,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,3,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[3,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[4,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,1],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,2],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,4],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,3],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,4],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,4],[4]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,2],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,3],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,4],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,4],[4]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[3,3],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[3,4],[4]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1],[2],[4]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[[1],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[[1,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[6,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[6,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[7,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[6],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,8]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,8]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,2]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[2]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,3]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,3]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,3]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1,2]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,2,2]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,2,2]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,1],[2]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,2],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,2,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,3,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,2,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,3,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,1],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,3],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,3],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,3],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[1,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,2,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,3,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,2,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,3,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[2,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[3,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[4,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[[1,1],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,2],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,4],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,3],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,4],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1,4],[4]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,2],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,3],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,4],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[2,4],[4]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[3,3],[4]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[3,4],[4]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[[1],[2],[4]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[[1],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3
[[1,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[6,6]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[6,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[7,7]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[5],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[6],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,8]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2,8]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St000101
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000101: Semistandard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000101: Semistandard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[2,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[1],[2]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[2,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[3,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[1],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[1,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[2,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[1,1],[2]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[1,2],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[1,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[2,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[3,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[4,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[1],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[1,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[1,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[2,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[2,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[3,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[1,1],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[1,3],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[1,3],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[2,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[2,3],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 3 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 2 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ? = 2 - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ? = 2 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 2 - 1
[[1,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[2,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[3,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[4,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[5,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[1],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[1,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[1,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[1,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[2,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[2,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[2,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[3,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[3,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[4,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2 - 1
[[1,1],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[1,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[1,4],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[1,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[1,4],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[1,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[2,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[2,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[2,4],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[2,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[3,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2 - 1
[[3,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2 - 1
[[1],[2],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 3 - 1
[[1],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 3 - 1
[[1,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[2,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[3,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[4,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[5,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[6,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[1],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[2],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[3],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[4],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[5],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[1,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[2,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[3,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[4,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[5,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[6,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[7,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[1],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[2],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[3],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[4],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[5],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[6],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1 = 2 - 1
[[1,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
[[2,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1 = 2 - 1
Description
The cocharge of a semistandard tableau.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001409The maximal entry of a semistandard tableau. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph.
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