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Your data matches 64 different statistics following compositions of up to 3 maps.
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Matching statistic: St000030
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000030: 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] => 3
[[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
The sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
Matching statistic: St000029
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000029: Permutations ⟶ 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] => [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] => [3,1,2] => 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] => [3,1,2] => 2
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => 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] => [3,1,2] => 2
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 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] => [4,1,2,3] => 3
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => 3
[[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
Description
The depth of a permutation.
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Matching statistic: St000728
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000728: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000728: Set partitions ⟶ 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] => [2,1] => {{1,2}}
=> 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] => [2,1] => {{1,2}}
=> 1
[[2],[3]]
=> [2,1] => [2,1] => {{1,2}}
=> 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] => [3,1,2] => {{1,3},{2}}
=> 2
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => {{1,2},{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] => [2,1] => {{1,2}}
=> 1
[[2],[4]]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[[3],[4]]
=> [2,1] => [2,1] => {{1,2}}
=> 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] => [3,1,2] => {{1,3},{2}}
=> 2
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[[2,2],[3]]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => {{1,2,3}}
=> 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] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 2
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3
[[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] => [2,1] => {{1,2}}
=> 1
[[2],[5]]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[[3],[5]]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[[4],[5]]
=> [2,1] => [2,1] => {{1,2}}
=> 1
Description
The dimension of a set partition.
This is the sum of the lengths of the arcs of a set partition. Equivalently, one obtains that this is the sum of the maximal entries of the blocks minus the sum of the minimal entries of the blocks.
A slightly shifted definition of the dimension is [[St000572]].
Matching statistic: St000454
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Values
[[1,2]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[2,2]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,3]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[2,3]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[3,3]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[1],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,2,2]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[2,2,2]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,1],[2]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[[1,2],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[1,4]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[2,4]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[3,4]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[4,4]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[1],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,2,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,3,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[2,2,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[2,3,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[3,3,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,1],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2}
[[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2}
[[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[1,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[2,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2}
[[2,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,3}
[[1,1,2],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,3}
[[1,2,2],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1,5]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[2,5]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[3,5]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[4,5]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[5,5]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[1],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[3],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[4],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,4]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,2,4]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,3,4]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,4,4]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[2,2,4]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[2,3,4]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,1],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2}
[[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2}
[[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2}
[[2,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2}
[[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2}
[[3,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2}
[[1,1,1],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[1,1,3],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[1,1,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[2,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[2,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3}
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,4}
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {4,4}
[[1,1],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[1,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[1,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[1,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[2,2],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[2,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[2,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[3,3],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[3,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[4,4],[5]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2}
[[1,1,1],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,1,4],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,1,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,1,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,2,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,2,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {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}
[[1,3,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {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}
[[2,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {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}
[[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {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}
[[2,2,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {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}
[[2,2,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {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}
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001861
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001861: Signed permutations ⟶ ℤResult quality: 44% ●values known / values provided: 45%●distinct values known / distinct values provided: 44%
Mp00277: Permutations —catalanization⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001861: Signed permutations ⟶ ℤResult quality: 44% ●values known / values provided: 45%●distinct values known / distinct values provided: 44%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[2,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => [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] => [2,1] => [2,1] => 1
[[2],[3]]
=> [2,1] => [2,1] => [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] => [2,3,1] => [2,3,1] => 2
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => [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] => [2,1] => [2,1] => 1
[[2],[4]]
=> [2,1] => [2,1] => [2,1] => 1
[[3],[4]]
=> [2,1] => [2,1] => [2,1] => 1
[[1,1,3]]
=> [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] => [2,3,1] => [2,3,1] => 2
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => [2,3,1] => 2
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => [2,3,1] => [2,3,1] => 2
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [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] => [2,3,4,1] => [2,3,4,1] => 3
[[1,1,2],[2]]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 3
[[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] => [2,1] => [2,1] => 1
[[2],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[3],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[4],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,2,3,3,4,4}
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,2,3,3,4,4}
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,2,3,3,4,4}
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,2,3,3,4,4}
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? ∊ {1,2,3,3,4,4}
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,2,3,3,4,4}
[[1,1,1,1],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [4,3,5,1,2] => [4,3,5,1,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,4,2,5,1] => [3,4,2,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [5,3,4,2,1] => [5,3,4,2,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [5,2,4,3,1] => [5,2,4,3,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,1,2,3,3,4,4,5,5,5}
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,1,2,3,3,4,4,5,5,5}
Description
The number of Bruhat lower covers of a permutation.
This is, for a signed permutation $\pi$, the number of signed permutations $\tau$ having a reduced word which is obtained by deleting a letter from a reduced word from $\pi$.
Matching statistic: St001894
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 44% ●values known / values provided: 45%●distinct values known / distinct values provided: 44%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 44% ●values known / values provided: 45%●distinct values known / distinct values provided: 44%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[2,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => [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] => [2,1] => [2,1] => 1
[[2],[3]]
=> [2,1] => [2,1] => [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] => [2,3,1] => [2,3,1] => 2
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => [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] => [2,1] => [2,1] => 1
[[2],[4]]
=> [2,1] => [2,1] => [2,1] => 1
[[3],[4]]
=> [2,1] => [2,1] => [2,1] => 1
[[1,1,3]]
=> [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] => [2,3,1] => [2,3,1] => 2
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => [2,3,1] => 2
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => [2,3,1] => [2,3,1] => 2
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 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] => [2,3,4,1] => [2,3,4,1] => 3
[[1,1,2],[2]]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 3
[[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] => [2,1] => [2,1] => 1
[[2],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[3],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[4],[5]]
=> [2,1] => [2,1] => [2,1] => 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,2,3,3,4,4}
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,2,3,3,4,4}
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,2,3,3,4,4}
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,2,3,3,4,4}
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {1,2,3,3,4,4}
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {1,2,3,3,4,4}
[[1,1,1,1],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [3,1,4,5,2] => [3,1,4,5,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,1,2,3,3,4,4,5,5,5}
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,1,2,3,3,4,4,5,5,5}
Description
The depth of a signed permutation.
The depth of a positive root is its rank in the root poset. The depth of an element of a Coxeter group is the minimal sum of depths for any representation as product of reflections.
Matching statistic: St001498
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Values
[[1,2]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0}
[[2,2]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0}
[[1],[2]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[1,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0}
[[2,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0}
[[3,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0}
[[1],[3]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[2],[3]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[1,1,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1}
[[1,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1}
[[2,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1}
[[1,1],[2]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1}
[[1,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0,0}
[[2,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0,0}
[[3,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0,0}
[[4,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0,0}
[[1],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[2],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[3],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[1,1,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[[1,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[[2,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[[2,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[[3,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[[1,1],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[[1,3],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[[2,2],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[2,3],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,3}
[[1,1,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,3}
[[1,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,3}
[[2,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,3}
[[1,1,1],[2]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,1,2],[2]]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,3}
[[1,2,2],[2]]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,3}
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[1,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0,0,0}
[[2,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0,0,0}
[[3,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0,0,0}
[[4,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0,0,0}
[[5,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,0,0,0}
[[1],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[2],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[3],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[4],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[1,1,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[1,2,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[1,3,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[1,4,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[2,2,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[2,3,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[2,4,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[3,3,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[3,4,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[4,4,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[1,1],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,4],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[1,3],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,4],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[1,4],[4]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[2,2],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[2,3],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[2,4],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[2,4],[4]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[3,3],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[3,4],[4]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[[1],[2],[4]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[[2],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1,1,1,3]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,1,2],[3]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,2,2],[3]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[2,2,2],[3]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,1],[2,3]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[1,1],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2],[2,3]]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 0
[[1,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[2,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1],[6]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[2],[6]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[3],[6]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[4],[6]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[5],[6]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[1,1],[5]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[5]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,3],[5]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,4],[5]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[2,2],[5]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[2,3],[5]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001207
(load all 58 compositions to match this statistic)
(load all 58 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 44%
Mp00277: Permutations —catalanization⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 44%
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] => [4,3,2,1] => 3
[[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] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,3,5,2,1] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => ? ∊ {0,0,0,0,0,1,2,3,3,4,4}
[[1,1,1,1,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,3,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,3,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[3,3,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,1],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3,3],[2]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2,2],[3]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,2,3,3],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,3,5,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,3,5,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [4,3,5,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6}
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: St001232
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%
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]
=> ? = 1
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[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]
=> ? ∊ {1,1,1,2}
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,2}
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,2}
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,2}
[[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]
=> ? ∊ {1,2}
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,2}
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[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
[[1,1],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[2,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[3,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[1,1,3],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[1,1,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[1,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[2,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[2,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[1,2],[2,3]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3}
[[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]
=> ? ∊ {1,2,3,3}
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,3}
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,2,3,3}
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,2,3,3}
[[1,1],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[2,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[2,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[3,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[3,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[4,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[1],[2],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[1],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[1],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[2],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[2],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[3],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[1,1,1],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001877
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
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)
=> ? ∊ {3,3}
[[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)
=> ? ∊ {3,3}
[[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)
=> ? ∊ {2,2,2}
[[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)
=> ? ∊ {2,2,2}
[[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)
=> ? ∊ {2,2,2}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,3,3,4,4}
[[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)
=> ? ∊ {2,3,3,4,4}
[[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)
=> ? ∊ {2,3,3,4,4}
[[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)
=> ? ∊ {2,3,3,4,4}
[[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)
=> ? ∊ {2,3,3,4,4}
[[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)
=> ? ∊ {2,2,2,2,2,2}
[[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)
=> ? ∊ {2,2,2,2,2,2}
[[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)
=> ? ∊ {2,2,2,2,2,2}
[[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)
=> ? ∊ {2,2,2,2,2,2}
[[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)
=> ? ∊ {2,2,2,2,2,2}
[[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)
=> ? ∊ {2,2,2,2,2,2}
[[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)
=> ? ∊ {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}
[[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)
=> ? ∊ {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}
[[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)
=> ? ∊ {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}
[[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)
=> ? ∊ {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}
[[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)
=> ? ∊ {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}
[[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)
=> ? ∊ {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}
[[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)
=> ? ∊ {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}
[[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)
=> ? ∊ {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}
[[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)
=> ? ∊ {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],[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)
=> ? ∊ {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}
[[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)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
Number of indecomposable injective modules with projective dimension 2.
The following 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001060The distinguishing index of a graph. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000264The girth of a graph, which is not a tree. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000379The number of Hamiltonian cycles in a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001568The smallest positive integer that does not appear twice in the partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000736The last entry in the first row of a semistandard tableau. St000103The sum of the entries of a semistandard tableau. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000456The monochromatic index of a connected graph. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.
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