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Your data matches 71 different statistics following compositions of up to 3 maps.
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Matching statistic: St000008
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Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2] => 0
[2,1] => [2,1] => [1,1] => 1
[1,2,3] => [1,2,3] => [3] => 0
[1,3,2] => [3,1,2] => [1,2] => 1
[2,1,3] => [2,1,3] => [1,2] => 1
[2,3,1] => [1,3,2] => [2,1] => 2
[3,1,2] => [2,3,1] => [2,1] => 2
[3,2,1] => [3,2,1] => [1,1,1] => 3
[1,2,3,4] => [1,2,3,4] => [4] => 0
[1,2,4,3] => [4,1,2,3] => [1,3] => 1
[1,3,2,4] => [3,1,2,4] => [1,3] => 1
[1,3,4,2] => [2,4,1,3] => [2,2] => 2
[1,4,2,3] => [3,4,1,2] => [2,2] => 2
[1,4,3,2] => [4,3,1,2] => [1,1,2] => 3
[2,1,3,4] => [2,1,3,4] => [1,3] => 1
[2,1,4,3] => [1,4,2,3] => [2,2] => 2
[2,3,1,4] => [1,3,2,4] => [2,2] => 2
[2,3,4,1] => [1,2,4,3] => [3,1] => 3
[2,4,1,3] => [1,3,4,2] => [3,1] => 3
[3,1,2,4] => [2,3,1,4] => [2,2] => 2
[3,1,4,2] => [4,2,1,3] => [1,1,2] => 3
[3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[4,1,2,3] => [2,3,4,1] => [3,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5] => 0
[1,2,3,5,4] => [5,1,2,3,4] => [1,4] => 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,4] => 1
[1,2,4,5,3] => [3,5,1,2,4] => [2,3] => 2
[1,2,5,3,4] => [4,5,1,2,3] => [2,3] => 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 3
[1,3,2,4,5] => [3,1,2,4,5] => [1,4] => 1
[1,3,2,5,4] => [2,5,1,3,4] => [2,3] => 2
[1,3,4,2,5] => [2,4,1,3,5] => [2,3] => 2
[1,3,4,5,2] => [2,3,5,1,4] => [3,2] => 3
[1,3,5,2,4] => [2,4,5,1,3] => [3,2] => 3
[1,4,2,3,5] => [3,4,1,2,5] => [2,3] => 2
[1,4,2,5,3] => [5,3,1,2,4] => [1,1,3] => 3
[1,4,3,2,5] => [4,3,1,2,5] => [1,1,3] => 3
[1,5,2,3,4] => [3,4,5,1,2] => [3,2] => 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => 1
[2,1,3,5,4] => [1,5,2,3,4] => [2,3] => 2
[2,1,4,3,5] => [1,4,2,3,5] => [2,3] => 2
[2,1,4,5,3] => [1,3,5,2,4] => [3,2] => 3
[2,1,5,3,4] => [1,4,5,2,3] => [3,2] => 3
[2,3,1,4,5] => [1,3,2,4,5] => [2,3] => 2
[2,3,1,5,4] => [1,2,5,3,4] => [3,2] => 3
[2,3,4,1,5] => [1,2,4,3,5] => [3,2] => 3
[2,3,4,5,1] => [1,2,3,5,4] => [4,1] => 4
[2,3,5,1,4] => [1,2,4,5,3] => [4,1] => 4
[2,4,1,3,5] => [1,3,4,2,5] => [3,2] => 3
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000228
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Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000330
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(load all 5 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 0
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [3,1,2] => [[1,3],[2]]
=> 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
=> 2
[3,1,2] => [2,3,1] => [[1,2],[3]]
=> 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 3
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 2
[1,4,2,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 2
[1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,1,4,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 2
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 2
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[2,4,1,3] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
[3,1,2,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 2
[3,1,4,2] => [4,2,1,3] => [[1,4],[2],[3]]
=> 3
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[4,1,2,3] => [2,3,4,1] => [[1,2,3],[4]]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[1,2,4,3,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1
[1,2,4,5,3] => [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 2
[1,2,5,3,4] => [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 2
[1,2,5,4,3] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[1,3,2,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
[1,3,2,5,4] => [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 2
[1,3,4,2,5] => [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 2
[1,3,4,5,2] => [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 3
[1,3,5,2,4] => [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 3
[1,4,2,3,5] => [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 2
[1,4,2,5,3] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> 3
[1,4,3,2,5] => [4,3,1,2,5] => [[1,4,5],[2],[3]]
=> 3
[1,5,2,3,4] => [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[2,1,3,5,4] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 2
[2,1,4,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 2
[2,1,4,5,3] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 3
[2,1,5,3,4] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 3
[2,3,1,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
[2,3,1,5,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 3
[2,3,4,1,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
[2,3,4,5,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[2,3,5,1,4] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
[2,4,1,3,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 3
Description
The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St000384
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(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000459
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000784
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St001622
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St000063
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 1 = 0 + 1
[1,2] => ([],2)
=> []
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [1]
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> []
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [1]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4 = 3 + 1
[1,2,3,4] => ([],4)
=> []
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 3 = 2 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> []
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
Description
The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
Matching statistic: St000108
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 1 = 0 + 1
[1,2] => ([],2)
=> []
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [1]
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> []
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [1]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4 = 3 + 1
[1,2,3,4] => ([],4)
=> []
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 3 = 2 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> []
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
Description
The number of partitions contained in the given partition.
Matching statistic: St000532
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 1 = 0 + 1
[1,2] => ([],2)
=> []
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [1]
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> []
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [1]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4 = 3 + 1
[1,2,3,4] => ([],4)
=> []
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 3 = 2 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> []
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
Description
The total number of rook placements on a Ferrers board.
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001400The total number of Littlewood-Richardson tableaux of given shape. St000009The charge of a standard tableau. St000012The area of a Dyck path. St000081The number of edges of a graph. St000169The cocharge of a standard tableau. St000185The weighted size of a partition. St001161The major index north count of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000391The sum of the positions of the ones in a binary word. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000874The position of the last double rise in a Dyck path. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001527The cyclic permutation representation number of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000833The comajor index of a permutation. St000018The number of inversions of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000446The disorder of a permutation. St000795The mad of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St000454The largest eigenvalue of a graph if it is integral. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000246The number of non-inversions of a permutation. St001077The prefix exchange distance of a permutation. St000004The major index of a permutation. St001428The number of B-inversions of a signed permutation. St000224The sorting index of a permutation. St000305The inverse major index of a permutation. St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St000005The bounce statistic of a Dyck path. St000304The load of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001621The number of atoms of a lattice. St001875The number of simple modules with projective dimension at most 1. St000450The number of edges minus the number of vertices plus 2 of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001862The number of crossings of a signed permutation. St001645The pebbling number of a connected graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001892The flag excedance statistic of a signed permutation. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001433The flag major index of a signed permutation. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001866The nesting alignments of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2. 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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