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Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St000209
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(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 0
[[2,2]]
=> [1,2] => 0
[[1],[2]]
=> [2,1] => 1
[[1,3]]
=> [1,2] => 0
[[2,3]]
=> [1,2] => 0
[[3,3]]
=> [1,2] => 0
[[1],[3]]
=> [2,1] => 1
[[2],[3]]
=> [2,1] => 1
[[1,1,2]]
=> [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => 2
[[1,2],[2]]
=> [2,1,3] => 1
[[1,4]]
=> [1,2] => 0
[[2,4]]
=> [1,2] => 0
[[3,4]]
=> [1,2] => 0
[[4,4]]
=> [1,2] => 0
[[1],[4]]
=> [2,1] => 1
[[2],[4]]
=> [2,1] => 1
[[3],[4]]
=> [2,1] => 1
[[1,1,3]]
=> [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => 2
[[1,2],[3]]
=> [3,1,2] => 2
[[1,3],[2]]
=> [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => 2
[[2,3],[3]]
=> [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => 3
[[1,1,2],[2]]
=> [3,1,2,4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => 2
[[1,5]]
=> [1,2] => 0
[[2,5]]
=> [1,2] => 0
[[3,5]]
=> [1,2] => 0
[[4,5]]
=> [1,2] => 0
[[5,5]]
=> [1,2] => 0
[[1],[5]]
=> [2,1] => 1
[[2],[5]]
=> [2,1] => 1
[[3],[5]]
=> [2,1] => 1
[[4],[5]]
=> [2,1] => 1
Description
Maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the maximum of this value over all cycles in the permutation.
Matching statistic: St000503
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => {{1},{2}}
=> 0
[[2,2]]
=> [1,2] => {{1},{2}}
=> 0
[[1],[2]]
=> [2,1] => {{1,2}}
=> 1
[[1,3]]
=> [1,2] => {{1},{2}}
=> 0
[[2,3]]
=> [1,2] => {{1},{2}}
=> 0
[[3,3]]
=> [1,2] => {{1},{2}}
=> 0
[[1],[3]]
=> [2,1] => {{1,2}}
=> 1
[[2],[3]]
=> [2,1] => {{1,2}}
=> 1
[[1,1,2]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,2,2]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[2,2,2]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,1],[2]]
=> [3,1,2] => {{1,2,3}}
=> 2
[[1,2],[2]]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[1,4]]
=> [1,2] => {{1},{2}}
=> 0
[[2,4]]
=> [1,2] => {{1},{2}}
=> 0
[[3,4]]
=> [1,2] => {{1},{2}}
=> 0
[[4,4]]
=> [1,2] => {{1},{2}}
=> 0
[[1],[4]]
=> [2,1] => {{1,2}}
=> 1
[[2],[4]]
=> [2,1] => {{1,2}}
=> 1
[[3],[4]]
=> [2,1] => {{1,2}}
=> 1
[[1,1,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,2,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,3,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[2,2,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[2,3,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[3,3,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,1],[3]]
=> [3,1,2] => {{1,2,3}}
=> 2
[[1,2],[3]]
=> [3,1,2] => {{1,2,3}}
=> 2
[[1,3],[2]]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[1,3],[3]]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[2,2],[3]]
=> [3,1,2] => {{1,2,3}}
=> 2
[[2,3],[3]]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[1],[2],[3]]
=> [3,2,1] => {{1,3},{2}}
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => {{1,2,3,4}}
=> 3
[[1,1,2],[2]]
=> [3,1,2,4] => {{1,2,3},{4}}
=> 2
[[1,2,2],[2]]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2
[[1,5]]
=> [1,2] => {{1},{2}}
=> 0
[[2,5]]
=> [1,2] => {{1},{2}}
=> 0
[[3,5]]
=> [1,2] => {{1},{2}}
=> 0
[[4,5]]
=> [1,2] => {{1},{2}}
=> 0
[[5,5]]
=> [1,2] => {{1},{2}}
=> 0
[[1],[5]]
=> [2,1] => {{1,2}}
=> 1
[[2],[5]]
=> [2,1] => {{1,2}}
=> 1
[[3],[5]]
=> [2,1] => {{1,2}}
=> 1
[[4],[5]]
=> [2,1] => {{1,2}}
=> 1
[[1]]
=> [1] => {{1}}
=> ? = 0
[[2]]
=> [1] => {{1}}
=> ? = 0
[[3]]
=> [1] => {{1}}
=> ? = 0
[[4]]
=> [1] => {{1}}
=> ? = 0
[[5]]
=> [1] => {{1}}
=> ? = 0
[[6]]
=> [1] => {{1}}
=> ? = 0
Description
The maximal difference between two elements in a common block.
Matching statistic: St000956
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[2,2]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[1],[2]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[1,3]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[2,3]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[3,3]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[1],[3]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[2],[3]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[1,1,2]]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => {{1,2,3}}
=> [2,3,1] => 2
[[1,2],[2]]
=> [2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[[1,4]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[2,4]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[3,4]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[4,4]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[1],[4]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[2],[4]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[3],[4]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[1,1,3]]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => {{1,2,3}}
=> [2,3,1] => 2
[[1,2],[3]]
=> [3,1,2] => {{1,2,3}}
=> [2,3,1] => 2
[[1,3],[2]]
=> [2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => {{1,2,3}}
=> [2,3,1] => 2
[[2,3],[3]]
=> [2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => {{1,3},{2}}
=> [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[[1,1,2],[2]]
=> [3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 2
[[1,5]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[2,5]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[3,5]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[4,5]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[5,5]]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[[1],[5]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[2],[5]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[3],[5]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[4],[5]]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[[1]]
=> [1] => {{1}}
=> [1] => ? = 0
[[2]]
=> [1] => {{1}}
=> [1] => ? = 0
[[3]]
=> [1] => {{1}}
=> [1] => ? = 0
[[4]]
=> [1] => {{1}}
=> [1] => ? = 0
[[5]]
=> [1] => {{1}}
=> [1] => ? = 0
[[6]]
=> [1] => {{1}}
=> [1] => ? = 0
Description
The maximal displacement of a permutation.
This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$.
This statistic without the absolute value is the maximal drop size [[St000141]].
Matching statistic: St001207
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 67%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 67%
Values
[[1,2]]
=> [1,2] => [1,2] => 0
[[2,2]]
=> [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => 1
[[1,3]]
=> [1,2] => [1,2] => 0
[[2,3]]
=> [1,2] => [1,2] => 0
[[3,3]]
=> [1,2] => [1,2] => 0
[[1],[3]]
=> [2,1] => [2,1] => 1
[[2],[3]]
=> [2,1] => [2,1] => 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => [2,3,1] => 2
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => 1
[[1,4]]
=> [1,2] => [1,2] => 0
[[2,4]]
=> [1,2] => [1,2] => 0
[[3,4]]
=> [1,2] => [1,2] => 0
[[4,4]]
=> [1,2] => [1,2] => 0
[[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] => [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => [2,3,1] => 2
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => 2
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => [2,3,1] => 2
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => [2,3,4,1] => 3
[[1,1,2],[2]]
=> [3,1,2,4] => [2,3,1,4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,1,3] => 2
[[1,5]]
=> [1,2] => [1,2] => 0
[[2,5]]
=> [1,2] => [1,2] => 0
[[3,5]]
=> [1,2] => [1,2] => 0
[[4,5]]
=> [1,2] => [1,2] => 0
[[5,5]]
=> [1,2] => [1,2] => 0
[[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
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? = 4
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? = 3
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? = 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [2,3,5,1,4] => ? = 4
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,4,1,3,5] => ? = 2
[[1,1,1,1,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,1,1,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,1,1,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,1,2,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,1,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,1,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,2,2,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,2,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,2,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,3,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[2,2,2,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[2,2,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[2,2,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[2,3,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[3,3,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,1,1,1],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? = 4
[[1,1,1,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? = 4
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? = 3
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? = 3
[[1,1,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? = 4
[[1,1,2,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? = 2
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? = 3
[[1,1,3,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? = 2
[[1,1,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? = 2
[[1,2,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? = 4
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? = 3
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[[1,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? = 2
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[[2,2,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? = 4
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? = 3
[[2,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? = 2
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [2,3,5,1,4] => ? = 4
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [2,3,5,1,4] => ? = 4
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [2,4,5,1,3] => ? = 3
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,4,1,3,5] => ? = 2
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: St001877
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[2,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[2,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[3,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[1,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[2,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[3,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[4,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[2,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[3,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[4,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[5,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 3
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 4
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 3
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 4
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001232
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 33%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[3,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[2,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[3,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[[1,1,3],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[[1,1,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[[1,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[[2,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[[2,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[[1,1],[2,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[[1,1],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[[1,2],[2,3]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3
[[1,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[[2,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[[1,1],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[2,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[2,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[3,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[3,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[4,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1],[2],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[[1],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[2,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[1,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[2,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[3,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[2],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[1,1,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[1,2],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[1,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[2,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[3,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[4,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[2],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[3],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[1,1,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[1,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[2,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[2,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[1,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[2,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[3,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[4,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[5,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[2],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[3],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[4],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[1,1,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2,2,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[1,4],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[1,4],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[1,4],[4]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[2,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[2,4],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[2,4],[4]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[3,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[[3,4],[4]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[[1],[2],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[[1],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 4 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001816
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 3
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? = 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> ? = 2
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[1,1,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[1,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[1,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[1,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[2,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[2,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[2,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[3,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[[1,1,1,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[1,1,2,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[1,1,3,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[1,2,2,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[1,2,3,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[1,3,3,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[2,2,2,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[2,2,3,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[2,3,3,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[3,3,3,3]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 3
[[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 3
[[1,1,3],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[[1,1,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 3
[[1,2,3],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? = 1
[[1,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[[1,3,3],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? = 1
[[1,3,3],[3]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? = 1
[[2,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 3
[[2,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[[2,3,3],[3]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? = 1
[[1,1],[2,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> ? = 2
[[1,1],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> ? = 2
[[1,2],[2,3]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> ? = 3
[[1,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> ? = 2
[[2,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> ? = 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 3
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 3
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 0
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 0
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 0
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 0
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 0
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 4
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> ? = 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> ? = 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> ? = 4
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[[1,1,1,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St001645
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[2,2]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[1],[2]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,3]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[2,3]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[3,3]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[1],[3]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,2,2]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,2,2]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 2 + 1
[[1,2],[2]]
=> [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[1,4]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[2,4]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[3,4]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[4,4]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[1],[4]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,2,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,3,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,2,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,3,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[3,3,3]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 2 + 1
[[1,2],[3]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 2 + 1
[[1,3],[2]]
=> [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[1,3],[3]]
=> [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 2 + 1
[[2,3],[3]]
=> [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 3 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 2 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[[1,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[2,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[3,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[4,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[5,5]]
=> [1,2] => [2] => ([],2)
=> ? = 0 + 1
[[1],[5]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,1,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,2,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,3,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,4,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,2,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,3,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[2,4,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[3,3,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[3,4,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[4,4,4]]
=> [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[[1,1],[4]]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 2 + 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[5],[6]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[3],[4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[5],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[6],[7]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1],[2],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[3],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[4],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[3],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[4],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[3],[4],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[3],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[4],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[2],[3],[5]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1],[2],[4],[5]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1],[3],[4],[5]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[2],[3],[4],[5]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1],[8]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[2],[8]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[3],[8]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[4],[8]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000112
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000112: Semistandard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000112: Semistandard tableaux ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,2]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[2]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,3]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[3]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,1,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,2,2]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,1],[2]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,2],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[1,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[4,4]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[3],[4]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,1,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[3,3,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,1],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,3],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[1,3],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[2,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[2,3],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[1],[2],[3]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 2
[[1,1,1,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,1,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[2,2,2,2]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
[[1,1,1],[2]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ? = 3
[[1,1,2],[2]]
=> [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ? = 2
[[1,2,2],[2]]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 2
[[1,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[4,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[5,5]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[3],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[4],[5]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,1,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,2,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[2,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[3,3,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[3,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[4,4,4]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
[[1,1],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,4],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[1,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[1,4],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[1,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[2,2],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[2,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[2,4],[3]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[2,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[3,3],[4]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ? = 2
[[3,4],[4]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 1
[[1],[2],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 2
[[1],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 2
[[1,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[4,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[5,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[6,6]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[3],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[4],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[5],[6]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[3,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[4,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[5,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[6,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[7,7]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[2],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[3],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[4],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[5],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[6],[7]]
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[1,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[2,8]]
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
Description
The sum of the entries reduced by the index of their row in a semistandard tableau.
This is also the depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is the shape of $T$, independent of the Cartan rank.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000736The last entry in the first row of a semistandard tableau. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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