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Your data matches 54 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
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Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,4],[2]]
=> [[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,4],[3]]
=> [[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2,4],[3]]
=> [[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[4]]
=> [[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,5],[2]]
=> [[1,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,5],[3]]
=> [[1,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,5],[4]]
=> [[1,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2,5],[3]]
=> [[2,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2,5],[4]]
=> [[2,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[3,5],[4]]
=> [[3,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[5]]
=> [[1,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,6],[2]]
=> [[1,2],[6]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,6],[3]]
=> [[1,3],[6]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000659
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,4],[2]]
=> [[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,5],[2]]
=> [[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[1,6],[2]]
=> [[1,2],[6]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,6],[3]]
=> [[1,3],[6]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1]]
=> [[1]]
=> [1] => [1,0]
=> ? = 0 - 1
[[2]]
=> [[2]]
=> [1] => [1,0]
=> ? = 0 - 1
[[3]]
=> [[3]]
=> [1] => [1,0]
=> ? = 0 - 1
[[4]]
=> [[4]]
=> [1] => [1,0]
=> ? = 0 - 1
[[5]]
=> [[5]]
=> [1] => [1,0]
=> ? = 0 - 1
[[6]]
=> [[6]]
=> [1] => [1,0]
=> ? = 0 - 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000989
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1,4],[2]]
=> [[1,2],[4]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1,4],[3]]
=> [[1,3],[4]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[2,4],[3]]
=> [[2,3],[4]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1],[2],[4]]
=> [[1,2],[4]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,5],[2]]
=> [[1,2],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1,5],[3]]
=> [[1,3],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1,5],[4]]
=> [[1,4],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[2,5],[3]]
=> [[2,3],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[2,5],[4]]
=> [[2,4],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[3,5],[4]]
=> [[3,4],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1],[2],[5]]
=> [[1,2],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 0 = 2 - 2
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 0 = 2 - 2
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[[1,6],[2]]
=> [[1,2],[6]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1,6],[3]]
=> [[1,3],[6]]
=> [3,1,2] => [1,3,2] => 0 = 2 - 2
[[1]]
=> [[1]]
=> [1] => [1] => ? = 0 - 2
[[2]]
=> [[2]]
=> [1] => [1] => ? = 0 - 2
[[3]]
=> [[3]]
=> [1] => [1] => ? = 0 - 2
[[4]]
=> [[4]]
=> [1] => [1] => ? = 0 - 2
[[5]]
=> [[5]]
=> [1] => [1] => ? = 0 - 2
[[6]]
=> [[6]]
=> [1] => [1] => ? = 0 - 2
Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 90%●distinct values known / distinct values provided: 67%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 90%●distinct values known / distinct values provided: 67%
Values
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1,4],[2]]
=> [[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1,4],[3]]
=> [[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[2,4],[3]]
=> [[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1],[2],[4]]
=> [[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,5],[2]]
=> [[1,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1,5],[3]]
=> [[1,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1,5],[4]]
=> [[1,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[2,5],[3]]
=> [[2,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[2,5],[4]]
=> [[2,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[3,5],[4]]
=> [[3,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1],[2],[5]]
=> [[1,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,6],[2]]
=> [[1,2],[6]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1,6],[3]]
=> [[1,3],[6]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,4,5],[2],[4]]
=> [[1,2,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,4,5],[3],[4]]
=> [[1,3,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[2,4,5],[3],[4]]
=> [[2,3,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[[1,3],[2],[3],[5]]
=> [[1,2,3],[3],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,4],[2],[4],[5]]
=> [[1,2,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,4],[3],[4],[5]]
=> [[1,3,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[2,4],[3],[4],[5]]
=> [[2,3,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,1,3,4],[2,3]]
=> [[1,1,2,3],[3,4]]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 2
[[1,2,3,4],[2,3]]
=> [[1,2,2,3],[3,4]]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 2
[[1,1,3,4],[2],[3]]
=> [[1,1,2,3],[3],[4]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,3,4],[2],[3]]
=> [[1,2,2,3],[3],[4]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,3,4],[2],[3]]
=> [[1,2,3,3],[3],[4]]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,1,3],[2,3],[4]]
=> [[1,1,2,3],[3,4]]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 2
[[1,1,4],[2,3],[3]]
=> [[1,1,2,3],[3],[4]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,3],[2,3],[4]]
=> [[1,2,2,3],[3,4]]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 2
[[1,2,4],[2,3],[3]]
=> [[1,2,2,3],[3],[4]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,1,3],[2],[3],[4]]
=> [[1,1,2,3],[3],[4]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,3],[2],[3],[4]]
=> [[1,2,2,3],[3],[4]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,3],[2],[3],[4]]
=> [[1,2,3,3],[3],[4]]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,1],[2,3],[3,4]]
=> [[1,1,2,3],[3,4]]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 2
[[1,2],[2,3],[3,4]]
=> [[1,2,2,3],[3,4]]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 2
[[1,1],[2,3],[3],[4]]
=> [[1,1,2,3],[3],[4]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2],[2,3],[3],[4]]
=> [[1,2,2,3],[3],[4]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1]]
=> [[1]]
=> [1] => ([],1)
=> ? = 0 - 2
[[2]]
=> [[2]]
=> [1] => ([],1)
=> ? = 0 - 2
[[3]]
=> [[3]]
=> [1] => ([],1)
=> ? = 0 - 2
[[4]]
=> [[4]]
=> [1] => ([],1)
=> ? = 0 - 2
[[5]]
=> [[5]]
=> [1] => ([],1)
=> ? = 0 - 2
[[6]]
=> [[6]]
=> [1] => ([],1)
=> ? = 0 - 2
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 2
[[1,2,4],[3,5],[6]]
=> [[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 2
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,4],[3],[5],[6]]
=> [[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,4],[2,5],[6]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,6],[2,4],[5]]
=> [[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3],[2,4],[5],[6]]
=> [[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,4],[2],[5],[6]]
=> [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,6],[2],[4],[5]]
=> [[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3],[2],[4],[5],[6]]
=> [[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,4,6],[2],[3],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,4],[2],[3],[5],[6]]
=> [[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001811
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 2 - 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 2 - 2
[[1,4],[2]]
=> [[1,2],[4]]
=> [3,1,2] => 0 = 2 - 2
[[1,4],[3]]
=> [[1,3],[4]]
=> [3,1,2] => 0 = 2 - 2
[[2,4],[3]]
=> [[2,3],[4]]
=> [3,1,2] => 0 = 2 - 2
[[1],[2],[4]]
=> [[1,2],[4]]
=> [3,1,2] => 0 = 2 - 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [3,1,2] => 0 = 2 - 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [3,1,2] => 0 = 2 - 2
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,5],[2]]
=> [[1,2],[5]]
=> [3,1,2] => 0 = 2 - 2
[[1,5],[3]]
=> [[1,3],[5]]
=> [3,1,2] => 0 = 2 - 2
[[1,5],[4]]
=> [[1,4],[5]]
=> [3,1,2] => 0 = 2 - 2
[[2,5],[3]]
=> [[2,3],[5]]
=> [3,1,2] => 0 = 2 - 2
[[2,5],[4]]
=> [[2,4],[5]]
=> [3,1,2] => 0 = 2 - 2
[[3,5],[4]]
=> [[3,4],[5]]
=> [3,1,2] => 0 = 2 - 2
[[1],[2],[5]]
=> [[1,2],[5]]
=> [3,1,2] => 0 = 2 - 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [3,1,2] => 0 = 2 - 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [3,1,2] => 0 = 2 - 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [3,1,2] => 0 = 2 - 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [3,1,2] => 0 = 2 - 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [3,1,2] => 0 = 2 - 2
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 2 - 2
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => 0 = 2 - 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 2 - 2
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => 0 = 2 - 2
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => 0 = 2 - 2
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => 0 = 2 - 2
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => 0 = 2 - 2
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => 0 = 2 - 2
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => 0 = 2 - 2
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => 0 = 2 - 2
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => 0 = 2 - 2
[[1,6],[2]]
=> [[1,2],[6]]
=> [3,1,2] => 0 = 2 - 2
[[1,6],[3]]
=> [[1,3],[6]]
=> [3,1,2] => 0 = 2 - 2
[[1,1,1,1,3],[2]]
=> [[1,1,1,1,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,2,3],[2]]
=> [[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,2,2,3],[2]]
=> [[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,2,2,2,3],[2]]
=> [[1,2,2,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,3],[2,2]]
=> [[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,2,3],[2,2]]
=> [[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => ? = 2 - 2
[[1,1,1,1],[2],[3]]
=> [[1,1,1,1,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,2],[2],[3]]
=> [[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,2,2],[2],[3]]
=> [[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,2,2,2],[2],[3]]
=> [[1,2,2,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1],[2,2],[3]]
=> [[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,2],[2,2],[3]]
=> [[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => ? = 2 - 2
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => ? = 2 - 2
[[1,1,1,1,4],[2]]
=> [[1,1,1,1,2],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,1,4],[3]]
=> [[1,1,1,1,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,2,4],[2]]
=> [[1,1,1,2,2],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,2,4],[3]]
=> [[1,1,1,2,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,3,4],[3]]
=> [[1,1,1,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,2,2,4],[2]]
=> [[1,1,2,2,2],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,2,2,4],[3]]
=> [[1,1,2,2,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,2,3,4],[3]]
=> [[1,1,2,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,3,3,4],[3]]
=> [[1,1,3,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,2,2,2,4],[2]]
=> [[1,2,2,2,2],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,2,2,2,4],[3]]
=> [[1,2,2,2,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,2,2,3,4],[3]]
=> [[1,2,2,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,2,3,3,4],[3]]
=> [[1,2,3,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,3,3,3,4],[3]]
=> [[1,3,3,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[2,2,2,2,4],[3]]
=> [[2,2,2,2,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[2,2,2,3,4],[3]]
=> [[2,2,2,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[2,2,3,3,4],[3]]
=> [[2,2,3,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[2,3,3,3,4],[3]]
=> [[2,3,3,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,4],[2,2]]
=> [[1,1,1,2,2],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,4],[2,3]]
=> [[1,1,1,2,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,1,4],[3,3]]
=> [[1,1,1,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,2,4],[2,2]]
=> [[1,1,2,2,2],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,2,4],[2,3]]
=> [[1,1,2,2,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,3,4],[2,2]]
=> [[1,1,2,2],[3,4]]
=> [5,6,1,2,3,4] => ? = 2 - 2
[[1,1,4,4],[2,2]]
=> [[1,1,2,2],[4,4]]
=> [5,6,1,2,3,4] => ? = 2 - 2
[[1,1,2,4],[3,3]]
=> [[1,1,2,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,3,4],[2,3]]
=> [[1,1,2,3],[3,4]]
=> [4,6,1,2,3,5] => ? = 3 - 2
[[1,1,4,4],[2,3]]
=> [[1,1,2,3],[4,4]]
=> [5,6,1,2,3,4] => ? = 2 - 2
[[1,1,3,4],[3,3]]
=> [[1,1,3,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,1,4,4],[3,3]]
=> [[1,1,3,3],[4,4]]
=> [5,6,1,2,3,4] => ? = 2 - 2
[[1,2,2,4],[2,3]]
=> [[1,2,2,2,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,2,2,4],[3,3]]
=> [[1,2,2,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
[[1,2,3,4],[2,3]]
=> [[1,2,2,3],[3,4]]
=> [4,6,1,2,3,5] => ? = 3 - 2
[[1,2,4,4],[2,3]]
=> [[1,2,2,3],[4,4]]
=> [5,6,1,2,3,4] => ? = 2 - 2
[[1,2,3,4],[3,3]]
=> [[1,2,3,3,3],[4]]
=> [6,1,2,3,4,5] => ? = 2 - 2
Description
The Castelnuovo-Mumford regularity of a permutation.
The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$.
Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St001948
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1,4],[2]]
=> [[1,2],[4]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1,4],[3]]
=> [[1,3],[4]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[2,4],[3]]
=> [[2,3],[4]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1],[2],[4]]
=> [[1,2],[4]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,5],[2]]
=> [[1,2],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1,5],[3]]
=> [[1,3],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1,5],[4]]
=> [[1,4],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[2,5],[3]]
=> [[2,3],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[2,5],[4]]
=> [[2,4],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[3,5],[4]]
=> [[3,4],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1],[2],[5]]
=> [[1,2],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 0 = 2 - 2
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 0 = 2 - 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 0 = 2 - 2
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0 = 2 - 2
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0 = 2 - 2
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0 = 2 - 2
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0 = 2 - 2
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0 = 2 - 2
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0 = 2 - 2
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0 = 2 - 2
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0 = 2 - 2
[[1,6],[2]]
=> [[1,2],[6]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1,6],[3]]
=> [[1,3],[6]]
=> [3,1,2] => [3,1,2] => 0 = 2 - 2
[[1,1,1,1,3],[2]]
=> [[1,1,1,1,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,2,3],[2]]
=> [[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,2,2,3],[2]]
=> [[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,2,2,2,3],[2]]
=> [[1,2,2,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,3],[2,2]]
=> [[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,2,3],[2,2]]
=> [[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => ? = 2 - 2
[[1,1,1,1],[2],[3]]
=> [[1,1,1,1,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,2],[2],[3]]
=> [[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,2,2],[2],[3]]
=> [[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,2,2,2],[2],[3]]
=> [[1,2,2,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1],[2,2],[3]]
=> [[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,2],[2,2],[3]]
=> [[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => ? = 2 - 2
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => ? = 2 - 2
[[1,1,1,1,4],[2]]
=> [[1,1,1,1,2],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,1,4],[3]]
=> [[1,1,1,1,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,2,4],[2]]
=> [[1,1,1,2,2],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,2,4],[3]]
=> [[1,1,1,2,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,3,4],[3]]
=> [[1,1,1,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,2,2,4],[2]]
=> [[1,1,2,2,2],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,2,2,4],[3]]
=> [[1,1,2,2,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,2,3,4],[3]]
=> [[1,1,2,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,3,3,4],[3]]
=> [[1,1,3,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,2,2,2,4],[2]]
=> [[1,2,2,2,2],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,2,2,2,4],[3]]
=> [[1,2,2,2,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,2,2,3,4],[3]]
=> [[1,2,2,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,2,3,3,4],[3]]
=> [[1,2,3,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,3,3,3,4],[3]]
=> [[1,3,3,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[2,2,2,2,4],[3]]
=> [[2,2,2,2,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[2,2,2,3,4],[3]]
=> [[2,2,2,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[2,2,3,3,4],[3]]
=> [[2,2,3,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[2,3,3,3,4],[3]]
=> [[2,3,3,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,4],[2,2]]
=> [[1,1,1,2,2],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,4],[2,3]]
=> [[1,1,1,2,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,1,4],[3,3]]
=> [[1,1,1,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,2,4],[2,2]]
=> [[1,1,2,2,2],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,2,4],[2,3]]
=> [[1,1,2,2,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,3,4],[2,2]]
=> [[1,1,2,2],[3,4]]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => ? = 2 - 2
[[1,1,4,4],[2,2]]
=> [[1,1,2,2],[4,4]]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => ? = 2 - 2
[[1,1,2,4],[3,3]]
=> [[1,1,2,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,3,4],[2,3]]
=> [[1,1,2,3],[3,4]]
=> [4,6,1,2,3,5] => [4,6,1,5,3,2] => ? = 3 - 2
[[1,1,4,4],[2,3]]
=> [[1,1,2,3],[4,4]]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => ? = 2 - 2
[[1,1,3,4],[3,3]]
=> [[1,1,3,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,1,4,4],[3,3]]
=> [[1,1,3,3],[4,4]]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => ? = 2 - 2
[[1,2,2,4],[2,3]]
=> [[1,2,2,2,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,2,2,4],[3,3]]
=> [[1,2,2,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
[[1,2,3,4],[2,3]]
=> [[1,2,2,3],[3,4]]
=> [4,6,1,2,3,5] => [4,6,1,5,3,2] => ? = 3 - 2
[[1,2,4,4],[2,3]]
=> [[1,2,2,3],[4,4]]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => ? = 2 - 2
[[1,2,3,4],[3,3]]
=> [[1,2,3,3,3],[4]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => ? = 2 - 2
Description
The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$.
A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Matching statistic: St001805
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St001805: Semistandard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 47%●distinct values known / distinct values provided: 33%
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St001805: Semistandard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 47%●distinct values known / distinct values provided: 33%
Values
[[1,3],[2]]
=> [[1,2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 0 = 2 - 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 0 = 2 - 2
[[1,4],[2]]
=> [[1,2],[4]]
=> [[1,2,4]]
=> [[1,2,4]]
=> 0 = 2 - 2
[[1,4],[3]]
=> [[1,3],[4]]
=> [[1,3,4]]
=> [[1,3,4]]
=> 0 = 2 - 2
[[2,4],[3]]
=> [[2,3],[4]]
=> [[2,3,4]]
=> [[2,3,4]]
=> 0 = 2 - 2
[[1],[2],[4]]
=> [[1,2],[4]]
=> [[1,2,4]]
=> [[1,2,4]]
=> 0 = 2 - 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [[1,3,4]]
=> [[1,3,4]]
=> 0 = 2 - 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [[2,3,4]]
=> [[2,3,4]]
=> 0 = 2 - 2
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> 0 = 2 - 2
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> 0 = 2 - 2
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> 0 = 2 - 2
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> 0 = 2 - 2
[[1,5],[2]]
=> [[1,2],[5]]
=> [[1,2,5]]
=> [[1,2,5]]
=> 0 = 2 - 2
[[1,5],[3]]
=> [[1,3],[5]]
=> [[1,3,5]]
=> [[1,3,5]]
=> 0 = 2 - 2
[[1,5],[4]]
=> [[1,4],[5]]
=> [[1,4,5]]
=> [[1,4,5]]
=> 0 = 2 - 2
[[2,5],[3]]
=> [[2,3],[5]]
=> [[2,3,5]]
=> [[2,3,5]]
=> 0 = 2 - 2
[[2,5],[4]]
=> [[2,4],[5]]
=> [[2,4,5]]
=> [[2,4,5]]
=> 0 = 2 - 2
[[3,5],[4]]
=> [[3,4],[5]]
=> [[3,4,5]]
=> [[3,4,5]]
=> 0 = 2 - 2
[[1],[2],[5]]
=> [[1,2],[5]]
=> [[1,2,5]]
=> [[1,2,5]]
=> 0 = 2 - 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [[1,3,5]]
=> [[1,3,5]]
=> 0 = 2 - 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [[1,4,5]]
=> [[1,4,5]]
=> 0 = 2 - 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [[2,3,5]]
=> [[2,3,5]]
=> 0 = 2 - 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [[2,4,5]]
=> [[2,4,5]]
=> 0 = 2 - 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [[3,4,5]]
=> [[3,4,5]]
=> 0 = 2 - 2
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [[1,1,2,4]]
=> [[1,1,2,4]]
=> 0 = 2 - 2
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [[1,1,3,4]]
=> [[1,1,3,4]]
=> 0 = 2 - 2
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [[1,2,2,4]]
=> [[1,2,2,4]]
=> 0 = 2 - 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> 0 = 2 - 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [[1,3,3,4]]
=> [[1,3,3,4]]
=> 0 = 2 - 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [[2,2,3,4]]
=> [[2,2,3,4]]
=> 0 = 2 - 2
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [[2,3,3,4]]
=> [[2,3,3,4]]
=> 0 = 2 - 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [[1,1,2,4]]
=> [[1,1,2,4]]
=> 0 = 2 - 2
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [[1,1,3,4]]
=> [[1,1,3,4]]
=> 0 = 2 - 2
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [[1,2,2,4]]
=> [[1,2,2,4]]
=> 0 = 2 - 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> 0 = 2 - 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> 0 = 2 - 2
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [[1,3,3,4]]
=> [[1,3,3,4]]
=> 0 = 2 - 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [[2,2,3,4]]
=> [[2,2,3,4]]
=> 0 = 2 - 2
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [[2,3,3,4]]
=> [[2,3,3,4]]
=> 0 = 2 - 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> 0 = 2 - 2
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [[1,1,1,2,3]]
=> [[1,1,1,2,3]]
=> 0 = 2 - 2
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [[1,1,2,2,3]]
=> [[1,1,2,2,3]]
=> 0 = 2 - 2
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [[1,2,2,2,3]]
=> [[1,2,2,2,3]]
=> 0 = 2 - 2
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [[1,1,2,2,3]]
=> [[1,1,2,2,3]]
=> 0 = 2 - 2
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [[1,1,1,2,3]]
=> [[1,1,1,2,3]]
=> 0 = 2 - 2
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [[1,1,2,2,3]]
=> [[1,1,2,2,3]]
=> 0 = 2 - 2
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [[1,2,2,2,3]]
=> [[1,2,2,2,3]]
=> 0 = 2 - 2
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [[1,1,2,2,3]]
=> [[1,1,2,2,3]]
=> 0 = 2 - 2
[[1,6],[2]]
=> [[1,2],[6]]
=> [[1,2,6]]
=> [[1,2,6]]
=> 0 = 2 - 2
[[1,6],[3]]
=> [[1,3],[6]]
=> [[1,3,6]]
=> [[1,3,6]]
=> 0 = 2 - 2
[[1,1,1,5],[2]]
=> [[1,1,1,2],[5]]
=> [[1,1,1,2,5]]
=> [[1,1,1,2,5]]
=> ? = 2 - 2
[[1,1,1,5],[3]]
=> [[1,1,1,3],[5]]
=> [[1,1,1,3,5]]
=> [[1,1,1,3,5]]
=> ? = 2 - 2
[[1,1,1,5],[4]]
=> [[1,1,1,4],[5]]
=> [[1,1,1,4,5]]
=> [[1,1,1,4,5]]
=> ? = 2 - 2
[[1,1,2,5],[2]]
=> [[1,1,2,2],[5]]
=> [[1,1,2,2,5]]
=> [[1,1,2,2,5]]
=> ? = 2 - 2
[[1,1,2,5],[3]]
=> [[1,1,2,3],[5]]
=> [[1,1,2,3,5]]
=> [[1,1,2,3,5]]
=> ? = 2 - 2
[[1,1,2,5],[4]]
=> [[1,1,2,4],[5]]
=> [[1,1,2,4,5]]
=> [[1,1,2,4,5]]
=> ? = 2 - 2
[[1,1,3,5],[3]]
=> [[1,1,3,3],[5]]
=> [[1,1,3,3,5]]
=> [[1,1,3,3,5]]
=> ? = 2 - 2
[[1,1,3,5],[4]]
=> [[1,1,3,4],[5]]
=> [[1,1,3,4,5]]
=> [[1,1,3,4,5]]
=> ? = 2 - 2
[[1,1,4,5],[4]]
=> [[1,1,4,4],[5]]
=> [[1,1,4,4,5]]
=> [[1,1,4,4,5]]
=> ? = 2 - 2
[[1,2,2,5],[2]]
=> [[1,2,2,2],[5]]
=> [[1,2,2,2,5]]
=> [[1,2,2,2,5]]
=> ? = 2 - 2
[[1,2,2,5],[3]]
=> [[1,2,2,3],[5]]
=> [[1,2,2,3,5]]
=> [[1,2,2,3,5]]
=> ? = 2 - 2
[[1,2,2,5],[4]]
=> [[1,2,2,4],[5]]
=> [[1,2,2,4,5]]
=> [[1,2,2,4,5]]
=> ? = 2 - 2
[[1,2,3,5],[3]]
=> [[1,2,3,3],[5]]
=> [[1,2,3,3,5]]
=> [[1,2,3,3,5]]
=> ? = 2 - 2
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> ? = 2 - 2
[[1,2,4,5],[4]]
=> [[1,2,4,4],[5]]
=> [[1,2,4,4,5]]
=> [[1,2,4,4,5]]
=> ? = 2 - 2
[[1,3,3,5],[3]]
=> [[1,3,3,3],[5]]
=> [[1,3,3,3,5]]
=> [[1,3,3,3,5]]
=> ? = 2 - 2
[[1,3,3,5],[4]]
=> [[1,3,3,4],[5]]
=> [[1,3,3,4,5]]
=> [[1,3,3,4,5]]
=> ? = 2 - 2
[[1,3,4,5],[4]]
=> [[1,3,4,4],[5]]
=> [[1,3,4,4,5]]
=> [[1,3,4,4,5]]
=> ? = 2 - 2
[[1,4,4,5],[4]]
=> [[1,4,4,4],[5]]
=> [[1,4,4,4,5]]
=> [[1,4,4,4,5]]
=> ? = 2 - 2
[[2,2,2,5],[3]]
=> [[2,2,2,3],[5]]
=> [[2,2,2,3,5]]
=> [[2,2,2,3,5]]
=> ? = 2 - 2
[[2,2,2,5],[4]]
=> [[2,2,2,4],[5]]
=> [[2,2,2,4,5]]
=> [[2,2,2,4,5]]
=> ? = 2 - 2
[[2,2,3,5],[3]]
=> [[2,2,3,3],[5]]
=> [[2,2,3,3,5]]
=> [[2,2,3,3,5]]
=> ? = 2 - 2
[[2,2,3,5],[4]]
=> [[2,2,3,4],[5]]
=> [[2,2,3,4,5]]
=> [[2,2,3,4,5]]
=> ? = 2 - 2
[[2,2,4,5],[4]]
=> [[2,2,4,4],[5]]
=> [[2,2,4,4,5]]
=> [[2,2,4,4,5]]
=> ? = 2 - 2
[[2,3,3,5],[3]]
=> [[2,3,3,3],[5]]
=> [[2,3,3,3,5]]
=> [[2,3,3,3,5]]
=> ? = 2 - 2
[[2,3,3,5],[4]]
=> [[2,3,3,4],[5]]
=> [[2,3,3,4,5]]
=> [[2,3,3,4,5]]
=> ? = 2 - 2
[[2,3,4,5],[4]]
=> [[2,3,4,4],[5]]
=> [[2,3,4,4,5]]
=> [[2,3,4,4,5]]
=> ? = 2 - 2
[[2,4,4,5],[4]]
=> [[2,4,4,4],[5]]
=> [[2,4,4,4,5]]
=> [[2,4,4,4,5]]
=> ? = 2 - 2
[[3,3,3,5],[4]]
=> [[3,3,3,4],[5]]
=> [[3,3,3,4,5]]
=> [[3,3,3,4,5]]
=> ? = 2 - 2
[[3,3,4,5],[4]]
=> [[3,3,4,4],[5]]
=> [[3,3,4,4,5]]
=> [[3,3,4,4,5]]
=> ? = 2 - 2
[[3,4,4,5],[4]]
=> [[3,4,4,4],[5]]
=> [[3,4,4,4,5]]
=> [[3,4,4,4,5]]
=> ? = 2 - 2
[[1,1,5],[2,2]]
=> [[1,1,2,2],[5]]
=> [[1,1,2,2,5]]
=> [[1,1,2,2,5]]
=> ? = 2 - 2
[[1,1,5],[2,3]]
=> [[1,1,2,3],[5]]
=> [[1,1,2,3,5]]
=> [[1,1,2,3,5]]
=> ? = 2 - 2
[[1,1,5],[2,4]]
=> [[1,1,2,4],[5]]
=> [[1,1,2,4,5]]
=> [[1,1,2,4,5]]
=> ? = 2 - 2
[[1,1,5],[3,3]]
=> [[1,1,3,3],[5]]
=> [[1,1,3,3,5]]
=> [[1,1,3,3,5]]
=> ? = 2 - 2
[[1,1,5],[3,4]]
=> [[1,1,3,4],[5]]
=> [[1,1,3,4,5]]
=> [[1,1,3,4,5]]
=> ? = 2 - 2
[[1,1,5],[4,4]]
=> [[1,1,4,4],[5]]
=> [[1,1,4,4,5]]
=> [[1,1,4,4,5]]
=> ? = 2 - 2
[[1,2,5],[2,3]]
=> [[1,2,2,3],[5]]
=> [[1,2,2,3,5]]
=> [[1,2,2,3,5]]
=> ? = 2 - 2
[[1,2,5],[2,4]]
=> [[1,2,2,4],[5]]
=> [[1,2,2,4,5]]
=> [[1,2,2,4,5]]
=> ? = 2 - 2
[[1,2,5],[3,3]]
=> [[1,2,3,3],[5]]
=> [[1,2,3,3,5]]
=> [[1,2,3,3,5]]
=> ? = 2 - 2
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> ? = 2 - 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> ? = 3 - 2
[[1,2,5],[4,4]]
=> [[1,2,4,4],[5]]
=> [[1,2,4,4,5]]
=> [[1,2,4,4,5]]
=> ? = 2 - 2
[[1,3,5],[3,4]]
=> [[1,3,3,4],[5]]
=> [[1,3,3,4,5]]
=> [[1,3,3,4,5]]
=> ? = 2 - 2
[[1,3,5],[4,4]]
=> [[1,3,4,4],[5]]
=> [[1,3,4,4,5]]
=> [[1,3,4,4,5]]
=> ? = 2 - 2
[[2,2,5],[3,3]]
=> [[2,2,3,3],[5]]
=> [[2,2,3,3,5]]
=> [[2,2,3,3,5]]
=> ? = 2 - 2
[[2,2,5],[3,4]]
=> [[2,2,3,4],[5]]
=> [[2,2,3,4,5]]
=> [[2,2,3,4,5]]
=> ? = 2 - 2
[[2,2,5],[4,4]]
=> [[2,2,4,4],[5]]
=> [[2,2,4,4,5]]
=> [[2,2,4,4,5]]
=> ? = 2 - 2
[[2,3,5],[3,4]]
=> [[2,3,3,4],[5]]
=> [[2,3,3,4,5]]
=> [[2,3,3,4,5]]
=> ? = 2 - 2
[[2,3,5],[4,4]]
=> [[2,3,4,4],[5]]
=> [[2,3,4,4,5]]
=> [[2,3,4,4,5]]
=> ? = 2 - 2
Description
The maximal overlap of a cylindrical tableau associated with a semistandard tableau.
A cylindrical tableau associated with a semistandard Young tableau $T$ is the skew semistandard tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St001207
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 67%
Values
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,4],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,4],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[2,4],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,1,3],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,2,3],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,5],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,5],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,5],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[2,5],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[2,5],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[3,5],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,1,4],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,1,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,2,4],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,2,4],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,3,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[2,2,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[2,3,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1,2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,1,1,3],[2]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 2 + 1
[[1,1,2,3],[2]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[[1,2,2,3],[2]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 2 + 1
[[1,1,3],[2,2]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 2 + 1
[[1,1,2],[2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 2 + 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[[1,6],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,6],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,6],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,6],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[2,6],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[2,6],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,2,5],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,2,5],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,3,5],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[2,3,5],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[2,3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,1,2,4],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2,2,4],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2,3,4],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,1,4],[2,3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2,4],[2,3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2,4],[3,3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,1,2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2,2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2,3],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,1],[2,3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2],[2,3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2],[3,3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 2 + 1
[[1,1,1,1,3],[2]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 2 + 1
[[1,2,2,2,3],[2]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 2 + 1
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ? = 2 + 1
[[1,1,1,1],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 2 + 1
[[1,2,2,2],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 2 + 1
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ? = 2 + 1
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ? = 2 + 1
[[1,2,6],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,2,6],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,2,6],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,3,6],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
[[1,3,6],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 2 + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St000744
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 67%
Values
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,4],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,4],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[2,4],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,1,3],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,2,3],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,5],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,5],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,5],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[2,5],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[2,5],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[3,5],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,1,4],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,1,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,2,4],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,2,4],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,3,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[2,2,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[2,3,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1,2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 2 + 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,1,1,3],[2]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 2 + 2
[[1,1,2,3],[2]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 4 = 2 + 2
[[1,2,2,3],[2]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 2 + 2
[[1,1,3],[2,2]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 4 = 2 + 2
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 2 + 2
[[1,1,2],[2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 4 = 2 + 2
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 2 + 2
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 4 = 2 + 2
[[1,6],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,6],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,6],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,6],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[2,6],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[2,6],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 2 + 2
[[1,2,5],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,2,5],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,3,5],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[2,3,5],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[2,3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,1,2,4],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2,2,4],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2,3,4],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,1,4],[2,3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2,4],[2,3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2,4],[3,3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,1,2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2,2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2,3],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,1],[2,3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2],[2,3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2],[3,3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2 + 2
[[1,1,1,1,3],[2]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 2 + 2
[[1,2,2,2,3],[2]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 2 + 2
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 2 + 2
[[1,1,1,1],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 2 + 2
[[1,2,2,2],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 2 + 2
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 2 + 2
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 2 + 2
[[1,2,6],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,2,6],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,2,6],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,3,6],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
[[1,3,6],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2 + 2
Description
The length of the path to the largest entry in a standard Young tableau.
Matching statistic: St001515
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 67%
Values
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,4],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,4],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[2,4],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,1,3],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,2,3],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,5],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,5],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,5],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[2,5],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[2,5],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[3,5],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,1,4],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,1,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,2,4],[2]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,2,4],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,3,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[2,2,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[2,3,4],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,1,1,3],[2]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[[1,1,2,3],[2]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[[1,2,2,3],[2]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[[1,1,3],[2,2]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[[1,1,2],[2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[[1,6],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,6],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,6],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,6],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[2,6],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[2,6],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,2,5],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,2,5],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,3,5],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[2,3,5],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[2,3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,1,2,4],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2,2,4],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2,3,4],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,1,4],[2,3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2,4],[2,3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2,4],[3,3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,1,2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2,2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2,3],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,1],[2,3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2],[2,3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2],[3,3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[[1,1,1,1,3],[2]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 2
[[1,2,2,2,3],[2]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 2
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 2
[[1,1,1,1],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 2
[[1,2,2,2],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 2
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 2
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 2
[[1,2,6],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,2,6],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,2,6],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,3,6],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,3,6],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000044The number of vertices of the unicellular map given by a perfect matching. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000914The sum of the values of the Möbius function of a poset. St000640The rank of the largest boolean interval in a poset. St000717The number of ordinal summands of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001927Sparre Andersen's number of positives of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001875The number of simple modules with projective dimension at most 1. St001890The maximum magnitude of the Möbius function of a poset. St001410The minimal entry of a semistandard tableau. St001621The number of atoms of a lattice. St000168The number of internal nodes of an ordered tree. St000632The jump number of the poset. St001623The number of doubly irreducible elements of a lattice. St000075The orbit size of a standard tableau under promotion. St000527The width of the poset. St000845The maximal number of elements covered by an element in a poset. St001625The Möbius invariant of a lattice. St000528The height of a poset. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001626The number of maximal proper sublattices of a lattice. St001926Sparre Andersen's position of the maximum of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000327The number of cover relations in a poset. St000635The number of strictly order preserving maps of a poset into itself. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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