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Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St000260
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(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1],[2]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,2],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[3,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,2,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[5],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,3],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,4],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,2],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,3],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001553
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[3,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[2,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1,2],[2,3]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[[1,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[1],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[3],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[4],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[5],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,1,1,1],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,2,2,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[2,2,2,2,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2],[2,3],[3]]
=> [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[1,2,2],[2,3],[3]]
=> [5,2,6,1,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,1,1],[2]]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,1],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,2],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,2,2],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,2,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,3,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,2,2],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,2,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,3,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,3,3,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 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: St001431
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[3,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[2,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1,2],[2,3]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[[1,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[1],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[3],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[4],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[5],[6]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,1,1,1],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,2,2,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[2,2,2,2,2],[3]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2],[2,3],[3]]
=> [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[1,2,2],[2,3],[3]]
=> [5,2,6,1,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,1,1],[2]]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,1],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,2],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,2,2],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,2,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,1,3,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,2,2],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,2,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,2,3,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,1,3,3,3],[4]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 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: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[1],[3]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[2],[3]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[1,1],[2]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[2],[4]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[3],[4]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[1,1],[3]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2],[3]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[2],[5]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[3],[5]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[4],[5]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[1,1],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1,2],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1,3],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,3],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,3],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[1],[3],[4]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[2],[3],[4]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[3]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,1,2],[3]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,2,2],[3]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[2,2,2],[3]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[1],[6]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[2],[6]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[3],[6]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[4],[6]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[5],[6]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[1,1],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1,2],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1,3],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,3],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,3],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[4,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[1],[3],[5]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[1],[4],[5]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[2],[3],[5]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[2],[4],[5]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[3],[4],[5]]
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[4]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,1,2],[4]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,1,3],[4]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,2,2],[4]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,3,3],[4]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[2,2,2],[4]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[[1],[7]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[2],[7]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[3],[7]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[4],[7]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[5],[7]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[6],[7]]
=> [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[[1,2],[2,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,3],[2,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,4],[2,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,3],[3,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,4],[3,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,4],[4,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[2,3],[3,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[2,4],[3,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[2,4],[4,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[3,4],[4,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[[1,1],[2],[5]]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[[1,1],[3],[5]]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
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: St001330
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1],[2]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1],[3]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,3],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,2],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,3],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,3],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,1,1],[3]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[1],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[5],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,3],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,2],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,3],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,3],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[4,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[[1,1],[2,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[3,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[4,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[5,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[2,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,2],[3,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,3],[2,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,2],[4,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,4],[2,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,2],[5,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,3],[3,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,3],[4,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,4],[3,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,3],[5,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,4],[4,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,4],[5,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,2],[3,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 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: St001904
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001904: Parking functions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001904: Parking functions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[1],[3]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[2],[3]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1],[4]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[2],[4]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[3],[4]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[1],[5]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[2],[5]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[3],[5]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[4],[5]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[1,1],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,2],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,3],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[2,2],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[2,3],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[3,3],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,1,1],[3]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[1,1,2],[3]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[1,2,2],[3]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[2,2,2],[3]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1],[6]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[2],[6]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[3],[6]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[4],[6]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[5],[6]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[1,1],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,2],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,3],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,4],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[2,2],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[2,3],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[2,4],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[3,3],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,1,1,1],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,1,2],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,2,2,2],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[2,2,2,2],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 1 + 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 1 + 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 1 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,5,3,2,6,4] => ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [1,4,2,5,3,6] => [1,4,2,5,3,6] => ? = 1 + 1
[[1,1,1,1],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,1,2],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,1,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,2,2],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,2,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,2,2,2],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,2,2,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,2,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,3,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[2,2,2,2],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[2,2,2,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[2,2,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[2,3,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[3,3,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1 + 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1 + 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1 + 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2 + 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2 + 1
Description
The length of the initial strictly increasing segment of a parking function.
Matching statistic: St001582
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 67%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 1
[[1],[3]]
=> [2,1] => [1,2] => [1,2] => 1
[[2],[3]]
=> [2,1] => [1,2] => [1,2] => 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[4]]
=> [2,1] => [1,2] => [1,2] => 1
[[2],[4]]
=> [2,1] => [1,2] => [1,2] => 1
[[3],[4]]
=> [2,1] => [1,2] => [1,2] => 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1],[5]]
=> [2,1] => [1,2] => [1,2] => 1
[[2],[5]]
=> [2,1] => [1,2] => [1,2] => 1
[[3],[5]]
=> [2,1] => [1,2] => [1,2] => 1
[[4],[5]]
=> [2,1] => [1,2] => [1,2] => 1
[[1,1],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1,2],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1,3],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[2,2],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[2,3],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[3,3],[4]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[4]]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[[1],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[[2],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[[1,1,1],[3]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[[1,1,2],[3]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[[1,2,2],[3]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[[2,2,2],[3]]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,2,4,3] => [1,4,3,2] => 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1],[6]]
=> [2,1] => [1,2] => [1,2] => 1
[[2],[6]]
=> [2,1] => [1,2] => [1,2] => 1
[[3],[6]]
=> [2,1] => [1,2] => [1,2] => 1
[[4],[6]]
=> [2,1] => [1,2] => [1,2] => 1
[[5],[6]]
=> [2,1] => [1,2] => [1,2] => 1
[[1,1],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1,2],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1,3],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1,4],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[2,2],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[2,3],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[2,4],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[3,3],[5]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1,1,1,1],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,1,2],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,2,2],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,2,2,2],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[2,2,2,2],[3]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,5,4,3,2] => ? = 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [1,3,5,4,2] => [1,5,4,3,2] => ? = 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [1,4,5,3,2] => [1,5,4,3,2] => ? = 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [1,3,2,5,4] => [1,5,4,3,2] => ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [2,3,1,5,4] => [2,5,1,4,3] => ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,5,1,4,3] => ? = 1
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ? = 1
[[1,1,1,1],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,1,2],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,1,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,2,2],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,2,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,2,2,2],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,2,2,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,2,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,3,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[2,2,2,2],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[2,2,2,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[2,2,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[2,3,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[3,3,3,3],[4]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001860
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001860: Signed permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 67%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001860: Signed permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 1
[[1],[3]]
=> [2,1] => [2,1] => [2,1] => 1
[[2],[3]]
=> [2,1] => [2,1] => [2,1] => 1
[[1,1],[2]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1],[4]]
=> [2,1] => [2,1] => [2,1] => 1
[[2],[4]]
=> [2,1] => [2,1] => [2,1] => 1
[[3],[4]]
=> [2,1] => [2,1] => [2,1] => 1
[[1,1],[3]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[2,2],[3]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,2,3,1] => [4,2,3,1] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[2],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[3],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[4],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[1,1],[4]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1,2],[4]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1,3],[4]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[2,2],[4]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[2,3],[4]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[3,3],[4]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1,1,1],[3]]
=> [4,1,2,3] => [4,2,3,1] => [4,2,3,1] => 1
[[1,1,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => [4,2,3,1] => 1
[[1,2,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => [4,2,3,1] => 1
[[2,2,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => [4,2,3,1] => 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,4,1,2] => [3,4,1,2] => 2
[[1,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1],[6]]
=> [2,1] => [2,1] => [2,1] => 1
[[2],[6]]
=> [2,1] => [2,1] => [2,1] => 1
[[3],[6]]
=> [2,1] => [2,1] => [2,1] => 1
[[4],[6]]
=> [2,1] => [2,1] => [2,1] => 1
[[5],[6]]
=> [2,1] => [2,1] => [2,1] => 1
[[1,1],[5]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1,2],[5]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1,3],[5]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1,4],[5]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[2,2],[5]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[2,3],[5]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[2,4],[5]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[3,3],[5]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[[1,1,1,1],[3]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,1,2],[3]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,2,2],[3]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,2,2,2],[3]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[2,2,2,2],[3]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 1
[[1,1,1,1],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,1,2],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,1,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,2,2],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,2,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,3,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,2,2,2],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,2,2,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,2,3,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,3,3,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[2,2,2,2],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[2,2,2,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[2,2,3,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[2,3,3,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[3,3,3,3],[4]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
Description
The number of factors of the Stanley symmetric function associated with a signed permutation.
Matching statistic: St000914
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 34%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 34%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 1
[[1],[3]]
=> [2,1] => ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => ([(0,1)],2)
=> 1
[[1,1],[2]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[4]]
=> [2,1] => ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => ([(0,1)],2)
=> 1
[[1,1],[3]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[[1],[5]]
=> [2,1] => ([(0,1)],2)
=> 1
[[2],[5]]
=> [2,1] => ([(0,1)],2)
=> 1
[[3],[5]]
=> [2,1] => ([(0,1)],2)
=> 1
[[4],[5]]
=> [2,1] => ([(0,1)],2)
=> 1
[[1,1],[4]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2],[4]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,3],[4]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,2],[4]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,3],[4]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,3],[4]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[2],[4]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1],[3],[4]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[2],[3],[4]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1,1,1],[3]]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1,2],[3]]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2,2],[3]]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,2,2],[3]]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 1
[[1],[6]]
=> [2,1] => ([(0,1)],2)
=> 1
[[2],[6]]
=> [2,1] => ([(0,1)],2)
=> 1
[[3],[6]]
=> [2,1] => ([(0,1)],2)
=> 1
[[4],[6]]
=> [2,1] => ([(0,1)],2)
=> 1
[[5],[6]]
=> [2,1] => ([(0,1)],2)
=> 1
[[1,1],[5]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2],[5]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,3],[5]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,4],[5]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,2],[5]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,3],[5]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,4],[5]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,3],[5]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,4],[5]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[2,3],[3,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,1,1,1],[3]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[2,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 2
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => ([(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)
=> ? = 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)
=> ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => ([(0,1),(0,2),(1,4),(1,10),(2,3),(2,10),(3,5),(3,8),(4,5),(4,9),(5,11),(7,6),(8,7),(8,11),(9,7),(9,11),(10,8),(10,9),(11,6)],12)
=> ? = 1
[[1,2],[2,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,3],[2,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,4],[2,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,3],[3,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,4],[3,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,4],[4,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[2,3],[3,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[2,4],[3,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[2,4],[4,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[3,4],[4,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,1,1,1],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,1,1,2],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,1,1,3],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,1,2,2],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,1,2,3],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,1,3,3],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,2,2],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,2,3],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,3,3],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,3,3,3],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[2,2,2,2],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[2,2,2,3],[4]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\
-\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\
0&\text{otherwise}.
\end{cases}
$$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$
\sum_{x\leq y} \mu(x,y).
$$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
Matching statistic: St001890
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1],[2]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1],[3]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[2,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1],[4]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1,2],[4]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[2,2],[4]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[2,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[3,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,1],[3]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2],[3]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2],[3]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2],[3]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[5],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1,2],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1,3],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1,4],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[2,2],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[2,3],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[2,4],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[3,3],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[3,4],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[4,4],[5]]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1
[[1,1],[2,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
Description
The maximum magnitude of the Möbius function of a poset.
The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001624The breadth of a lattice. 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. St001488The number of corners of a skew partition. St000679The pruning number of an ordered tree. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001926Sparre Andersen's position of the maximum of a signed permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001621The number of atoms of a lattice. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001407The number of minimal entries in a semistandard tableau. St001409The maximal entry of a semistandard tableau.
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