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Your data matches 54 different statistics following compositions of up to 3 maps.
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Matching statistic: St000060
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(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000060: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000060: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 1
[[2,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 1
[[1,3]]
=> [1,2] => 1
[[2,3]]
=> [1,2] => 1
[[3,3]]
=> [1,2] => 1
[[1],[3]]
=> [2,1] => 1
[[2],[3]]
=> [2,1] => 1
[[1,1,2]]
=> [1,2,3] => 2
[[1,2,2]]
=> [1,2,3] => 2
[[2,2,2]]
=> [1,2,3] => 2
[[1,1],[2]]
=> [3,1,2] => 1
[[1,2],[2]]
=> [2,1,3] => 1
[[1,4]]
=> [1,2] => 1
[[2,4]]
=> [1,2] => 1
[[3,4]]
=> [1,2] => 1
[[4,4]]
=> [1,2] => 1
[[1],[4]]
=> [2,1] => 1
[[2],[4]]
=> [2,1] => 1
[[3],[4]]
=> [2,1] => 1
[[1,1,3]]
=> [1,2,3] => 2
[[1,2,3]]
=> [1,2,3] => 2
[[1,3,3]]
=> [1,2,3] => 2
[[2,2,3]]
=> [1,2,3] => 2
[[2,3,3]]
=> [1,2,3] => 2
[[3,3,3]]
=> [1,2,3] => 2
[[1,1],[3]]
=> [3,1,2] => 1
[[1,2],[3]]
=> [3,1,2] => 1
[[1,3],[2]]
=> [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => 1
[[2,3],[3]]
=> [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => 3
[[1,1,2,2]]
=> [1,2,3,4] => 3
[[1,2,2,2]]
=> [1,2,3,4] => 3
[[2,2,2,2]]
=> [1,2,3,4] => 3
[[1,1,1],[2]]
=> [4,1,2,3] => 1
[[1,1,2],[2]]
=> [3,1,2,4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => 3
[[1,1],[2,2]]
=> [3,4,1,2] => 3
[[1,5]]
=> [1,2] => 1
[[2,5]]
=> [1,2] => 1
[[3,5]]
=> [1,2] => 1
[[4,5]]
=> [1,2] => 1
[[5,5]]
=> [1,2] => 1
[[1],[5]]
=> [2,1] => 1
[[2],[5]]
=> [2,1] => 1
[[3],[5]]
=> [2,1] => 1
[[4],[5]]
=> [2,1] => 1
Description
The greater neighbor of the maximum.
Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the smallest path leaf label of the binary tree associated to a permutation ([[St000724]]), see also [3].
Matching statistic: St000771
(load all 5 compositions to match this statistic)
(load all 5 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
St000771: Graphs ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1}
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1}
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1}
[[2,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1}
[[3,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1}
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2}
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2}
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2}
[[1,1],[2]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,2,2}
[[1,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1}
[[2,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1}
[[3,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1}
[[4,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1}
[[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]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1,1],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[2,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,3,3,3,3}
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,3,3,3,3}
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,3,3,3,3}
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,3,3,3,3}
[[1,1,1],[2]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,1,2],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,3,3,3,3,3}
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,3,3,3,3,3}
[[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)
=> 3
[[1,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[2,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[3,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[4,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[5,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[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]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[4,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,2],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,4],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,4],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,4],[4]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,2],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[2,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[2,4],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,4],[4]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[3,4],[4]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,1,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[2,2,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[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)
=> 3
[[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)
=> 3
[[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)
=> 3
[[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)
=> 3
[[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)
=> 3
[[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)
=> 3
[[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)
=> 2
[[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)
=> 4
[[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)
=> 2
[[1,2],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,3],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,4],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[2,2],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[2,3],[5]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St001060
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 80%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 80%
Values
[[1,2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1}
[[2,2]]
=> [[2,2]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1}
[[1],[2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1}
[[1,3]]
=> [[1,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[2,3]]
=> [[2,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[3,3]]
=> [[3,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[1],[3]]
=> [[1,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[2],[3]]
=> [[2,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[1,1,2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [[2,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,4]]
=> [[1,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[2,4]]
=> [[2,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[3,4]]
=> [[3,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[4,4]]
=> [[4,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[1],[4]]
=> [[1,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[2],[4]]
=> [[2,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[3],[4]]
=> [[3,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[1,1,3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[1,2,3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[1,3,3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[2,2,3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[2,3,3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[3,3,3]]
=> [[3,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[1,1],[3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[1,2],[3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,3],[3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[2,2],[3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[2,3],[3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,5]]
=> [[1,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[2,5]]
=> [[2,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[3,5]]
=> [[3,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[4,5]]
=> [[4,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[5,5]]
=> [[5,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[1],[5]]
=> [[1,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[2],[5]]
=> [[2,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[3],[5]]
=> [[3,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[4],[5]]
=> [[4,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[1,1,4]]
=> [[1,1,4]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1,2,4]]
=> [[1,2,4]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1,4],[2]]
=> [[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,4],[3]]
=> [[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2,4],[3]]
=> [[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[4]]
=> [[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,3],[2],[3]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,5],[2]]
=> [[1,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,5],[3]]
=> [[1,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,5],[4]]
=> [[1,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2,5],[3]]
=> [[2,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2,5],[4]]
=> [[2,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[3,5],[4]]
=> [[3,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[5]]
=> [[1,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,4,4],[2]]
=> [[1,2,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,4,4],[3]]
=> [[1,3,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[2,4,4],[3]]
=> [[2,3,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,4],[2],[4]]
=> [[1,2,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,4],[3],[4]]
=> [[1,3,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[2,4],[3],[4]]
=> [[2,3,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000772
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 80%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 80%
Values
[[1,2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1}
[[2,2]]
=> [[2,2]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1}
[[1],[2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1}
[[1,3]]
=> [[1,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[2,3]]
=> [[2,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[3,3]]
=> [[3,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[1],[3]]
=> [[1,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[2],[3]]
=> [[2,3]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1}
[[1,1,2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [[2,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,4]]
=> [[1,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[2,4]]
=> [[2,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[3,4]]
=> [[3,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[4,4]]
=> [[4,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[1],[4]]
=> [[1,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[2],[4]]
=> [[2,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[3],[4]]
=> [[3,4]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[1,1,3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[1,2,3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[1,3,3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[2,2,3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[2,3,3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[3,3,3]]
=> [[3,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[1,1],[3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[1,2],[3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[2,2],[3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[2,3],[3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,5]]
=> [[1,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[2,5]]
=> [[2,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[3,5]]
=> [[3,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[4,5]]
=> [[4,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[5,5]]
=> [[5,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[1],[5]]
=> [[1,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[2],[5]]
=> [[2,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[3],[5]]
=> [[3,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[4],[5]]
=> [[4,5]]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[1,1,4]]
=> [[1,1,4]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,2,4]]
=> [[1,2,4]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,4],[2]]
=> [[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,4],[3]]
=> [[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2,4],[3]]
=> [[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1],[3],[4]]
=> [[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2],[3],[4]]
=> [[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,5],[2]]
=> [[1,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,5],[3]]
=> [[1,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,5],[4]]
=> [[1,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2,5],[3]]
=> [[2,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2,5],[4]]
=> [[2,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[3,5],[4]]
=> [[3,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1],[3],[5]]
=> [[1,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1],[4],[5]]
=> [[1,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2],[3],[5]]
=> [[2,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2],[4],[5]]
=> [[2,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[3],[4],[5]]
=> [[3,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,6],[2]]
=> [[1,2],[6]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,6],[3]]
=> [[1,3],[6]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001880
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 60%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1}
[[2,2]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1}
[[1],[2]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1}
[[1,3]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1}
[[2,3]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1}
[[3,3]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1}
[[1],[3]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1}
[[2],[3]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1}
[[1,1,2]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [2,1,3] => [2,1] => ([],2)
=> ? ∊ {1,1,2,2,2}
[[1,4]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1}
[[2,4]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1}
[[3,4]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1}
[[4,4]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1}
[[1],[4]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1}
[[2],[4]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1}
[[3],[4]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1}
[[1,1,3]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1,2,3]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1,3,3]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[2,2,3]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[2,3,3]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[3,3,3]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1,1],[3]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1,2],[3]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1,3],[2]]
=> [2,1,3] => [2,1] => ([],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1,3],[3]]
=> [2,1,3] => [2,1] => ([],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[2,2],[3]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[2,3],[3]]
=> [2,1,3] => [2,1] => ([],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1],[2],[3]]
=> [3,2,1] => [2,1] => ([],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2}
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,2,3}
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {1,2,3}
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,2,3}
[[1,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[2,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[3,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[4,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[5,5]]
=> [1,2] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[1],[5]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[2],[5]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[3],[5]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[4],[5]]
=> [2,1] => [1] => ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[1,1,4]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,2,4]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,3,4]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,4,4]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[2,2,4]]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,1],[3]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,2],[3]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,2,2],[3]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,2,2],[3]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,1,1,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,2,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,3,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,2,2,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,2,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,3,3,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,3,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,4,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,2,2,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,2,3,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,2,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,3,3,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,3,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,4,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,3,3,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,3,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,4,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[4,4,4,4]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,1],[4]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,2],[4]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,1,3],[4]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,2,2],[4]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 80%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 80%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1}
[[2,2]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1}
[[1],[2]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1}
[[1,3]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1}
[[2,3]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1}
[[3,3]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1}
[[1],[3]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1}
[[2],[3]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1}
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1}
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1}
[[1,4]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[2,4]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[3,4]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[4,4]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[1],[4]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[2],[4]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[3],[4]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1}
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1}
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1}
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1}
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1}
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1}
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1}
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,3,3}
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,3,3}
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,3,3}
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,3,3}
[[1,5]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[2,5]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[3,5]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[4,5]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[5,5]]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[1],[5]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[2],[5]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[3],[5]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[4],[5]]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[[1,1,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,2,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,3,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,4,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[2,2,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[2,3,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[2,4,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[3,3,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[3,4,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[4,4,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,1],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[1,3],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[1,4],[4]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[2,2],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[2,3],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[2,4],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[2,4],[4]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[3,3],[4]]
=> [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[3,4],[4]]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[[1],[2],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
[[1],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
[[2],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
[[1,1,1,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,1,2,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,1,3,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,2,2,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,2,3,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,3,3,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[2,2,2,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[2,2,3,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[2,3,3,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[3,3,3,3]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,1,1],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[1,1,5]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,2,5]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,3,5]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,4,5]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[1,5,5]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[2,2,5]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[2,3,5]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[[2,4,5]]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
Description
The 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.
Matching statistic: St000522
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000522: Ordered trees ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 60%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000522: Ordered trees ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 4 = 3 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 4 = 3 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 4 = 3 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 4 = 3 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,2,3,3} + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,2,3,3} + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,2,3,3} + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,2,3,3} + 1
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[1,1,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[1,2,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[1,3,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[1,4,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 3 = 2 + 1
[[1,1,1],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,1,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,1,3],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,1,3],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,2,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,2,3],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,2,3],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,3,3],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,3,3],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[2,2,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[2,2,3],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[2,3,3],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,2],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,4],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,4],[4]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,2],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,4],[4]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,4,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,3,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,3,4],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
Description
The number of 1-protected nodes of a rooted tree.
This is the number of nodes with minimal distance one to a leaf.
Matching statistic: St000521
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 60%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 1 + 2
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 1 + 2
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 1 + 2
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 1 + 2
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 1 + 2
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 1 + 2
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 1 + 2
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 1 + 2
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 4 = 2 + 2
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 5 = 3 + 2
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 5 = 3 + 2
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 5 = 3 + 2
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 5 = 3 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,2,3,3} + 2
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,2,3,3} + 2
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,2,3,3} + 2
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,2,3,3} + 2
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3 = 1 + 2
[[1,1,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[1,2,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[1,3,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[1,4,4]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4 = 2 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,1,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,1,3],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,1,3],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,2,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,2,3],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,2,3],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,3,3],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,3,3],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[2,2,2],[3]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[2,2,3],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[2,3,3],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,1],[2,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,1],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,2],[2,3]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[2,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 2
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 2
[[1,1,1],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,1,2],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,1,4],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,1,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,1,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,1,4],[4]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,2,2],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,2,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,2,4],[4]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,4,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,3,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
[[1,3,4],[3]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
Description
The number of distinct subtrees of an ordered tree.
A subtree is specified by a node of the tree. Thus, the tree consisting of a single path has as many subtrees as nodes, whereas the tree of height two, having all leaves attached to the root, has only two distinct subtrees. Because we consider ordered trees, the tree $[[[[]], []], [[], [[]]]]$ on nine nodes has five distinct subtrees.
Matching statistic: St001207
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 40%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[1,3]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[2,3]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[1,1,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[[1,4]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[2,4]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[3,4]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[1,1,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[2,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[2,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 2 + 1
[[1,1,1,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,3,3,3,3,3,3} + 1
[[1,1,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,3,3,3,3,3,3} + 1
[[1,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,3,3,3,3,3,3} + 1
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,3,3,3,3,3,3} + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,3,3,3,3,3,3} + 1
[[1,1,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,3,3,3,3,3,3} + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,3,3,3,3,3,3} + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[[1,5]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[2,5]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[3,5]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[4,5]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[[1,1,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,2,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,3,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,4,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[2,2,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[2,3,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[2,4,4]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,1,1,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,2,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,3,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,2,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,3,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,3,3,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[2,2,2,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[2,2,3,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[2,3,3,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[3,3,3,3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,1],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,3,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,3,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[2,2,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[2,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[2,3,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,1,1,2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1,2,2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,2,2,2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,2,2,2,2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[2,2,2,2,2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1,1],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,2,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,2,2,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {1,2,3,4,4,4,4,4,4,4,4} + 1
[[1,1,1,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,2,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,3,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,1,4,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,2,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,3,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[[1,2,4,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001633
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 40%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,2]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[2]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,3]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,3]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,3]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,2]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,2,2]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[2,2,2]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,1],[2]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[1,2],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[1,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[4,4]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,2,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,3,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[2,2,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[2,3,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[3,3,3]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,1],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[1,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[1,3],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[1,3],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[2,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[2,3],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,2,3,3,3,3,3,3}
[[1,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[4,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[5,5]]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[4],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,2,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,3,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[2,2,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[2,3,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[2,4,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[3,3,4]]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[[1,1,1,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[3]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3],[3]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,2]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,1,1,2,2]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,1,2,2,2]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[2,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? ∊ {1,2,3,4,4,4,4,4,4,4,4}
[[1,1,1,4]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,4]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001095The number of non-isomorphic posets with precisely one further covering relation. St000906The length of the shortest maximal chain in a poset. St000550The number of modular elements of a lattice. St000259The diameter of a connected graph. St000166The depth minus 1 of an ordered tree. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000080The rank of the poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000260The radius of a connected graph. St000264The girth of a graph, which is not a tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000422The energy of a graph, if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000101The cocharge of a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St000102The charge of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001409The maximal entry of a semistandard tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001964The interval resolution global dimension of a poset. St000327The number of cover relations in a poset. St000635The number of strictly order preserving maps of a poset into itself. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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