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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
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Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,3]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,3]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,1,1,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,1,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[4,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,3,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,2,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,3,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[3,3,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[3,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[4,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001431
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 80%●distinct values known / distinct values provided: 71%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 80%●distinct values known / distinct values provided: 71%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,4]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [4,1,2,5,3,6] => [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [3,1,4,2,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [4,1,6,3,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,1,1,1],[2],[4]]
=> [6,5,1,2,3,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 1
[[1,1,1,1],[3],[4]]
=> [6,5,1,2,3,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 1
[[1,1,1,2],[2],[4]]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[[1,1,1,2],[3],[4]]
=> [6,5,1,2,3,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 1
[[1,1,1,3],[2],[4]]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[[1,1,1,4],[2],[3]]
=> [5,4,1,2,3,6] => [4,1,2,5,3,6] => [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[[1,1,1,4],[2],[4]]
=> [5,4,1,2,3,6] => [4,1,2,5,3,6] => [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[[1,1,1,3],[3],[4]]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[[1,1,1,4],[3],[4]]
=> [5,4,1,2,3,6] => [4,1,2,5,3,6] => [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[[1,1,2,2],[2],[4]]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,1,2,2],[3],[4]]
=> [6,5,1,2,3,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 1
[[1,1,2,3],[2],[4]]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,1,2,4],[2],[3]]
=> [5,3,1,2,4,6] => [3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,1,2,4],[2],[4]]
=> [5,3,1,2,4,6] => [3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,1,2,3],[3],[4]]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[[1,1,3,3],[2],[4]]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,1,3,4],[2],[3]]
=> [4,3,1,2,5,6] => [3,1,4,2,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,1,2,4],[3],[4]]
=> [5,4,1,2,3,6] => [4,1,2,5,3,6] => [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[[1,1,3,4],[2],[4]]
=> [5,3,1,2,4,6] => [3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,1,4,4],[2],[3]]
=> [4,3,1,2,5,6] => [3,1,4,2,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,1,4,4],[2],[4]]
=> [4,3,1,2,5,6] => [3,1,4,2,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,1,3,3],[3],[4]]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,1,3,4],[3],[4]]
=> [5,3,1,2,4,6] => [3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,1,4,4],[3],[4]]
=> [4,3,1,2,5,6] => [3,1,4,2,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,2,2],[2],[4]]
=> [6,2,1,3,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,2,2],[3],[4]]
=> [6,5,1,2,3,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 1
[[1,2,2,3],[2],[4]]
=> [6,2,1,3,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,2,4],[2],[3]]
=> [5,2,1,3,4,6] => [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,2,4],[2],[4]]
=> [5,2,1,3,4,6] => [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Matching statistic: St001152
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001152: Perfect matchings ⟶ ℤResult quality: 71% ●values known / values provided: 79%●distinct values known / distinct values provided: 71%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001152: Perfect matchings ⟶ ℤResult quality: 71% ●values known / values provided: 79%●distinct values known / distinct values provided: 71%
Values
[[1,2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[1,3]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[2,3]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[1,1,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1,4]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[2,4]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[3,4]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[1,1,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[2,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[2,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2 = 3 - 1
[[1,1,1,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[[1,1,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[[1,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 2 - 1
[[1,5]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[2,5]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[3,5]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[4,5]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1 = 2 - 1
[[1,1,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1,2,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1,3,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1,4,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[2,2,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[2,3,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[2,4,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[3,3,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[3,4,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[4,4,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2 = 3 - 1
[[1,1,1,1,1,2]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,1,2,2]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,2,2,2]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,2,2,2,2]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,2,2,2,2,2]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[2,2,2,2,2,2]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,1,1,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,1,2,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,1,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,2,2,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,2,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,2,2,2,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,2,2,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,2,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,3,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,2,2,2,2,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,2,2,2,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,2,2,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,2,3,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,3,3,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[2,2,2,2,2,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[2,2,2,2,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[2,2,2,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[2,2,3,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[2,3,3,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[3,3,3,3,3,3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,1],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,1,1,2],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,1,1,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,1,2,2],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,1,2,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,1,3,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,2,2,2],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,2,2,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,2,3,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,3,3,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 5 - 1
[[1,1,1,1,1,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,1,2,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,1,3,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,1,4,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,2,2,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,2,3,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,2,4,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,3,3,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,3,4,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,1,4,4,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,2,2,2,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,2,2,3,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
[[1,1,2,2,4,4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 1 - 1
Description
The number of pairs with even minimum in a perfect matching.
Matching statistic: St001000
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 76%●distinct values known / distinct values provided: 71%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 76%●distinct values known / distinct values provided: 71%
Values
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,3]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,3]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,1,1,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,1,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[4,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,3,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,2,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,3,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[3,3,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[3,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[4,4,4]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,1,1,1,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,2,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,2,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,2,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[2,2,2,2,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,1,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,2,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,2,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,3,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,2,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,3,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[2,2,2,2,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[2,2,2,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[2,2,3,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[2,3,3,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[3,3,3,3,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,1,1,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,1,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[2,2,2,2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,1,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,2,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,3,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,2,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,3,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,3,3,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,3,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,4,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,2,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,3,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,3,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,4,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,3,3,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,3,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,4,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[1,4,4,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[2,2,2,2,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[2,2,2,3,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[2,2,2,4,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[[2,2,3,3,4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001553
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 72%●distinct values known / distinct values provided: 71%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 72%●distinct values known / distinct values provided: 71%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,4]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,1,1,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,1,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,1,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2,2,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2,2,2,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2,2,2,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2,2,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[3,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [4,1,2,5,3,6] => [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [3,1,4,2,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [4,1,6,3,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,1,1,1,1,4]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,1,2,4]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
Matching statistic: St001896
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 57% ●values known / values provided: 65%●distinct values known / distinct values provided: 57%
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 57% ●values known / values provided: 65%●distinct values known / distinct values provided: 57%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[3]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[2],[3]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[4]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[2],[4]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[3],[4]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[[1,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[2],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[3],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[4],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3 - 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3 - 1
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2 - 1
[[1,1,1],[2],[4]]
=> [5,4,1,2,3] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3 - 1
[[1,1,1],[3],[4]]
=> [5,4,1,2,3] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3 - 1
[[1,1,2],[2],[4]]
=> [5,3,1,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3 - 1
[[1,1,2],[3],[4]]
=> [5,4,1,2,3] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3 - 1
[[1,1,3],[2],[4]]
=> [5,3,1,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3 - 1
[[1,1,3],[3],[4]]
=> [5,3,1,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3 - 1
[[1,2,2],[2],[4]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3 - 1
[[1,2,2],[3],[4]]
=> [5,4,1,2,3] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3 - 1
[[1,2,3],[2],[4]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3 - 1
[[1,2,3],[3],[4]]
=> [5,3,1,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3 - 1
[[1,3,3],[2],[4]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3 - 1
[[1,3,3],[3],[4]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3 - 1
[[2,2,2],[3],[4]]
=> [5,4,1,2,3] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3 - 1
[[2,2,3],[3],[4]]
=> [5,3,1,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3 - 1
[[2,3,3],[3],[4]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3 - 1
[[1,1,1,1,1,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,2,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,2,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,2,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,2,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[3,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => ? = 5 - 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [5,2,3,4,6,1] => [5,2,3,4,6,1] => ? = 5 - 1
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [1,6,5,2,3,4] => [1,6,5,2,3,4] => ? = 5 - 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [4,2,3,5,6,1] => [4,2,3,5,6,1] => ? = 5 - 1
Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
Matching statistic: St001864
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 57% ●values known / values provided: 58%●distinct values known / distinct values provided: 57%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 57% ●values known / values provided: 58%●distinct values known / distinct values provided: 57%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[3]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[2],[3]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[4]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[2],[4]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[3],[4]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2 = 3 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1 = 2 - 1
[[1,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[2],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[3],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[4],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[4,4,4]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2 = 3 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3 - 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 3 - 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 3 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 3 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 3 - 1
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [6,1,2,4,5,3] => ? = 2 - 1
[[1,1,1],[2],[4]]
=> [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3 - 1
[[1,1,1],[3],[4]]
=> [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3 - 1
[[1,1,2],[2],[4]]
=> [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 3 - 1
[[1,1,2],[3],[4]]
=> [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3 - 1
[[1,1,3],[2],[4]]
=> [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 3 - 1
[[1,1,4],[2],[3]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 3 - 1
[[1,1,4],[2],[4]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 3 - 1
[[1,1,3],[3],[4]]
=> [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 3 - 1
[[1,1,4],[3],[4]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 3 - 1
[[1,2,2],[2],[4]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3 - 1
[[1,2,2],[3],[4]]
=> [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3 - 1
[[1,2,3],[2],[4]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3 - 1
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 3 - 1
[[1,2,4],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 3 - 1
[[1,2,3],[3],[4]]
=> [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 3 - 1
[[1,3,3],[2],[4]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3 - 1
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 3 - 1
[[1,2,4],[3],[4]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 3 - 1
[[1,3,4],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 3 - 1
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 3 - 1
[[1,4,4],[2],[4]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 3 - 1
[[1,3,3],[3],[4]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3 - 1
[[1,3,4],[3],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 3 - 1
[[1,4,4],[3],[4]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 3 - 1
[[2,2,2],[3],[4]]
=> [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3 - 1
[[2,2,3],[3],[4]]
=> [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 3 - 1
[[2,2,4],[3],[4]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 3 - 1
[[2,3,3],[3],[4]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3 - 1
[[2,3,4],[3],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 3 - 1
[[2,4,4],[3],[4]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 3 - 1
[[1,1,1,1,1,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,1,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,1,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,1,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Matching statistic: St001330
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 86%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 86%
Values
[[1,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[4,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[3],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[4,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[5,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[3],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[4],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[4,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,1],[2,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1],[3,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1],[4,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1],[5,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[2,5]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[3,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3],[2,5]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[4,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,4],[2,5]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[5,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3],[3,5]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3],[4,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,4],[3,5]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3],[5,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,4],[4,5]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,4],[5,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,2],[3,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,2],[4,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,2],[5,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,3],[3,5]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,3],[4,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,4],[3,5]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[2,3],[5,5]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001235
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 57%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 57%
Values
[[1,2]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[2,2]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[1],[2]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[1,3]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[2,3]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[3,3]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[1],[3]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[1,1,2]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[1,2,2]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[2,2,2]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[1,4]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[2,4]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[3,4]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[4,4]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[1],[4]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[1,1,3]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[1,2,3]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[1,3,3]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[2,2,3]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[2,3,3]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[3,3,3]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1]
=> 1110 => [1,1,1,2] => 4 = 3 + 1
[[1,1,1,2]]
=> [4]
=> 10000 => [1,5] => 2 = 1 + 1
[[1,1,2,2]]
=> [4]
=> 10000 => [1,5] => 2 = 1 + 1
[[1,2,2,2]]
=> [4]
=> 10000 => [1,5] => 2 = 1 + 1
[[2,2,2,2]]
=> [4]
=> 10000 => [1,5] => 2 = 1 + 1
[[1,1],[2,2]]
=> [2,2]
=> 1100 => [1,1,3] => 3 = 2 + 1
[[1,5]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[2,5]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[3,5]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[4,5]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[5,5]]
=> [2]
=> 100 => [1,3] => 2 = 1 + 1
[[1],[5]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[[1,1,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[1,2,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[1,3,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[1,4,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[2,2,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[2,3,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[2,4,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[3,3,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[3,4,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[4,4,4]]
=> [3]
=> 1000 => [1,4] => 2 = 1 + 1
[[1],[2],[4]]
=> [1,1,1]
=> 1110 => [1,1,1,2] => 4 = 3 + 1
[[1,1,1,1,2]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,1,2,2]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,2,2,2]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,2,2,2]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[2,2,2,2,2]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,1,1,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,1,2,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,1,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,2,2,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,2,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,3,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,2,2,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,2,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,3,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,3,3,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[2,2,2,2,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[2,2,2,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[2,2,3,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[2,3,3,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[3,3,3,3,3]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 3 + 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 3 + 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 3 + 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 3 + 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 3 + 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 3 + 1
[[1,1,1,1,1,2]]
=> [6]
=> 1000000 => [1,7] => ? = 1 + 1
[[1,1,1,1,2,2]]
=> [6]
=> 1000000 => [1,7] => ? = 1 + 1
[[1,1,1,2,2,2]]
=> [6]
=> 1000000 => [1,7] => ? = 1 + 1
[[1,1,2,2,2,2]]
=> [6]
=> 1000000 => [1,7] => ? = 1 + 1
[[1,2,2,2,2,2]]
=> [6]
=> 1000000 => [1,7] => ? = 1 + 1
[[2,2,2,2,2,2]]
=> [6]
=> 1000000 => [1,7] => ? = 1 + 1
[[1,1,1,1,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,1,2,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,1,3,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,1,4,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,2,2,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,2,3,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,2,4,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,3,3,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,3,4,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,1,4,4,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,2,2,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,2,3,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,2,4,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,3,3,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,3,4,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,2,4,4,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,3,3,3,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
[[1,3,3,4,4]]
=> [5]
=> 100000 => [1,6] => ? = 1 + 1
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001621
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 29% ●values known / values provided: 51%●distinct values known / distinct values provided: 29%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 29% ●values known / values provided: 51%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[2,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[3,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[2,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[3,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[4,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[2,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[3,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[4,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[5,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[3,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[3,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[4,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1,1,1,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[2,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[[2,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[3,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 4
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 3
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 3
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 3
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 3
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 3
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 2
[[1],[2],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1],[3],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1],[4],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1],[5],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[2],[3],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[2],[4],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[2],[5],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[3],[4],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[3],[5],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[4],[5],[6]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[1,1],[2,5]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,1],[3,5]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001946The number of descents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001937The size of the center of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001645The pebbling number of a connected graph. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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