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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St000533
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St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 2
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 2
[3,2]
=> 2
[3,1,1]
=> 3
[2,2,1]
=> 2
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 2
[4,2]
=> 2
[4,1,1]
=> 3
[3,3]
=> 2
[3,2,1]
=> 3
[3,1,1,1]
=> 3
[2,2,2]
=> 2
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 1
[6,1]
=> 2
[5,2]
=> 2
[5,1,1]
=> 3
[4,3]
=> 2
[4,2,1]
=> 3
[4,1,1,1]
=> 4
[3,3,1]
=> 3
[3,2,2]
=> 3
[3,2,1,1]
=> 3
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 1
[7,1]
=> 2
[6,2]
=> 2
[6,1,1]
=> 3
[5,3]
=> 2
[5,2,1]
=> 3
Description
The minimum of the number of parts and the size of the first part of an integer partition.
This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000183
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 83%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 83%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> [1]
=> 1
[2]
=> [[2],[]]
=> [[2],[]]
=> [2]
=> 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 1
[3]
=> [[3],[]]
=> [[3],[]]
=> [3]
=> 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1
[4]
=> [[4],[]]
=> [[4],[]]
=> [4]
=> 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 2
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 1
[5]
=> [[5],[]]
=> [[5],[]]
=> [5]
=> 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 2
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 3
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 2
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 1
[6]
=> [[6],[]]
=> [[6],[]]
=> [6]
=> 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [5,5]
=> 2
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [4,4]
=> 2
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> 3
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> [3,3]
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 3
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 3
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 2
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> 1
[7]
=> [[7],[]]
=> [[7],[]]
=> [7]
=> 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [6,6]
=> 2
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [5,5]
=> 2
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [5,5,5]
=> 3
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [4,4]
=> 2
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> 3
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> 4
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 3
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 3
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> 3
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> 3
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [2,2,2,2,2]
=> 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> 2
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1,1]
=> 1
[8]
=> [[8],[]]
=> [[8],[]]
=> [8]
=> 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [7,7]
=> 2
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [6,6]
=> 2
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [6,6,6]
=> 3
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [5,5]
=> 2
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [5,5,5]
=> 3
[7,1,1]
=> [[7,1,1],[]]
=> [[7,7,7],[6,6]]
=> [7,7,7]
=> ? = 3
[7,2,1]
=> [[7,2,1],[]]
=> [[7,7,7],[6,5]]
=> [7,7,7]
=> ? ∊ {3,4,4}
[7,1,1,1]
=> [[7,1,1,1],[]]
=> [[7,7,7,7],[6,6,6]]
=> [7,7,7,7]
=> ? ∊ {3,4,4}
[4,1,1,1,1,1,1]
=> [[4,1,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [4,4,4,4,4,4,4]
=> ? ∊ {3,4,4}
[11]
=> [[11],[]]
=> [[11],[]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[10,1]
=> [[10,1],[]]
=> [[10,10],[9]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[9,2]
=> [[9,2],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[9,1,1]
=> [[9,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[8,3]
=> [[8,3],[]]
=> [[8,8],[5]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[8,2,1]
=> [[8,2,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[8,1,1,1]
=> [[8,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,4]
=> [[7,4],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,3,1]
=> [[7,3,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,2,2]
=> [[7,2,2],[]]
=> [[7,7,7],[5,5]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,2,1,1]
=> [[7,2,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,1,1,1,1]
=> [[7,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,4,1]
=> [[6,4,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,3,2]
=> [[6,3,2],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,3,1,1]
=> [[6,3,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,2,2,1]
=> [[6,2,2,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,2,1,1,1]
=> [[6,2,1,1,1],[]]
=> [[6,6,6,6,6],[5,5,5,4]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,1,1,1,1,1]
=> [[6,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[5,2,1,1,1,1]
=> [[5,2,1,1,1,1],[]]
=> [[5,5,5,5,5,5],[4,4,4,4,3]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[5,1,1,1,1,1,1]
=> [[5,1,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[4,3,1,1,1,1]
=> [[4,3,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[4,2,2,1,1,1]
=> [[4,2,2,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[4,2,1,1,1,1,1]
=> [[4,2,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[4,1,1,1,1,1,1,1]
=> [[4,1,1,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,3,2,1,1,1]
=> [[3,3,2,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,3,1,1,1,1,1]
=> [[3,3,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,2,2,2,1,1]
=> [[3,2,2,2,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,2,2,1,1,1,1]
=> [[3,2,2,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,2,1,1,1,1,1,1]
=> [[3,2,1,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,1,1,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[2,2,2,2,2,1]
=> [[2,2,2,2,2,1],[]]
=> [[2,2,2,2,2,2],[1]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[2,2,2,2,1,1,1]
=> [[2,2,2,2,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[2,2,2,1,1,1,1,1]
=> [[2,2,2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[2,2,1,1,1,1,1,1,1]
=> [[2,2,1,1,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[2,1,1,1,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1,1]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[12]
=> [[12],[]]
=> [[12],[]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[11,1]
=> [[11,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[10,2]
=> [[10,2],[]]
=> [[10,10],[8]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[10,1,1]
=> [[10,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[9,3]
=> [[9,3],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[9,2,1]
=> [[9,2,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[9,1,1,1]
=> [[9,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[8,4]
=> [[8,4],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[8,3,1]
=> [[8,3,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[8,2,2]
=> [[8,2,2],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000875
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 83%
Mp00224: Binary words —runsort⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 83%
Values
[1]
=> 10 => 01 => 10 => 1
[2]
=> 100 => 001 => 100 => 1
[1,1]
=> 110 => 011 => 110 => 1
[3]
=> 1000 => 0001 => 1000 => 1
[2,1]
=> 1010 => 0011 => 1100 => 2
[1,1,1]
=> 1110 => 0111 => 1110 => 1
[4]
=> 10000 => 00001 => 10000 => 1
[3,1]
=> 10010 => 00011 => 11000 => 2
[2,2]
=> 1100 => 0011 => 1100 => 2
[2,1,1]
=> 10110 => 00111 => 11100 => 2
[1,1,1,1]
=> 11110 => 01111 => 11110 => 1
[5]
=> 100000 => 000001 => 100000 => 1
[4,1]
=> 100010 => 000011 => 110000 => 2
[3,2]
=> 10100 => 00011 => 11000 => 2
[3,1,1]
=> 100110 => 000111 => 111000 => 3
[2,2,1]
=> 11010 => 00111 => 11100 => 2
[2,1,1,1]
=> 101110 => 001111 => 111100 => 2
[1,1,1,1,1]
=> 111110 => 011111 => 111110 => 1
[6]
=> 1000000 => 0000001 => 1000000 => 1
[5,1]
=> 1000010 => 0000011 => 1100000 => 2
[4,2]
=> 100100 => 000011 => 110000 => 2
[4,1,1]
=> 1000110 => 0000111 => 1110000 => 3
[3,3]
=> 11000 => 00011 => 11000 => 2
[3,2,1]
=> 101010 => 001011 => 110100 => 3
[3,1,1,1]
=> 1001110 => 0001111 => 1111000 => 3
[2,2,2]
=> 11100 => 00111 => 11100 => 2
[2,2,1,1]
=> 110110 => 001111 => 111100 => 2
[2,1,1,1,1]
=> 1011110 => 0011111 => 1111100 => 2
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 1111110 => 1
[7]
=> 10000000 => 00000001 => 10000000 => 1
[6,1]
=> 10000010 => 00000011 => 11000000 => 2
[5,2]
=> 1000100 => 0000011 => 1100000 => 2
[5,1,1]
=> 10000110 => 00000111 => 11100000 => 3
[4,3]
=> 101000 => 000011 => 110000 => 2
[4,2,1]
=> 1001010 => 0001011 => 1101000 => 3
[4,1,1,1]
=> 10001110 => 00001111 => 11110000 => 4
[3,3,1]
=> 110010 => 000111 => 111000 => 3
[3,2,2]
=> 101100 => 000111 => 111000 => 3
[3,2,1,1]
=> 1010110 => 0010111 => 1110100 => 3
[3,1,1,1,1]
=> 10011110 => 00011111 => 11111000 => 3
[2,2,2,1]
=> 111010 => 001111 => 111100 => 2
[2,2,1,1,1]
=> 1101110 => 0011111 => 1111100 => 2
[2,1,1,1,1,1]
=> 10111110 => 00111111 => 11111100 => 2
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 11111110 => 1
[8]
=> 100000000 => 000000001 => 100000000 => 1
[7,1]
=> 100000010 => 000000011 => 110000000 => 2
[6,2]
=> 10000100 => 00000011 => 11000000 => 2
[6,1,1]
=> 100000110 => 000000111 => 111000000 => 3
[5,3]
=> 1001000 => 0000011 => 1100000 => 2
[5,2,1]
=> 10001010 => 00001011 => 11010000 => 3
[8,1]
=> 1000000010 => 0000000011 => 1100000000 => ? ∊ {2,2,3,3,4,4}
[7,1,1]
=> 1000000110 => 0000000111 => 1110000000 => ? ∊ {2,2,3,3,4,4}
[6,1,1,1]
=> 1000001110 => 0000001111 => 1111000000 => ? ∊ {2,2,3,3,4,4}
[4,1,1,1,1,1]
=> 1000111110 => 0000111111 => 1111110000 => ? ∊ {2,2,3,3,4,4}
[3,1,1,1,1,1,1]
=> 1001111110 => 0001111111 => 1111111000 => ? ∊ {2,2,3,3,4,4}
[2,1,1,1,1,1,1,1]
=> 1011111110 => 0011111111 => 1111111100 => ? ∊ {2,2,3,3,4,4}
[9,1]
=> 10000000010 => 00000000011 => 11000000000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[8,2]
=> 1000000100 => 0000000011 => 1100000000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[8,1,1]
=> 10000000110 => 00000000111 => 11100000000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[7,2,1]
=> 1000001010 => 0000001011 => 1101000000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[7,1,1,1]
=> 10000001110 => 00000001111 => 11110000000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[6,2,1,1]
=> 1000010110 => 0000010111 => 1110100000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[6,1,1,1,1]
=> 10000011110 => 00000011111 => 11111000000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[5,1,1,1,1,1]
=> 10000111110 => 00000111111 => 11111100000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[4,2,1,1,1,1]
=> 1001011110 => 0001011111 => 1111101000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[4,1,1,1,1,1,1]
=> 10001111110 => 00001111111 => 11111110000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[3,2,1,1,1,1,1]
=> 1010111110 => 0010111111 => 1111110100 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[3,1,1,1,1,1,1,1]
=> 10011111110 => 00011111111 => 11111111000 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[2,2,1,1,1,1,1,1]
=> 1101111110 => 0011111111 => 1111111100 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 00111111111 => 11111111100 => ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5}
[11]
=> 100000000000 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[10,1]
=> 100000000010 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[9,2]
=> 10000000100 => 00000000011 => 11000000000 => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[9,1,1]
=> 100000000110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[8,3]
=> 1000001000 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[8,2,1]
=> 10000001010 => 00000001011 => 11010000000 => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[8,1,1,1]
=> 100000001110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,3,1]
=> 1000010010 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,2,2]
=> 1000001100 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,2,1,1]
=> 10000010110 => 00000010111 => 11101000000 => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,1,1,1,1]
=> 100000011110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,3,1,1]
=> 1000100110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,2,2,1]
=> 1000011010 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,2,1,1,1]
=> 10000101110 => 00000101111 => 11110100000 => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,1,1,1,1,1]
=> 100000111110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[5,2,1,1,1,1]
=> 10001011110 => 00001011111 => 11111010000 => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[4,3,1,1,1,1]
=> 1010011110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[4,2,2,1,1,1]
=> 1001101110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[4,2,1,1,1,1,1]
=> 10010111110 => 00010111111 => 11111101000 => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,2,1,1,1,1,1,1]
=> 10101111110 => 00101111111 => 11111110100 => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0011111111 => 1111111100 => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 00111111111 => 11111111100 => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[12]
=> 1000000000000 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001515
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 67%
Values
[1]
=> []
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> []
=> ? = 1
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 2
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4]
=> []
=> []
=> []
=> ? = 2
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[5]
=> []
=> []
=> []
=> ? = 3
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[6]
=> []
=> []
=> []
=> ? = 3
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[7]
=> []
=> []
=> []
=> ? ∊ {3,3}
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {3,3}
[8]
=> []
=> []
=> []
=> ? ∊ {2,2,3,4}
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,4}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,4}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,3,4}
[9]
=> []
=> []
=> []
=> ? ∊ {2,2,3,3,3,4,4,5}
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,4,4,5}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,4,4,5}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,4,4,5}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,4,4,5}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,4,4,5}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,4,4,5}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,4,4,5}
[10]
=> []
=> []
=> []
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,5,5}
[11]
=> []
=> []
=> []
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,5,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,4,2,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,3,1,1]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St001526
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 83%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 83%
Values
[1]
=> []
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> []
=> ? = 1
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 1
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[4]
=> []
=> []
=> []
=> ? = 1
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[5]
=> []
=> []
=> []
=> ? = 1
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[6]
=> []
=> []
=> []
=> ? = 1
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[7]
=> []
=> []
=> []
=> ? ∊ {1,4}
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,4}
[8]
=> []
=> []
=> []
=> ? ∊ {1,2,3,4}
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,3,4}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,4}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,4}
[9]
=> []
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,4,5}
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,2,3,3,3,4,4,5}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5}
[10]
=> []
=> []
=> []
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,4,4,4,4,4,5,5}
[11]
=> []
=> []
=> []
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[5,5,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[5,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[5,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[4,4,2,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[4,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[4,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[3,3,3,1,1]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[3,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000141
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 83%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,3,4] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3,5,4,1] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1,4] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [2,3,4,6,5,1] => 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,5,3,4,1] => 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,4,5,2,3] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,5,1] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2,5] => 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [3,2,4,5,6,1] => 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [2,3,4,5,7,6,1] => ? = 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,3,4,6,5,2] => 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [2,3,6,4,5,1] => 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,4,2,1] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,5,2,1,3] => 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,1,3,2] => 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,5,2,1] => 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [4,2,3,5,6,1] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [3,2,4,5,1,6] => 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [2,1,3,4,5,6,7,8] => 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [2,3,4,5,6,8,7,1] => ? ∊ {2,2,2,3}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [1,3,4,5,7,6,2] => ? ∊ {2,2,2,3}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [2,3,4,7,5,6,1] => ? ∊ {2,2,2,3}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,4,6,5,3] => 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [3,4,6,5,2,1] => 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [2,6,3,4,5,1] => 4
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,2,5,6,3,4] => 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,5,4,3,1] => 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,5,1,4,2] => 3
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => [3,2,4,5,6,1,7] => ? ∊ {2,2,2,3}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [2,3,4,5,6,7,9,8,1] => ? ∊ {2,2,2,2,2,3,3,3,4}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => [1,3,4,5,6,8,7,2] => ? ∊ {2,2,2,2,2,3,3,3,4}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,1,4,5,6,7] => [2,3,4,5,8,6,7,1] => ? ∊ {2,2,2,2,2,3,3,3,4}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => [1,2,4,5,7,6,3] => ? ∊ {2,2,2,2,2,3,3,3,4}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => [3,4,5,7,6,2,1] => ? ∊ {2,2,2,2,2,3,3,3,4}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,1,5,6] => [2,3,7,4,5,6,1] => ? ∊ {2,2,2,2,2,3,3,3,4}
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => [4,3,5,6,7,2,1] => ? ∊ {2,2,2,2,2,3,3,3,4}
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [3,2,4,5,1,6,7] => ? ∊ {2,2,2,2,2,3,3,3,4}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => [3,2,4,5,6,7,1,8] => ? ∊ {2,2,2,2,2,3,3,3,4}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [2,3,4,5,6,7,8,10,9,1] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [1,3,4,5,6,7,9,8,2] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => [2,3,4,5,6,9,7,8,1] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [8,4,1,2,3,5,6,7] => [1,2,4,5,6,8,7,3] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => [3,4,5,6,8,7,2,1] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,2,3,4,6] => [1,2,3,5,7,6,4] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => [2,4,5,7,6,3,1] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => [3,4,5,7,1,6,2] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,1,5,6] => [3,4,7,5,6,2,1] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => [2,7,3,4,5,6,1] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [1,2,3,6,7,4,5] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,7,1] => [5,3,4,6,7,2,1] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,7,1] => [4,2,5,6,7,1,3] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [4,3,5,6,2,7,1] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => [4,3,5,6,7,8,2,1] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [3,2,4,1,5,6,7] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,1] => [3,2,4,5,6,1,7,8] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1,9] => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,12,11] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,1,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,11,10,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => [1,3,4,5,6,7,8,10,9,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,1,4,5,6,7,8,9] => [2,3,4,5,6,7,10,8,9,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => [1,2,4,5,6,7,9,8,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,1,4,5,6,7,8] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,1,5,6,7,8] => [2,3,4,5,9,6,7,8,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [8,5,1,2,3,4,6,7] => [1,2,3,5,6,8,7,4] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [8,4,2,1,3,5,6,7] => [2,4,5,6,8,7,3,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,1,2,5,6,7] => [3,4,5,6,8,1,7,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,1,5,6,7] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,5,1,6,7] => [2,3,8,4,5,6,7,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => [2,3,5,7,6,4,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,2,5,6] => [1,4,5,7,6,3,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,3,1,5,6] => [2,4,7,5,6,3,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => [3,4,7,5,2,6,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,1,6] => [3,7,4,5,6,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6}
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St001200
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 50%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,1}
[1,1]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,1}
[2,1]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1}
[1,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2}
[3,1]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2}
[2,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {1,1,2}
[2,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,3}
[4,1]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,3}
[3,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {1,1,2,3}
[3,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[2,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,2,3}
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2}
[5,1]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2}
[4,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {1,1,2,2}
[4,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[3,3]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,2,2}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2,3}
[6,1]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3}
[5,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {1,1,2,2,3}
[5,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[4,3]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[4,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,3}
[2,2,2,1]
=> [2,2,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,3}
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2,4,4}
[7,1]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,4,4}
[6,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {1,1,2,2,4,4}
[6,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[5,3]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[5,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[4,3,1]
=> [3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,4,4}
[3,3,2]
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,2,2,4,4}
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,4,4}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[8,1]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[7,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[7,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[6,3]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[6,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[6,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[5,4]
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[5,3,1]
=> [3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[5,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
[5,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[5,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[4,4,1]
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[4,3,2]
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[4,3,1,1]
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,3,3]
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[3,3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[3,2,2,2]
=> [2,2,2]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[2,2,2,2,1]
=> [2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,5}
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[9,1]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[8,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[6,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[4,4,2]
=> [4,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[4,4,1,1]
=> [4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[4,3,3]
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[4,3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[4,2,2,2]
=> [2,2,2]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[3,3,3,1]
=> [3,3,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[3,3,2,1,1]
=> [3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001924
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001924: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001924: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> [1]
=> 1
[2]
=> [[2],[]]
=> [[2],[]]
=> [2]
=> 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 1
[3]
=> [[3],[]]
=> [[3],[]]
=> [3]
=> 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1
[4]
=> [[4],[]]
=> [[4],[]]
=> [4]
=> 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 2
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 1
[5]
=> [[5],[]]
=> [[5],[]]
=> [5]
=> 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 2
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 3
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 2
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 1
[6]
=> [[6],[]]
=> [[6],[]]
=> [6]
=> 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [5,5]
=> 2
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [4,4]
=> 2
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> 3
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> [3,3]
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 3
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 3
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 2
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> 1
[7]
=> [[7],[]]
=> [[7],[]]
=> [7]
=> 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [6,6]
=> 2
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [5,5]
=> 2
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [5,5,5]
=> 3
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [4,4]
=> 2
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> 3
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> 4
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 3
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 3
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> 3
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> 3
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [2,2,2,2,2]
=> 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> 2
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1,1]
=> 1
[8]
=> [[8],[]]
=> [[8],[]]
=> [8]
=> 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [7,7]
=> 2
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [6,6]
=> 2
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [6,6,6]
=> ? ∊ {3,3,4,4}
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [5,5]
=> 2
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [5,5,5]
=> 3
[5,1,1,1]
=> [[5,1,1,1],[]]
=> [[5,5,5,5],[4,4,4]]
=> [5,5,5,5]
=> ? ∊ {3,3,4,4}
[4,4]
=> [[4,4],[]]
=> [[4,4],[]]
=> [4,4]
=> 2
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [4,4,4,4,4]
=> ? ∊ {3,3,4,4}
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [3,3,3,3,3,3]
=> ? ∊ {3,3,4,4}
[7,1,1]
=> [[7,1,1],[]]
=> [[7,7,7],[6,6]]
=> [7,7,7]
=> ? ∊ {3,3,3,3,4,4,4,4,5}
[6,2,1]
=> [[6,2,1],[]]
=> [[6,6,6],[5,4]]
=> [6,6,6]
=> ? ∊ {3,3,3,3,4,4,4,4,5}
[6,1,1,1]
=> [[6,1,1,1],[]]
=> [[6,6,6,6],[5,5,5]]
=> [6,6,6,6]
=> ? ∊ {3,3,3,3,4,4,4,4,5}
[5,2,1,1]
=> [[5,2,1,1],[]]
=> [[5,5,5,5],[4,4,3]]
=> [5,5,5,5]
=> ? ∊ {3,3,3,3,4,4,4,4,5}
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> [[5,5,5,5,5],[4,4,4,4]]
=> [5,5,5,5,5]
=> ? ∊ {3,3,3,3,4,4,4,4,5}
[4,2,1,1,1]
=> [[4,2,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,2]]
=> [4,4,4,4,4]
=> ? ∊ {3,3,3,3,4,4,4,4,5}
[4,1,1,1,1,1]
=> [[4,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [4,4,4,4,4,4]
=> ? ∊ {3,3,3,3,4,4,4,4,5}
[3,2,1,1,1,1]
=> [[3,2,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,1]]
=> [3,3,3,3,3,3]
=> ? ∊ {3,3,3,3,4,4,4,4,5}
[3,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> [3,3,3,3,3,3,3]
=> ? ∊ {3,3,3,3,4,4,4,4,5}
[9,1]
=> [[9,1],[]]
=> [[9,9],[8]]
=> [9,9]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[8,1,1]
=> [[8,1,1],[]]
=> [[8,8,8],[7,7]]
=> [8,8,8]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[7,2,1]
=> [[7,2,1],[]]
=> [[7,7,7],[6,5]]
=> [7,7,7]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[7,1,1,1]
=> [[7,1,1,1],[]]
=> [[7,7,7,7],[6,6,6]]
=> [7,7,7,7]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[6,3,1]
=> [[6,3,1],[]]
=> [[6,6,6],[5,3]]
=> [6,6,6]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[6,2,2]
=> [[6,2,2],[]]
=> [[6,6,6],[4,4]]
=> [6,6,6]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[6,2,1,1]
=> [[6,2,1,1],[]]
=> [[6,6,6,6],[5,5,4]]
=> [6,6,6,6]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[6,1,1,1,1]
=> [[6,1,1,1,1],[]]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [6,6,6,6,6]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[5,3,1,1]
=> [[5,3,1,1],[]]
=> [[5,5,5,5],[4,4,2]]
=> [5,5,5,5]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[5,2,2,1]
=> [[5,2,2,1],[]]
=> [[5,5,5,5],[4,3,3]]
=> [5,5,5,5]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> [[5,5,5,5,5],[4,4,4,3]]
=> [5,5,5,5,5]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[5,1,1,1,1,1]
=> [[5,1,1,1,1,1],[]]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [5,5,5,5,5,5]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[4,3,1,1,1]
=> [[4,3,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,1]]
=> [4,4,4,4,4]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[4,2,2,1,1]
=> [[4,2,2,1,1],[]]
=> [[4,4,4,4,4],[3,3,2,2]]
=> [4,4,4,4,4]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[4,2,1,1,1,1]
=> [[4,2,1,1,1,1],[]]
=> [[4,4,4,4,4,4],[3,3,3,3,2]]
=> [4,4,4,4,4,4]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[4,1,1,1,1,1,1]
=> [[4,1,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [4,4,4,4,4,4,4]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[3,3,1,1,1,1]
=> [[3,3,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3,3]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[3,2,2,1,1,1]
=> [[3,2,2,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,1,1]]
=> [3,3,3,3,3,3]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[3,2,1,1,1,1,1]
=> [[3,2,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,1]]
=> [3,3,3,3,3,3,3]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[3,1,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [3,3,3,3,3,3,3,3]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[2,1,1,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2,2]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[11]
=> [[11],[]]
=> [[11],[]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[10,1]
=> [[10,1],[]]
=> [[10,10],[9]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[9,2]
=> [[9,2],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[9,1,1]
=> [[9,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,3]
=> [[8,3],[]]
=> [[8,8],[5]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,2,1]
=> [[8,2,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,1,1,1]
=> [[8,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,4]
=> [[7,4],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,3,1]
=> [[7,3,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,2,2]
=> [[7,2,2],[]]
=> [[7,7,7],[5,5]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,2,1,1]
=> [[7,2,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,1,1,1,1]
=> [[7,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,4,1]
=> [[6,4,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,3,2]
=> [[6,3,2],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,3,1,1]
=> [[6,3,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,2,2,1]
=> [[6,2,2,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
Description
The number of cells in an integer partition whose arm and leg length coincide.
Matching statistic: St000783
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> [1]
=> 1
[2]
=> [[2],[]]
=> [[2],[]]
=> [2]
=> 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 1
[3]
=> [[3],[]]
=> [[3],[]]
=> [3]
=> 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1
[4]
=> [[4],[]]
=> [[4],[]]
=> [4]
=> 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 2
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 1
[5]
=> [[5],[]]
=> [[5],[]]
=> [5]
=> 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 2
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 3
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 2
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 1
[6]
=> [[6],[]]
=> [[6],[]]
=> [6]
=> 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [5,5]
=> 2
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [4,4]
=> 2
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> 3
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> [3,3]
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 3
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 3
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 2
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> 1
[7]
=> [[7],[]]
=> [[7],[]]
=> [7]
=> 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [6,6]
=> 2
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [5,5]
=> 2
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [5,5,5]
=> 3
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [4,4]
=> 2
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> 3
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> 4
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 3
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 3
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> 3
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> 3
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [2,2,2,2,2]
=> 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> 2
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1,1]
=> 1
[8]
=> [[8],[]]
=> [[8],[]]
=> [8]
=> 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [7,7]
=> ? ∊ {2,2,3,3,4,4}
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [6,6]
=> 2
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [6,6,6]
=> ? ∊ {2,2,3,3,4,4}
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [5,5]
=> 2
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [5,5,5]
=> 3
[5,1,1,1]
=> [[5,1,1,1],[]]
=> [[5,5,5,5],[4,4,4]]
=> [5,5,5,5]
=> ? ∊ {2,2,3,3,4,4}
[4,4]
=> [[4,4],[]]
=> [[4,4],[]]
=> [4,4]
=> 2
[4,3,1]
=> [[4,3,1],[]]
=> [[4,4,4],[3,1]]
=> [4,4,4]
=> 3
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [4,4,4,4,4]
=> ? ∊ {2,2,3,3,4,4}
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [3,3,3,3,3,3]
=> ? ∊ {2,2,3,3,4,4}
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? ∊ {2,2,3,3,4,4}
[8,1]
=> [[8,1],[]]
=> [[8,8],[7]]
=> [8,8]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[7,2]
=> [[7,2],[]]
=> [[7,7],[5]]
=> [7,7]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[7,1,1]
=> [[7,1,1],[]]
=> [[7,7,7],[6,6]]
=> [7,7,7]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[6,2,1]
=> [[6,2,1],[]]
=> [[6,6,6],[5,4]]
=> [6,6,6]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[6,1,1,1]
=> [[6,1,1,1],[]]
=> [[6,6,6,6],[5,5,5]]
=> [6,6,6,6]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[5,2,1,1]
=> [[5,2,1,1],[]]
=> [[5,5,5,5],[4,4,3]]
=> [5,5,5,5]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> [[5,5,5,5,5],[4,4,4,4]]
=> [5,5,5,5,5]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[4,2,1,1,1]
=> [[4,2,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,2]]
=> [4,4,4,4,4]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[4,1,1,1,1,1]
=> [[4,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [4,4,4,4,4,4]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[3,2,1,1,1,1]
=> [[3,2,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,1]]
=> [3,3,3,3,3,3]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[3,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> [3,3,3,3,3,3,3]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[2,2,1,1,1,1,1]
=> [[2,2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[2,1,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5}
[9,1]
=> [[9,1],[]]
=> [[9,9],[8]]
=> [9,9]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[8,2]
=> [[8,2],[]]
=> [[8,8],[6]]
=> [8,8]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[8,1,1]
=> [[8,1,1],[]]
=> [[8,8,8],[7,7]]
=> [8,8,8]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[7,3]
=> [[7,3],[]]
=> [[7,7],[4]]
=> [7,7]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[7,2,1]
=> [[7,2,1],[]]
=> [[7,7,7],[6,5]]
=> [7,7,7]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[7,1,1,1]
=> [[7,1,1,1],[]]
=> [[7,7,7,7],[6,6,6]]
=> [7,7,7,7]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[6,3,1]
=> [[6,3,1],[]]
=> [[6,6,6],[5,3]]
=> [6,6,6]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[6,2,2]
=> [[6,2,2],[]]
=> [[6,6,6],[4,4]]
=> [6,6,6]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[6,2,1,1]
=> [[6,2,1,1],[]]
=> [[6,6,6,6],[5,5,4]]
=> [6,6,6,6]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[6,1,1,1,1]
=> [[6,1,1,1,1],[]]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [6,6,6,6,6]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[5,3,1,1]
=> [[5,3,1,1],[]]
=> [[5,5,5,5],[4,4,2]]
=> [5,5,5,5]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[5,2,2,1]
=> [[5,2,2,1],[]]
=> [[5,5,5,5],[4,3,3]]
=> [5,5,5,5]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> [[5,5,5,5,5],[4,4,4,3]]
=> [5,5,5,5,5]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[5,1,1,1,1,1]
=> [[5,1,1,1,1,1],[]]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [5,5,5,5,5,5]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[4,3,1,1,1]
=> [[4,3,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,1]]
=> [4,4,4,4,4]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[4,2,2,1,1]
=> [[4,2,2,1,1],[]]
=> [[4,4,4,4,4],[3,3,2,2]]
=> [4,4,4,4,4]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[4,2,1,1,1,1]
=> [[4,2,1,1,1,1],[]]
=> [[4,4,4,4,4,4],[3,3,3,3,2]]
=> [4,4,4,4,4,4]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[4,1,1,1,1,1,1]
=> [[4,1,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [4,4,4,4,4,4,4]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[3,3,1,1,1,1]
=> [[3,3,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3,3]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[3,2,2,1,1,1]
=> [[3,2,2,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,1,1]]
=> [3,3,3,3,3,3]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[3,2,1,1,1,1,1]
=> [[3,2,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,1]]
=> [3,3,3,3,3,3,3]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[3,1,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [3,3,3,3,3,3,3,3]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[2,2,2,1,1,1,1]
=> [[2,2,2,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[2,2,1,1,1,1,1,1]
=> [[2,2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[2,1,1,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2,2]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5}
[11]
=> [[11],[]]
=> [[11],[]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[10,1]
=> [[10,1],[]]
=> [[10,10],[9]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[9,2]
=> [[9,2],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[9,1,1]
=> [[9,1,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,3]
=> [[8,3],[]]
=> [[8,8],[5]]
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,2,1]
=> [[8,2,1],[]]
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St000316
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 83%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,3,4] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3,5,4,1] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1,4] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => ? ∊ {1,1}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [2,3,4,6,5,1] => 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,5,3,4,1] => 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,4,5,2,3] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,5,1] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2,5] => 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [3,2,4,5,6,1] => 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => ? ∊ {1,1}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => ? ∊ {1,1,2,2}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [2,3,4,5,7,6,1] => ? ∊ {1,1,2,2}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,3,4,6,5,2] => 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [2,3,6,4,5,1] => 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,4,2,1] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,5,2,1,3] => 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,1,3,2] => 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,5,2,1] => 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [4,2,3,5,6,1] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [3,2,4,5,1,6] => 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => ? ∊ {1,1,2,2}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [2,1,3,4,5,6,7,8] => ? ∊ {1,1,2,2}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => ? ∊ {1,1,2,2,2,2,3,3}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [2,3,4,5,6,8,7,1] => ? ∊ {1,1,2,2,2,2,3,3}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [1,3,4,5,7,6,2] => ? ∊ {1,1,2,2,2,2,3,3}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [2,3,4,7,5,6,1] => ? ∊ {1,1,2,2,2,2,3,3}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,4,6,5,3] => 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [3,4,6,5,2,1] => 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [2,6,3,4,5,1] => 4
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,2,5,6,3,4] => 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,5,4,3,1] => 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,5,1,4,2] => 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,4,2,1] => 4
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [5,2,3,4,6,1] => 4
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,5,3,1,2] => 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,5,1,3] => 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,2,5,1] => 3
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [4,3,5,6,2,1] => 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [4,2,3,5,6,7,1] => ? ∊ {1,1,2,2,2,2,3,3}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => [3,2,4,5,6,1,7] => ? ∊ {1,1,2,2,2,2,3,3}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,8,1] => ? ∊ {1,1,2,2,2,2,3,3}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,1,3,4,5,6,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [2,3,4,5,6,7,9,8,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => [1,3,4,5,6,8,7,2] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,1,4,5,6,7] => [2,3,4,5,8,6,7,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => [1,2,4,5,7,6,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => [3,4,5,7,6,2,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,1,5,6] => [2,3,7,4,5,6,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [5,2,3,4,6,7,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => [4,3,5,6,7,2,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,8,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [3,2,4,5,1,6,7] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => [3,2,4,5,6,7,1,8] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,9,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,1,3,4,5,6,7,8,9,10] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,4,4}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,11,10] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [2,3,4,5,6,7,8,10,9,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [1,3,4,5,6,7,9,8,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => [2,3,4,5,6,9,7,8,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [8,4,1,2,3,5,6,7] => [1,2,4,5,6,8,7,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => [3,4,5,6,8,7,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,1,5,6,7] => [2,3,4,8,5,6,7,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,2,3,4,6] => [1,2,3,5,7,6,4] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => [2,4,5,7,6,3,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => [3,4,5,7,1,6,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,1,5,6] => [3,4,7,5,6,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => [2,7,3,4,5,6,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [1,2,3,6,7,4,5] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => [6,2,3,4,5,7,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,7,1] => [5,3,4,6,7,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,8,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,7,1] => [4,2,5,6,7,1,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [4,3,5,6,2,7,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => [4,3,5,6,7,8,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [4,2,3,5,6,7,8,9,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [3,4,1,2,5,6,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [3,2,4,1,5,6,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000836The number of descents of distance 2 of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001432The order dimension of the partition. St000619The number of cyclic descents of a permutation. St000711The number of big exceedences of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001322The size of a minimal independent dominating set in a graph. St000647The number of big descents of a permutation. St000837The number of ascents of distance 2 of a permutation. St000242The number of indices that are not cyclical small weak excedances. St000099The number of valleys of a permutation, including the boundary. St000392The length of the longest run of ones in a binary word. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. St001829The common independence number of a graph. St000023The number of inner peaks of a permutation. St001388The number of non-attacking neighbors of a permutation. St000075The orbit size of a standard tableau under promotion. St001394The genus of a permutation. St000260The radius of a connected graph. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St001469The holeyness of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001083The number of boxed occurrences of 132 in a permutation. St000308The height of the tree associated to a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000662The staircase size of the code of a permutation. St000317The cycle descent number of a permutation. St001569The maximal modular displacement of a permutation. St001729The number of visible descents of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001948The number of augmented double ascents of a permutation. St000646The number of big ascents of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000863The length of the first row of the shifted shape of a permutation. 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. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001928The number of non-overlapping descents in a permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001557The number of inversions of the second entry of a permutation. St000092The number of outer peaks of a permutation. St000353The number of inner valleys of a permutation. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001863The number of weak excedances of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001868The number of alignments of type NE of a signed permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001487The number of inner corners of a skew partition. St001864The number of excedances of a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation.
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