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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000225
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> 0
[[2],[]]
=> [2]
=> 0
[[1,1],[]]
=> [1,1]
=> 0
[[2,1],[1]]
=> [2,1]
=> 1
[[3],[]]
=> [3]
=> 0
[[2,1],[]]
=> [2,1]
=> 1
[[3,1],[1]]
=> [3,1]
=> 2
[[2,2],[1]]
=> [2,2]
=> 0
[[3,2],[2]]
=> [3,2]
=> 1
[[1,1,1],[]]
=> [1,1,1]
=> 0
[[2,2,1],[1,1]]
=> [2,2,1]
=> 1
[[2,1,1],[1]]
=> [2,1,1]
=> 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> 2
[[4],[]]
=> [4]
=> 0
[[3,1],[]]
=> [3,1]
=> 2
[[4,1],[1]]
=> [4,1]
=> 3
[[2,2],[]]
=> [2,2]
=> 0
[[3,2],[1]]
=> [3,2]
=> 1
[[4,2],[2]]
=> [4,2]
=> 2
[[2,1,1],[]]
=> [2,1,1]
=> 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> 2
[[3,1,1],[1]]
=> [3,1,1]
=> 2
[[4,2,1],[2,1]]
=> [4,2,1]
=> 3
[[3,3],[2]]
=> [3,3]
=> 0
[[4,3],[3]]
=> [4,3]
=> 1
[[2,2,1],[1]]
=> [2,2,1]
=> 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> 2
[[3,2,1],[2]]
=> [3,2,1]
=> 2
[[4,3,1],[3,1]]
=> [4,3,1]
=> 3
[[2,2,2],[1,1]]
=> [2,2,2]
=> 0
[[3,3,2],[2,2]]
=> [3,3,2]
=> 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> 2
[[1,1,1,1],[]]
=> [1,1,1,1]
=> 0
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> 2
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> 2
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> 2
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> 3
[[5],[]]
=> [5]
=> 0
[[4,1],[]]
=> [4,1]
=> 3
[[5,1],[1]]
=> [5,1]
=> 4
[[3,2],[]]
=> [3,2]
=> 1
[[4,2],[1]]
=> [4,2]
=> 2
[[5,2],[2]]
=> [5,2]
=> 3
[[3,1,1],[]]
=> [3,1,1]
=> 2
[[4,2,1],[1,1]]
=> [4,2,1]
=> 3
[[4,1,1],[1]]
=> [4,1,1]
=> 3
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000147
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [[1],[]]
=> []
=> 0
[[2],[]]
=> [[2],[]]
=> []
=> 0
[[1,1],[]]
=> [[1,1],[]]
=> []
=> 0
[[2,1],[1]]
=> [[2,1],[1]]
=> [1]
=> 1
[[3],[]]
=> [[3],[]]
=> []
=> 0
[[2,1],[]]
=> [[2,2],[1]]
=> [1]
=> 1
[[3,1],[1]]
=> [[3,2],[2]]
=> [2]
=> 2
[[2,2],[1]]
=> [[2,1],[]]
=> []
=> 0
[[3,2],[2]]
=> [[3,1],[1]]
=> [1]
=> 1
[[1,1,1],[]]
=> [[1,1,1],[]]
=> []
=> 0
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> [1]
=> 1
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> [1,1]
=> 1
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [2,1]
=> 2
[[4],[]]
=> [[4],[]]
=> []
=> 0
[[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> 2
[[4,1],[1]]
=> [[4,3],[3]]
=> [3]
=> 3
[[2,2],[]]
=> [[2,2],[]]
=> []
=> 0
[[3,2],[1]]
=> [[3,2],[1]]
=> [1]
=> 1
[[4,2],[2]]
=> [[4,2],[2]]
=> [2]
=> 2
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> [2,1]
=> 2
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> [2,2]
=> 2
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> [3,2]
=> 3
[[3,3],[2]]
=> [[3,1],[]]
=> []
=> 0
[[4,3],[3]]
=> [[4,1],[1]]
=> [1]
=> 1
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> [2]
=> 2
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> [2,1]
=> 2
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> [3,1]
=> 3
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> []
=> 0
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> [1]
=> 1
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> [1,1]
=> 1
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> [2,1]
=> 2
[[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> [1]
=> 1
[[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> [1,1]
=> 1
[[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> [2,1]
=> 2
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 2
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 2
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 3
[[5],[]]
=> [[5],[]]
=> []
=> 0
[[4,1],[]]
=> [[4,4],[3]]
=> [3]
=> 3
[[5,1],[1]]
=> [[5,4],[4]]
=> [4]
=> 4
[[3,2],[]]
=> [[3,3],[1]]
=> [1]
=> 1
[[4,2],[1]]
=> [[4,3],[2]]
=> [2]
=> 2
[[5,2],[2]]
=> [[5,3],[3]]
=> [3]
=> 3
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> [3,2]
=> 3
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> [3,3]
=> 3
[[7,1,1,1],[]]
=> [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> ? = 6
[[6,1,1,1,1],[]]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> ? = 5
[[5,1,1,1,1,1],[]]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> ? = 4
[[4,1,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> ? = 3
[[2,2,1,1,1,1,1],[1]]
=> ?
=> ?
=> ? = 1
[[2,2,2,1,1,1,1],[1,1]]
=> ?
=> ?
=> ? = 1
[[3,2,1,1,1,1],[1]]
=> ?
=> ?
=> ? = 2
[[3,2,2,1,1,1],[1,1]]
=> ?
=> ?
=> ? = 2
[[4,3,1,1,1],[2]]
=> ?
=> ?
=> ? = 3
[[3,3,2,2,1,1],[2,1,1]]
=> ?
=> ?
=> ? = 2
[[4,3,2,1,1],[2,1]]
=> ?
=> ?
=> ? = 3
[[4,4,2,1,1],[3,1]]
=> ?
=> ?
=> ? = 3
[[4,3,3,1,1],[2,2]]
=> ?
=> ?
=> ? = 3
[[4,4,3,1,1],[3,2]]
=> ?
=> ?
=> ? = 3
[[5,3,1,1],[2]]
=> ?
=> ?
=> ? = 4
[[4,3,2,2,1],[2,1,1]]
=> ?
=> ?
=> ? = 3
[[4,4,2,2,1],[3,1,1]]
=> ?
=> ?
=> ? = 3
[[4,3,3,2,1],[2,2,1]]
=> ?
=> ?
=> ? = 3
[[4,4,3,2,1],[3,2,1]]
=> ?
=> ?
=> ? = 3
[[5,3,2,1],[2,1]]
=> ?
=> ?
=> ? = 4
[[5,4,2,1],[3,1]]
=> ?
=> ?
=> ? = 4
[[5,3,3,1],[2,2]]
=> ?
=> ?
=> ? = 4
[[5,4,3,1],[3,2]]
=> ?
=> ?
=> ? = 4
[[5,3,2,2],[2,1,1]]
=> ?
=> ?
=> ? = 3
[[5,4,2,2],[3,1,1]]
=> ?
=> ?
=> ? = 3
[[5,3,3,2],[2,2,1]]
=> ?
=> ?
=> ? = 3
[[5,4,3,2],[3,2,1]]
=> ?
=> ?
=> ? = 3
[[6,4,2],[3,1]]
=> ?
=> ?
=> ? = 4
[[4,4,3,1],[1]]
=> ?
=> ?
=> ? = 3
[[6,4],[2]]
=> ?
=> ?
=> ? = 2
[[6,3],[1]]
=> ?
=> ?
=> ? = 3
[[5,4,3],[2,1]]
=> ?
=> ?
=> ? = 2
[[5,3,3],[1,1]]
=> ?
=> ?
=> ? = 2
[[5,4,3],[1,1]]
=> ?
=> ?
=> ? = 2
[[6,4],[1]]
=> ?
=> ?
=> ? = 2
[[5,4,2],[2]]
=> ?
=> ?
=> ? = 3
[[5,3,2],[1]]
=> ?
=> ?
=> ? = 3
[[4,4,3,2],[2,1]]
=> ?
=> ?
=> ? = 2
[[4,3,3,2],[1,1]]
=> ?
=> ?
=> ? = 2
[[4,4,3,2],[1,1]]
=> ?
=> ?
=> ? = 2
[[5,4,2],[1]]
=> ?
=> ?
=> ? = 3
[[4,4,2,2],[2]]
=> ?
=> ?
=> ? = 2
[[4,3,2,2],[1]]
=> ?
=> ?
=> ? = 2
[[4,4,2,2],[1]]
=> ?
=> ?
=> ? = 2
[[4,4,3,2],[2]]
=> ?
=> ?
=> ? = 2
[[4,3,3,2],[1]]
=> ?
=> ?
=> ? = 2
[[4,4,3,2],[1]]
=> ?
=> ?
=> ? = 2
[[5,4,3],[2]]
=> ?
=> ?
=> ? = 2
[[5,3,3],[1]]
=> ?
=> ?
=> ? = 2
[[5,4,3],[1]]
=> ?
=> ?
=> ? = 2
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00187: Skew partitions —conjugate⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [[1],[]]
=> [[1],[]]
=> []
=> 0
[[2],[]]
=> [[1,1],[]]
=> [[1,1],[]]
=> []
=> 0
[[1,1],[]]
=> [[2],[]]
=> [[2],[]]
=> []
=> 0
[[2,1],[1]]
=> [[2,1],[1]]
=> [[2,1],[1]]
=> [1]
=> 1
[[3],[]]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> []
=> 0
[[2,1],[]]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [1]
=> 1
[[3,1],[1]]
=> [[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> [1,1]
=> 2
[[2,2],[1]]
=> [[2,2],[1]]
=> [[2,1],[]]
=> []
=> 0
[[3,2],[2]]
=> [[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> [1]
=> 1
[[1,1,1],[]]
=> [[3],[]]
=> [[3],[]]
=> []
=> 0
[[2,2,1],[1,1]]
=> [[3,2],[2]]
=> [[3,1],[1]]
=> [1]
=> 1
[[2,1,1],[1]]
=> [[3,1],[1]]
=> [[3,2],[2]]
=> [2]
=> 1
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [2,1]
=> 2
[[4],[]]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> []
=> 0
[[3,1],[]]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2
[[4,1],[1]]
=> [[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
[[2,2],[]]
=> [[2,2],[]]
=> [[2,2],[]]
=> []
=> 0
[[3,2],[1]]
=> [[2,2,1],[1]]
=> [[2,2,1],[1]]
=> [1]
=> 1
[[4,2],[2]]
=> [[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> [1,1]
=> 2
[[2,1,1],[]]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> 1
[[3,2,1],[1,1]]
=> [[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> [2,1]
=> 2
[[3,1,1],[1]]
=> [[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> [2,2]
=> 2
[[4,2,1],[2,1]]
=> [[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 3
[[3,3],[2]]
=> [[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> []
=> 0
[[4,3],[3]]
=> [[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> [1]
=> 1
[[2,2,1],[1]]
=> [[3,2],[1]]
=> [[3,2],[1]]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> [1,1]
=> 2
[[3,2,1],[2]]
=> [[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> [2,1]
=> 2
[[4,3,1],[3,1]]
=> [[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 3
[[2,2,2],[1,1]]
=> [[3,3],[2]]
=> [[3,1],[]]
=> []
=> 0
[[3,3,2],[2,2]]
=> [[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> [1]
=> 1
[[3,2,2],[2,1]]
=> [[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> [2]
=> 1
[[4,3,2],[3,2]]
=> [[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> [2,1]
=> 2
[[1,1,1,1],[]]
=> [[4],[]]
=> [[4],[]]
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [[4,3],[3]]
=> [[4,1],[1]]
=> [1]
=> 1
[[2,2,1,1],[1,1]]
=> [[4,2],[2]]
=> [[4,2],[2]]
=> [2]
=> 1
[[3,3,2,1],[2,2,1]]
=> [[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> [2,1]
=> 2
[[2,1,1,1],[1]]
=> [[4,1],[1]]
=> [[4,3],[3]]
=> [3]
=> 1
[[3,2,2,1],[2,1,1]]
=> [[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> [3,1]
=> 2
[[3,2,1,1],[2,1]]
=> [[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> [3,2]
=> 2
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 3
[[5],[]]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> []
=> 0
[[4,1],[]]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[[5,1],[1]]
=> [[2,1,1,1,1],[1]]
=> [[2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[[3,2],[]]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [1]
=> 1
[[4,2],[1]]
=> [[2,2,1,1],[1]]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[[5,2],[2]]
=> [[2,2,1,1,1],[1,1]]
=> [[2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 3
[[3,1,1],[]]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[[4,2,1],[1,1]]
=> [[3,2,1,1],[2]]
=> [[3,3,3,1],[2,2,1]]
=> [2,2,1]
=> 3
[[4,1,1],[1]]
=> [[3,1,1,1],[1]]
=> [[3,3,3,2],[2,2,2]]
=> [2,2,2]
=> 3
[[7,1,1,1],[]]
=> [[4,1,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> ? = 6
[[6,1,1,1,1],[]]
=> [[5,1,1,1,1,1],[]]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> ? = 5
[[5,1,1,1,1,1],[]]
=> [[6,1,1,1,1],[]]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> ? = 4
[[4,1,1,1,1,1,1],[]]
=> [[7,1,1,1],[]]
=> [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> ? = 3
[[2,2,1,1,1,1,1],[1]]
=> ?
=> ?
=> ?
=> ? = 1
[[2,2,2,1,1,1,1],[1,1]]
=> [[7,3],[2]]
=> ?
=> ?
=> ? = 1
[[3,2,1,1,1,1],[1]]
=> [[6,2,1],[1]]
=> ?
=> ?
=> ? = 2
[[3,2,2,1,1,1],[1,1]]
=> [[6,3,1],[2]]
=> ?
=> ?
=> ? = 2
[[4,3,1,1,1],[2]]
=> ?
=> ?
=> ?
=> ? = 3
[[3,3,2,2,1,1],[2,1,1]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,3,2,1,1],[2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,2,1,1],[3,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,3,3,1,1],[2,2]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,3,1,1],[3,2]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,3,1,1],[2]]
=> ?
=> ?
=> ?
=> ? = 4
[[4,3,2,2,1],[2,1,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,2,2,1],[3,1,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,3,3,2,1],[2,2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,3,2,1],[3,2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,3,2,1],[2,1]]
=> ?
=> ?
=> ?
=> ? = 4
[[5,4,2,1],[3,1]]
=> ?
=> ?
=> ?
=> ? = 4
[[5,3,3,1],[2,2]]
=> [[4,3,3,1,1],[2,2]]
=> ?
=> ?
=> ? = 4
[[5,4,3,1],[3,2]]
=> ?
=> ?
=> ?
=> ? = 4
[[5,3,2,2],[2,1,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,4,2,2],[3,1,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,3,3,2],[2,2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,4,3,2],[3,2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[6,4,2],[3,1]]
=> ?
=> ?
=> ?
=> ? = 4
[[4,4,3,1],[1]]
=> [[4,3,3,2],[1]]
=> ?
=> ?
=> ? = 3
[[6,4],[2]]
=> [[2,2,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? = 2
[[6,3],[1]]
=> [[2,2,2,1,1,1],[1]]
=> ?
=> ?
=> ? = 3
[[5,4,3],[2,1]]
=> [[3,3,3,2,1],[2,1]]
=> ?
=> ?
=> ? = 2
[[5,3,3],[1,1]]
=> [[3,3,3,1,1],[2]]
=> ?
=> ?
=> ? = 2
[[5,4,3],[1,1]]
=> [[3,3,3,2,1],[2]]
=> ?
=> ?
=> ? = 2
[[6,4],[1]]
=> [[2,2,2,2,1,1],[1]]
=> ?
=> ?
=> ? = 2
[[5,4,2],[2]]
=> [[3,3,2,2,1],[1,1]]
=> ?
=> ?
=> ? = 3
[[5,3,2],[1]]
=> [[3,3,2,1,1],[1]]
=> ?
=> ?
=> ? = 3
[[4,4,3,2],[2,1]]
=> [[4,4,3,2],[2,1]]
=> ?
=> ?
=> ? = 2
[[4,3,3,2],[1,1]]
=> [[4,4,3,1],[2]]
=> ?
=> ?
=> ? = 2
[[4,4,3,2],[1,1]]
=> [[4,4,3,2],[2]]
=> ?
=> ?
=> ? = 2
[[5,4,2],[1]]
=> [[3,3,2,2,1],[1]]
=> ?
=> ?
=> ? = 3
[[4,4,2,2],[2]]
=> [[4,4,2,2],[1,1]]
=> ?
=> ?
=> ? = 2
[[4,3,2,2],[1]]
=> [[4,4,2,1],[1]]
=> ?
=> ?
=> ? = 2
[[4,4,2,2],[1]]
=> [[4,4,2,2],[1]]
=> ?
=> ?
=> ? = 2
[[4,4,3,2],[2]]
=> [[4,4,3,2],[1,1]]
=> ?
=> ?
=> ? = 2
[[4,3,3,2],[1]]
=> [[4,4,3,1],[1]]
=> ?
=> ?
=> ? = 2
[[4,4,3,2],[1]]
=> [[4,4,3,2],[1]]
=> ?
=> ?
=> ? = 2
[[5,4,3],[2]]
=> [[3,3,3,2,1],[1,1]]
=> ?
=> ?
=> ? = 2
[[5,3,3],[1]]
=> [[3,3,3,1,1],[1]]
=> ?
=> ?
=> ? = 2
[[5,4,3],[1]]
=> [[3,3,3,2,1],[1]]
=> ?
=> ?
=> ? = 2
Description
The length of the partition.
Matching statistic: St000676
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [[1],[]]
=> []
=> []
=> 0
[[2],[]]
=> [[2],[]]
=> []
=> []
=> 0
[[1,1],[]]
=> [[1,1],[]]
=> []
=> []
=> 0
[[2,1],[1]]
=> [[2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3],[]]
=> [[3],[]]
=> []
=> []
=> 0
[[2,1],[]]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[[3,1],[1]]
=> [[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[2,2],[1]]
=> [[2,1],[]]
=> []
=> []
=> 0
[[3,2],[2]]
=> [[3,1],[1]]
=> [1]
=> [1,0]
=> 1
[[1,1,1],[]]
=> [[1,1,1],[]]
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4],[]]
=> [[4],[]]
=> []
=> []
=> 0
[[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[4,1],[1]]
=> [[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[2,2],[]]
=> [[2,2],[]]
=> []
=> []
=> 0
[[3,2],[1]]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2],[2]]
=> [[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[3,3],[2]]
=> [[3,1],[]]
=> []
=> []
=> 0
[[4,3],[3]]
=> [[4,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> []
=> []
=> 0
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[5],[]]
=> [[5],[]]
=> []
=> []
=> 0
[[4,1],[]]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[5,1],[1]]
=> [[5,4],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[3,2],[]]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2],[1]]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[5,2],[2]]
=> [[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[[8,1,1],[]]
=> [[8,8,8],[7,7]]
=> [7,7]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 7
[[7,1,1,1],[]]
=> [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6
[[6,1,1,1,1],[]]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 5
[[5,1,1,1,1,1],[]]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 4
[[4,1,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 3
[[3,1,1,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
[[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]]
=> [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
[[2,2,1,1,1,1,1],[1]]
=> ?
=> ?
=> ?
=> ? = 1
[[2,2,2,1,1,1,1],[1,1]]
=> ?
=> ?
=> ?
=> ? = 1
[[3,2,1,1,1,1],[1]]
=> ?
=> ?
=> ?
=> ? = 2
[[3,2,2,1,1,1],[1,1]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,3,1,1,1],[2]]
=> ?
=> ?
=> ?
=> ? = 3
[[3,3,2,2,1,1],[2,1,1]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,3,2,1,1],[2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,2,1,1],[3,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,3,3,1,1],[2,2]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,3,1,1],[3,2]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,3,1,1],[2]]
=> ?
=> ?
=> ?
=> ? = 4
[[4,3,2,2,1],[2,1,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,2,2,1],[3,1,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,3,3,2,1],[2,2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,3,2,1],[3,2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,3,2,1],[2,1]]
=> ?
=> ?
=> ?
=> ? = 4
[[5,4,2,1],[3,1]]
=> ?
=> ?
=> ?
=> ? = 4
[[5,3,3,1],[2,2]]
=> ?
=> ?
=> ?
=> ? = 4
[[5,4,3,1],[3,2]]
=> ?
=> ?
=> ?
=> ? = 4
[[5,3,2,2],[2,1,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,4,2,2],[3,1,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,3,3,2],[2,2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,4,3,2],[3,2,1]]
=> ?
=> ?
=> ?
=> ? = 3
[[6,4,2],[3,1]]
=> ?
=> ?
=> ?
=> ? = 4
[[4,4,3,1],[1]]
=> ?
=> ?
=> ?
=> ? = 3
[[6,4],[2]]
=> ?
=> ?
=> ?
=> ? = 2
[[6,3],[1]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,4,3],[2,1]]
=> ?
=> ?
=> ?
=> ? = 2
[[5,3,3],[1,1]]
=> ?
=> ?
=> ?
=> ? = 2
[[5,4,3],[1,1]]
=> ?
=> ?
=> ?
=> ? = 2
[[6,4],[1]]
=> ?
=> ?
=> ?
=> ? = 2
[[5,4,2],[2]]
=> ?
=> ?
=> ?
=> ? = 3
[[5,3,2],[1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,3,2],[2,1]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,3,3,2],[1,1]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,4,3,2],[1,1]]
=> ?
=> ?
=> ?
=> ? = 2
[[5,4,2],[1]]
=> ?
=> ?
=> ?
=> ? = 3
[[4,4,2,2],[2]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,3,2,2],[1]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,4,2,2],[1]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,4,3,2],[2]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,3,3,2],[1]]
=> ?
=> ?
=> ?
=> ? = 2
[[4,4,3,2],[1]]
=> ?
=> ?
=> ?
=> ? = 2
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000734
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Values
[[1],[]]
=> [[1],[]]
=> []
=> []
=> ? = 0
[[2],[]]
=> [[2],[]]
=> []
=> []
=> ? = 0
[[1,1],[]]
=> [[1,1],[]]
=> []
=> []
=> ? = 0
[[2,1],[1]]
=> [[2,1],[1]]
=> [1]
=> [[1]]
=> 1
[[3],[]]
=> [[3],[]]
=> []
=> []
=> ? = 0
[[2,1],[]]
=> [[2,2],[1]]
=> [1]
=> [[1]]
=> 1
[[3,1],[1]]
=> [[3,2],[2]]
=> [2]
=> [[1,2]]
=> 2
[[2,2],[1]]
=> [[2,1],[]]
=> []
=> []
=> ? = 0
[[3,2],[2]]
=> [[3,1],[1]]
=> [1]
=> [[1]]
=> 1
[[1,1,1],[]]
=> [[1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4],[]]
=> [[4],[]]
=> []
=> []
=> ? = 0
[[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> [[1,2]]
=> 2
[[4,1],[1]]
=> [[4,3],[3]]
=> [3]
=> [[1,2,3]]
=> 3
[[2,2],[]]
=> [[2,2],[]]
=> []
=> []
=> ? = 0
[[3,2],[1]]
=> [[3,2],[1]]
=> [1]
=> [[1]]
=> 1
[[4,2],[2]]
=> [[4,2],[2]]
=> [2]
=> [[1,2]]
=> 2
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[3,3],[2]]
=> [[3,1],[]]
=> []
=> []
=> ? = 0
[[4,3],[3]]
=> [[4,1],[1]]
=> [1]
=> [[1]]
=> 1
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> [2]
=> [[1,2]]
=> 2
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> []
=> []
=> ? = 0
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[[5],[]]
=> [[5],[]]
=> []
=> []
=> ? = 0
[[4,1],[]]
=> [[4,4],[3]]
=> [3]
=> [[1,2,3]]
=> 3
[[5,1],[1]]
=> [[5,4],[4]]
=> [4]
=> [[1,2,3,4]]
=> 4
[[3,2],[]]
=> [[3,3],[1]]
=> [1]
=> [[1]]
=> 1
[[4,2],[1]]
=> [[4,3],[2]]
=> [2]
=> [[1,2]]
=> 2
[[5,2],[2]]
=> [[5,3],[3]]
=> [3]
=> [[1,2,3]]
=> 3
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
[[5,2,1],[2,1]]
=> [[5,4,3],[4,3]]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 4
[[3,3],[1]]
=> [[3,2],[]]
=> []
=> []
=> ? = 0
[[4,3],[2]]
=> [[4,2],[1]]
=> [1]
=> [[1]]
=> 1
[[5,3],[3]]
=> [[5,2],[2]]
=> [2]
=> [[1,2]]
=> 2
[[2,2,1],[]]
=> [[2,2,2],[1]]
=> [1]
=> [[1]]
=> 1
[[3,3,1],[1,1]]
=> [[3,2,2],[2]]
=> [2]
=> [[1,2]]
=> 2
[[3,2,1],[1]]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4,3,1],[2,1]]
=> [[4,3,2],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[4,2,1],[2]]
=> [[4,4,2],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[5,3,1],[3,1]]
=> [[5,4,2],[4,2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
[[3,2,2],[1,1]]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[[4,3,2],[2,2]]
=> [[4,2,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[[4,2,2],[2,1]]
=> [[4,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[[4,4],[3]]
=> [[4,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2],[1]]
=> [[2,2,1],[]]
=> []
=> []
=> ? = 0
[[3,3,3],[2,2]]
=> [[3,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,2],[1,1,1]]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? = 0
[[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[6],[]]
=> [[6],[]]
=> []
=> []
=> ? = 0
[[3,3],[]]
=> [[3,3],[]]
=> []
=> []
=> ? = 0
[[4,4],[2]]
=> [[4,2],[]]
=> []
=> []
=> ? = 0
[[2,2,2],[]]
=> [[2,2,2],[]]
=> []
=> []
=> ? = 0
[[5,5],[4]]
=> [[5,1],[]]
=> []
=> []
=> ? = 0
[[3,3,3],[2,1]]
=> [[3,2,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,2],[1,1]]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,2,2],[1,1,1,1]]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[7],[]]
=> [[7],[]]
=> []
=> []
=> ? = 0
[[7,2,1],[2,1]]
=> [[7,6,5],[6,5]]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 6
[[6,2,1,1],[2,1]]
=> [[6,6,5,4],[5,5,4]]
=> [5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? = 5
[[4,4],[1]]
=> [[4,3],[]]
=> []
=> []
=> ? = 0
[[5,2,1,1,1],[2,1]]
=> [[5,5,5,4,3],[4,4,4,3]]
=> [4,4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15]]
=> ? = 4
[[5,5],[3]]
=> [[5,2],[]]
=> []
=> []
=> ? = 0
[[3,3,3],[1,1]]
=> [[3,2,2],[]]
=> []
=> []
=> ? = 0
[[5,3,1,1,1],[3,1]]
=> [[5,5,5,4,2],[4,4,4,2]]
=> [4,4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14]]
=> ? = 4
[[3,3,3],[2]]
=> [[3,3,1],[]]
=> []
=> []
=> ? = 0
[[5,4,1,1,1],[4,1]]
=> [[5,5,5,4,1],[4,4,4,1]]
=> [4,4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13]]
=> ? = 4
[[2,2,2,2],[1]]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,2,2],[1,1,1]]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? = 0
[[1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[8],[]]
=> [[8],[]]
=> []
=> []
=> ? = 0
[[4,4],[]]
=> [[4,4],[]]
=> []
=> []
=> ? = 0
[[2,2,2,2],[]]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? = 0
[[1,1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[9],[]]
=> [[9],[]]
=> []
=> []
=> ? = 0
[[6,1,1,1],[]]
=> [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 5
[[3,3,3],[]]
=> [[3,3,3],[]]
=> []
=> []
=> ? = 0
[[1,1,1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[10],[]]
=> [[10],[]]
=> []
=> []
=> ? = 0
[[7,2,1],[]]
=> [[7,7,7],[6,5]]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 6
Description
The last entry in the first row of a standard tableau.
Matching statistic: St001291
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 81%●distinct values known / distinct values provided: 56%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 81%●distinct values known / distinct values provided: 56%
Values
[[1],[]]
=> [[1],[]]
=> []
=> []
=> ? = 0 + 1
[[2],[]]
=> [[2],[]]
=> []
=> []
=> ? = 0 + 1
[[1,1],[]]
=> [[1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[2,1],[1]]
=> [[2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[3],[]]
=> [[3],[]]
=> []
=> []
=> ? = 0 + 1
[[2,1],[]]
=> [[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[3,1],[1]]
=> [[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[2,2],[1]]
=> [[2,1],[]]
=> []
=> []
=> ? = 0 + 1
[[3,2],[2]]
=> [[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[1,1,1],[]]
=> [[1,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[4],[]]
=> [[4],[]]
=> []
=> []
=> ? = 0 + 1
[[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[4,1],[1]]
=> [[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[2,2],[]]
=> [[2,2],[]]
=> []
=> []
=> ? = 0 + 1
[[3,2],[1]]
=> [[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[4,2],[2]]
=> [[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[3,3],[2]]
=> [[3,1],[]]
=> []
=> []
=> ? = 0 + 1
[[4,3],[3]]
=> [[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[5],[]]
=> [[5],[]]
=> []
=> []
=> ? = 0 + 1
[[4,1],[]]
=> [[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[5,1],[1]]
=> [[5,4],[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[3,2],[]]
=> [[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[4,2],[1]]
=> [[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[5,2],[2]]
=> [[5,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[5,2,1],[2,1]]
=> [[5,4,3],[4,3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 4 + 1
[[3,3],[1]]
=> [[3,2],[]]
=> []
=> []
=> ? = 0 + 1
[[4,3],[2]]
=> [[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[5,3],[3]]
=> [[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[2,2,1],[]]
=> [[2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[[3,3,1],[1,1]]
=> [[3,2,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[3,2,1],[1]]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[4,3,1],[2,1]]
=> [[4,3,2],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[4,2,1],[2]]
=> [[4,4,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[5,3,1],[3,1]]
=> [[5,4,2],[4,2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[[3,2,2],[1,1]]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[4,3,2],[2,2]]
=> [[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[4,2,2],[2,1]]
=> [[4,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[4,4],[3]]
=> [[4,1],[]]
=> []
=> []
=> ? = 0 + 1
[[2,2,2],[1]]
=> [[2,2,1],[]]
=> []
=> []
=> ? = 0 + 1
[[3,3,3],[2,2]]
=> [[3,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[2,2,2,2],[1,1,1]]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[6],[]]
=> [[6],[]]
=> []
=> []
=> ? = 0 + 1
[[3,3],[]]
=> [[3,3],[]]
=> []
=> []
=> ? = 0 + 1
[[4,4],[2]]
=> [[4,2],[]]
=> []
=> []
=> ? = 0 + 1
[[2,2,2],[]]
=> [[2,2,2],[]]
=> []
=> []
=> ? = 0 + 1
[[5,5],[4]]
=> [[5,1],[]]
=> []
=> []
=> ? = 0 + 1
[[3,3,3],[2,1]]
=> [[3,2,1],[]]
=> []
=> []
=> ? = 0 + 1
[[2,2,2,2],[1,1]]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[2,2,2,2,2],[1,1,1,1]]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[7],[]]
=> [[7],[]]
=> []
=> []
=> ? = 0 + 1
[[7,1],[1]]
=> [[7,6],[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[[6,1,1],[1]]
=> [[6,6,5],[5,5]]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5 + 1
[[7,2,1],[2,1]]
=> [[7,6,5],[6,5]]
=> [6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[5,1,1,1],[1]]
=> [[5,5,5,4],[4,4,4]]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[[6,2,1,1],[2,1]]
=> [[6,6,5,4],[5,5,4]]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[[4,4],[1]]
=> [[4,3],[]]
=> []
=> []
=> ? = 0 + 1
[[4,1,1,1,1],[1]]
=> [[4,4,4,4,3],[3,3,3,3]]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[[5,2,2,2,1],[2,1,1,1]]
=> [[5,4,4,4,3],[4,3,3,3]]
=> [4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 + 1
[[5,2,2,1,1],[2,1,1]]
=> [[5,5,4,4,3],[4,4,3,3]]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 4 + 1
[[5,2,1,1,1],[2,1]]
=> [[5,5,5,4,3],[4,4,4,3]]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 1
[[5,5],[3]]
=> [[5,2],[]]
=> []
=> []
=> ? = 0 + 1
[[3,3,3],[1,1]]
=> [[3,2,2],[]]
=> []
=> []
=> ? = 0 + 1
[[5,3,1,1,1],[3,1]]
=> [[5,5,5,4,2],[4,4,4,2]]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[[3,1,1,1,1,1],[1]]
=> [[3,3,3,3,3,2],[2,2,2,2,2]]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[[4,2,1,1,1,1],[2,1]]
=> [[4,4,4,4,3,2],[3,3,3,3,2]]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[[3,3,3],[2]]
=> [[3,3,1],[]]
=> []
=> []
=> ? = 0 + 1
[[5,4,1,1,1],[4,1]]
=> [[5,5,5,4,1],[4,4,4,1]]
=> [4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[[2,2,2,2],[1]]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? = 0 + 1
[[2,2,2,2,2],[1,1,1]]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? = 0 + 1
[[2,1,1,1,1,1,1],[1]]
=> [[2,2,2,2,2,2,1],[1,1,1,1,1,1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[[3,2,1,1,1,1,1],[2,1]]
=> [[3,3,3,3,3,2,1],[2,2,2,2,2,1]]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St001039
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 89%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 89%
Values
[[1],[]]
=> [[1],[]]
=> []
=> []
=> ? = 0
[[2],[]]
=> [[2],[]]
=> []
=> []
=> ? = 0
[[1,1],[]]
=> [[1,1],[]]
=> []
=> []
=> ? = 0
[[2,1],[1]]
=> [[2,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[3],[]]
=> [[3],[]]
=> []
=> []
=> ? = 0
[[2,1],[]]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[3,1],[1]]
=> [[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[2,2],[1]]
=> [[2,1],[]]
=> []
=> []
=> ? = 0
[[3,2],[2]]
=> [[3,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1],[]]
=> [[1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4],[]]
=> [[4],[]]
=> []
=> []
=> ? = 0
[[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[4,1],[1]]
=> [[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[2,2],[]]
=> [[2,2],[]]
=> []
=> []
=> ? = 0
[[3,2],[1]]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[4,2],[2]]
=> [[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[3,3],[2]]
=> [[3,1],[]]
=> []
=> []
=> ? = 0
[[4,3],[3]]
=> [[4,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> []
=> []
=> ? = 0
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[5],[]]
=> [[5],[]]
=> []
=> []
=> ? = 0
[[4,1],[]]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[5,1],[1]]
=> [[5,4],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[3,2],[]]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[4,2],[1]]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[5,2],[2]]
=> [[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[[5,2,1],[2,1]]
=> [[5,4,3],[4,3]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[[3,3],[1]]
=> [[3,2],[]]
=> []
=> []
=> ? = 0
[[4,3],[2]]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[5,3],[3]]
=> [[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[2,2,1],[]]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[3,3,1],[1,1]]
=> [[3,2,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[3,2,1],[1]]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,1],[2,1]]
=> [[4,3,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[4,2,1],[2]]
=> [[4,4,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[5,3,1],[3,1]]
=> [[5,4,2],[4,2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[[3,2,2],[1,1]]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4,3,2],[2,2]]
=> [[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,2,2],[2,1]]
=> [[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[5,3,2],[3,2]]
=> [[5,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[3,2,2,1],[1,1,1]]
=> [[3,2,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,2,1,1],[1,1]]
=> [[3,3,2,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[[4,3,2,1],[2,2,1]]
=> [[4,3,2,2],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[3,1,1,1],[1]]
=> [[3,3,3,2],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[[4,2,2,1],[2,1,1]]
=> [[4,3,3,2],[3,2,2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[[4,2,1,1],[2,1]]
=> [[4,4,3,2],[3,3,2]]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[[5,3,2,1],[3,2,1]]
=> [[5,4,3,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 4
[[4,4],[3]]
=> [[4,1],[]]
=> []
=> []
=> ? = 0
[[5,4],[4]]
=> [[5,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[3,3,1],[2]]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[4,4,1],[3,1]]
=> [[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[4,3,1],[3]]
=> [[4,4,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[2,2,2],[1]]
=> [[2,2,1],[]]
=> []
=> []
=> ? = 0
[[3,3,2],[2,1]]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[3,3,3],[2,2]]
=> [[3,1,1],[]]
=> []
=> []
=> ? = 0
[[4,4,3],[3,3]]
=> [[4,1,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[2,2,2,1],[1,1]]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[2,2,2,2],[1,1,1]]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? = 0
[[3,3,3,2],[2,2,2]]
=> [[3,1,1,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? = 0
[[2,2,2,2,1],[1,1,1,1]]
=> [[2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[6],[]]
=> [[6],[]]
=> []
=> []
=> ? = 0
[[3,3],[]]
=> [[3,3],[]]
=> []
=> []
=> ? = 0
[[4,3],[1]]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[4,4],[2]]
=> [[4,2],[]]
=> []
=> []
=> ? = 0
[[5,4],[3]]
=> [[5,2],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[2,2,2],[]]
=> [[2,2,2],[]]
=> []
=> []
=> ? = 0
[[3,3,2],[1,1]]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[5,5],[4]]
=> [[5,1],[]]
=> []
=> []
=> ? = 0
[[3,3,2],[2]]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[3,3,3],[2,1]]
=> [[3,2,1],[]]
=> []
=> []
=> ? = 0
[[4,4,3],[3,2]]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[2,2,2,1],[1]]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
[[2,2,2,2],[1,1]]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? = 0
[[3,3,3,2],[2,2,1]]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> ? = 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001804
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 78%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 78%
Values
[[1],[]]
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[[2],[]]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[[1,1],[]]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[[2,1],[1]]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[[3],[]]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[[2,1],[]]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[[3,1],[1]]
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[[2,2],[1]]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
[[3,2],[2]]
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 1 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 2 = 1 + 1
[[2,1,1],[1]]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 3 = 2 + 1
[[4],[]]
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[[3,1],[]]
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[[4,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[[2,2],[]]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
[[3,2],[1]]
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 1 + 1
[[4,2],[2]]
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 3 = 2 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 3 = 2 + 1
[[3,1,1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[[4,2,1],[2,1]]
=> [4,2,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 4 = 3 + 1
[[3,3],[2]]
=> [3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 1 = 0 + 1
[[4,3],[3]]
=> [4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 2 = 1 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 2 = 1 + 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 3 = 2 + 1
[[3,2,1],[2]]
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 3 = 2 + 1
[[4,3,1],[3,1]]
=> [4,3,1]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 4 = 3 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 0 + 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 2 = 1 + 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2 = 1 + 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 2 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 1 + 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 2 = 1 + 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 2 + 1
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 3 = 2 + 1
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 3 = 2 + 1
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? = 3 + 1
[[5],[]]
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[[4,1],[]]
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[[5,1],[1]]
=> [5,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 5 = 4 + 1
[[3,2],[]]
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 1 + 1
[[4,2],[1]]
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 3 = 2 + 1
[[5,2],[2]]
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 4 = 3 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[[4,2,1],[1,1]]
=> [4,2,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 4 = 3 + 1
[[4,1,1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 4 = 3 + 1
[[5,2,1],[2,1]]
=> [5,2,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> 5 = 4 + 1
[[3,3],[1]]
=> [3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 1 = 0 + 1
[[4,3],[2]]
=> [4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 2 = 1 + 1
[[5,3,1],[3,1]]
=> [5,3,1]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 4 + 1
[[4,3,2],[2,2]]
=> [4,3,2]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 2 + 1
[[5,3,2],[3,2]]
=> [5,3,2]
=> [3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> ? = 3 + 1
[[4,3,2,1],[2,2,1]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? = 3 + 1
[[4,2,2,1],[2,1,1]]
=> [4,2,2,1]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 3 + 1
[[5,3,2,1],[3,2,1]]
=> [5,3,2,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> ? = 4 + 1
[[5,4],[4]]
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 1 + 1
[[4,4,1],[3,1]]
=> [4,4,1]
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> ? = 3 + 1
[[5,4,1],[4,1]]
=> [5,4,1]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> ? = 4 + 1
[[4,4,2],[3,2]]
=> [4,4,2]
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> ? = 2 + 1
[[4,3,2],[3,1]]
=> [4,3,2]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 2 + 1
[[5,4,2],[4,2]]
=> [5,4,2]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> ? = 3 + 1
[[3,3,2,1],[2,1,1]]
=> [3,3,2,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 2 + 1
[[4,4,2,1],[3,2,1]]
=> [4,4,2,1]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> ? = 3 + 1
[[4,3,2,1],[3,1,1]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? = 3 + 1
[[4,3,1,1],[3,1]]
=> [4,3,1,1]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? = 3 + 1
[[5,4,2,1],[4,2,1]]
=> [5,4,2,1]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> ? = 4 + 1
[[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 0 + 1
[[4,4,3],[3,3]]
=> [4,4,3]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> ? = 1 + 1
[[4,3,3],[3,2]]
=> [4,3,3]
=> [3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> ? = 1 + 1
[[5,4,3],[4,3]]
=> [5,4,3]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 2 + 1
[[3,3,3,1],[2,2,1]]
=> [3,3,3,1]
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> ? = 2 + 1
[[3,3,2,1],[2,2]]
=> [3,3,2,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 2 + 1
[[4,4,3,1],[3,3,1]]
=> [4,4,3,1]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> ? = 3 + 1
[[4,3,3,1],[3,2,1]]
=> [4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> ? = 3 + 1
[[4,3,2,1],[3,2]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? = 3 + 1
[[5,4,3,1],[4,3,1]]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 4 + 1
[[3,3,3,2],[2,2,2]]
=> [3,3,3,2]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> ? = 1 + 1
[[3,3,2,2],[2,2,1]]
=> [3,3,2,2]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 1 + 1
[[4,4,3,2],[3,3,2]]
=> [4,4,3,2]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> ? = 2 + 1
[[3,2,2,2],[2,1,1]]
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 1 + 1
[[4,3,3,2],[3,2,2]]
=> [4,3,3,2]
=> [4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> ? = 2 + 1
[[4,3,2,2],[3,2,1]]
=> [4,3,2,2]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> ? = 2 + 1
[[5,4,3,2],[4,3,2]]
=> [5,4,3,2]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? = 3 + 1
[[2,2,2,2,1],[1,1,1,1]]
=> [2,2,2,2,1]
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> ? = 1 + 1
[[3,3,3,2,1],[2,2,2,1]]
=> [3,3,3,2,1]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> ? = 2 + 1
[[3,3,2,2,1],[2,2,1,1]]
=> [3,3,2,2,1]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> ? = 2 + 1
[[3,3,2,1,1],[2,2,1]]
=> [3,3,2,1,1]
=> [5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> ? = 2 + 1
[[4,4,3,2,1],[3,3,2,1]]
=> [4,4,3,2,1]
=> [5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> ? = 3 + 1
[[3,2,2,2,1],[2,1,1,1]]
=> [3,2,2,2,1]
=> [5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> ? = 2 + 1
[[3,2,2,1,1],[2,1,1]]
=> [3,2,2,1,1]
=> [5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> ? = 2 + 1
[[4,3,3,2,1],[3,2,2,1]]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
=> ? = 3 + 1
[[4,3,2,2,1],[3,2,1,1]]
=> [4,3,2,2,1]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> ? = 3 + 1
[[4,3,2,1,1],[3,2,1]]
=> [4,3,2,1,1]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> ? = 3 + 1
[[5,4,3,2,1],[4,3,2,1]]
=> [5,4,3,2,1]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? = 4 + 1
[[6,2,1],[2,1]]
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 5 + 1
[[6,3],[3]]
=> [6,3]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 3 + 1
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St001435
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00189: Skew partitions —rotate⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 56%
St001435: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 56%
Values
[[1],[]]
=> [[1],[]]
=> 0
[[2],[]]
=> [[2],[]]
=> 0
[[1,1],[]]
=> [[1,1],[]]
=> 0
[[2,1],[1]]
=> [[2,1],[1]]
=> 1
[[3],[]]
=> [[3],[]]
=> 0
[[2,1],[]]
=> [[2,2],[1]]
=> 1
[[3,1],[1]]
=> [[3,2],[2]]
=> 2
[[2,2],[1]]
=> [[2,1],[]]
=> 0
[[3,2],[2]]
=> [[3,1],[1]]
=> 1
[[1,1,1],[]]
=> [[1,1,1],[]]
=> 0
[[2,2,1],[1,1]]
=> [[2,1,1],[1]]
=> 1
[[2,1,1],[1]]
=> [[2,2,1],[1,1]]
=> 1
[[3,2,1],[2,1]]
=> [[3,2,1],[2,1]]
=> 2
[[4],[]]
=> [[4],[]]
=> 0
[[3,1],[]]
=> [[3,3],[2]]
=> 2
[[4,1],[1]]
=> [[4,3],[3]]
=> 3
[[2,2],[]]
=> [[2,2],[]]
=> 0
[[3,2],[1]]
=> [[3,2],[1]]
=> 1
[[4,2],[2]]
=> [[4,2],[2]]
=> 2
[[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> 1
[[3,2,1],[1,1]]
=> [[3,2,2],[2,1]]
=> 2
[[3,1,1],[1]]
=> [[3,3,2],[2,2]]
=> 2
[[4,2,1],[2,1]]
=> [[4,3,2],[3,2]]
=> 3
[[3,3],[2]]
=> [[3,1],[]]
=> 0
[[4,3],[3]]
=> [[4,1],[1]]
=> 1
[[2,2,1],[1]]
=> [[2,2,1],[1]]
=> 1
[[3,3,1],[2,1]]
=> [[3,2,1],[2]]
=> 2
[[3,2,1],[2]]
=> [[3,3,1],[2,1]]
=> 2
[[4,3,1],[3,1]]
=> [[4,3,1],[3,1]]
=> 3
[[2,2,2],[1,1]]
=> [[2,1,1],[]]
=> 0
[[3,3,2],[2,2]]
=> [[3,1,1],[1]]
=> 1
[[3,2,2],[2,1]]
=> [[3,2,1],[1,1]]
=> 1
[[4,3,2],[3,2]]
=> [[4,2,1],[2,1]]
=> 2
[[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> 0
[[2,2,2,1],[1,1,1]]
=> [[2,1,1,1],[1]]
=> 1
[[2,2,1,1],[1,1]]
=> [[2,2,1,1],[1,1]]
=> 1
[[3,3,2,1],[2,2,1]]
=> [[3,2,1,1],[2,1]]
=> 2
[[2,1,1,1],[1]]
=> [[2,2,2,1],[1,1,1]]
=> 1
[[3,2,2,1],[2,1,1]]
=> [[3,2,2,1],[2,1,1]]
=> 2
[[3,2,1,1],[2,1]]
=> [[3,3,2,1],[2,2,1]]
=> 2
[[4,3,2,1],[3,2,1]]
=> [[4,3,2,1],[3,2,1]]
=> 3
[[5],[]]
=> [[5],[]]
=> 0
[[4,1],[]]
=> [[4,4],[3]]
=> 3
[[5,1],[1]]
=> [[5,4],[4]]
=> 4
[[3,2],[]]
=> [[3,3],[1]]
=> 1
[[4,2],[1]]
=> [[4,3],[2]]
=> 2
[[5,2],[2]]
=> [[5,3],[3]]
=> 3
[[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> 2
[[4,2,1],[1,1]]
=> [[4,3,3],[3,2]]
=> 3
[[4,1,1],[1]]
=> [[4,4,3],[3,3]]
=> 3
[[6],[]]
=> [[6],[]]
=> ? = 0
[[5,1],[]]
=> [[5,5],[4]]
=> ? = 4
[[6,1],[1]]
=> [[6,5],[5]]
=> ? = 5
[[4,2],[]]
=> [[4,4],[2]]
=> ? = 2
[[5,2],[1]]
=> [[5,4],[3]]
=> ? = 3
[[6,2],[2]]
=> [[6,4],[4]]
=> ? = 4
[[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> ? = 3
[[5,2,1],[1,1]]
=> [[5,4,4],[4,3]]
=> ? = 4
[[5,1,1],[1]]
=> [[5,5,4],[4,4]]
=> ? = 4
[[6,2,1],[2,1]]
=> [[6,5,4],[5,4]]
=> ? = 5
[[3,3],[]]
=> [[3,3],[]]
=> ? = 0
[[4,3],[1]]
=> [[4,3],[1]]
=> ? = 1
[[5,3],[2]]
=> [[5,3],[2]]
=> ? = 2
[[6,3],[3]]
=> [[6,3],[3]]
=> ? = 3
[[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> ? = 2
[[4,3,1],[1,1]]
=> [[4,3,3],[3,1]]
=> ? = 3
[[4,2,1],[1]]
=> [[4,4,3],[3,2]]
=> ? = 3
[[5,3,1],[2,1]]
=> [[5,4,3],[4,2]]
=> ? = 4
[[5,2,1],[2]]
=> [[5,5,3],[4,3]]
=> ? = 4
[[6,3,1],[3,1]]
=> [[6,5,3],[5,3]]
=> ? = 5
[[4,2,2],[1,1]]
=> [[4,3,3],[2,2]]
=> ? = 2
[[5,3,2],[2,2]]
=> [[5,3,3],[3,2]]
=> ? = 3
[[5,2,2],[2,1]]
=> [[5,4,3],[3,3]]
=> ? = 3
[[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> ? = 2
[[4,2,2,1],[1,1,1]]
=> [[4,3,3,3],[3,2,2]]
=> ? = 3
[[4,2,1,1],[1,1]]
=> [[4,4,3,3],[3,3,2]]
=> ? = 3
[[5,3,2,1],[2,2,1]]
=> [[5,4,3,3],[4,3,2]]
=> ? = 4
[[4,1,1,1],[1]]
=> [[4,4,4,3],[3,3,3]]
=> ? = 3
[[5,2,2,1],[2,1,1]]
=> [[5,4,4,3],[4,3,3]]
=> ? = 4
[[5,2,1,1],[2,1]]
=> [[5,5,4,3],[4,4,3]]
=> ? = 4
[[4,4],[2]]
=> [[4,2],[]]
=> ? = 0
[[5,4],[3]]
=> [[5,2],[1]]
=> ? = 1
[[6,4],[4]]
=> [[6,2],[2]]
=> ? = 2
[[3,3,1],[1]]
=> [[3,3,2],[2]]
=> ? = 2
[[4,4,1],[2,1]]
=> [[4,3,2],[3]]
=> ? = 3
[[4,3,1],[2]]
=> [[4,4,2],[3,1]]
=> ? = 3
[[5,4,1],[3,1]]
=> [[5,4,2],[4,1]]
=> ? = 4
[[5,3,1],[3]]
=> [[5,5,2],[4,2]]
=> ? = 4
[[2,2,2],[]]
=> [[2,2,2],[]]
=> ? = 0
[[3,3,2],[1,1]]
=> [[3,2,2],[1]]
=> ? = 1
[[4,4,2],[2,2]]
=> [[4,2,2],[2]]
=> ? = 2
[[3,2,2],[1]]
=> [[3,3,2],[1,1]]
=> ? = 1
[[4,3,2],[2,1]]
=> [[4,3,2],[2,1]]
=> ? = 2
[[5,4,2],[3,2]]
=> [[5,3,2],[3,1]]
=> ? = 3
[[4,2,2],[2]]
=> [[4,4,2],[2,2]]
=> ? = 2
[[5,3,2],[3,1]]
=> [[5,4,2],[3,2]]
=> ? = 3
[[6,4,2],[4,2]]
=> [[6,4,2],[4,2]]
=> ? = 4
[[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> ? = 1
[[3,3,2,1],[1,1,1]]
=> [[3,2,2,2],[2,1]]
=> ? = 2
[[3,3,1,1],[1,1]]
=> [[3,3,2,2],[2,2]]
=> ? = 2
Description
The number of missing boxes in the first row.
Matching statistic: St000089
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000089: Integer compositions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000089: Integer compositions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[[1],[]]
=> [1]
=> [[1]]
=> [1] => 0
[[2],[]]
=> [2]
=> [[1,2]]
=> [2] => 0
[[1,1],[]]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0
[[2,1],[1]]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[[3],[]]
=> [3]
=> [[1,2,3]]
=> [3] => 0
[[2,1],[]]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[[3,1],[1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 2
[[2,2],[1]]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0
[[3,2],[2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[[1,1,1],[]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0
[[2,2,1],[1,1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[[2,1,1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 2
[[4],[]]
=> [4]
=> [[1,2,3,4]]
=> [4] => 0
[[3,1],[]]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 2
[[4,1],[1]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 3
[[2,2],[]]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0
[[3,2],[1]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[[4,2],[2]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 2
[[2,1,1],[]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 2
[[3,1,1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[[4,2,1],[2,1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 3
[[3,3],[2]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 0
[[4,3],[3]]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1
[[2,2,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2
[[3,2,1],[2]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 2
[[4,3,1],[3,1]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => ? = 3
[[2,2,2],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 0
[[3,3,2],[2,2]]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ? = 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => ? = 2
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => ? = 2
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ? = 2
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 2
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 3
[[5],[]]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 0
[[4,1],[]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 3
[[5,1],[1]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 4
[[3,2],[]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[[4,2],[1]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 2
[[5,2],[2]]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => 3
[[3,1,1],[]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[[4,2,1],[1,1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 3
[[4,1,1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 3
[[5,2,1],[2,1]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => ? = 4
[[3,3],[1]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 0
[[4,3],[2]]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1
[[5,3],[3]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => ? = 2
[[2,2,1],[]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[[3,3,1],[1,1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2
[[3,2,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 2
[[4,3,1],[2,1]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => ? = 3
[[4,2,1],[2]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 3
[[5,3,1],[3,1]]
=> [5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [5,3,1] => ? = 4
[[4,3,2],[2,2]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => ? = 2
[[4,2,2],[2,1]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[[5,3,2],[3,2]]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [5,3,2] => ? = 3
[[3,2,2,1],[1,1,1]]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ? = 2
[[4,3,2,1],[2,2,1]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 3
[[4,2,2,1],[2,1,1]]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [4,2,2,1] => ? = 3
[[4,2,1,1],[2,1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 3
[[5,3,2,1],[3,2,1]]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [5,3,2,1] => ? = 4
[[4,4],[3]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ? = 0
[[5,4],[4]]
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [5,4] => ? = 1
[[4,4,1],[3,1]]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [4,4,1] => ? = 3
[[4,3,1],[3]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => ? = 3
[[5,4,1],[4,1]]
=> [5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [5,4,1] => ? = 4
[[3,3,2],[2,1]]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ? = 1
[[4,4,2],[3,2]]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [4,4,2] => ? = 2
[[4,3,2],[3,1]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => ? = 2
[[5,4,2],[4,2]]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [5,4,2] => ? = 3
[[3,3,2,1],[2,1,1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => ? = 2
[[3,3,1,1],[2,1]]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => ? = 2
[[4,4,2,1],[3,2,1]]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [4,4,2,1] => ? = 3
[[4,3,2,1],[3,1,1]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 3
[[4,3,1,1],[3,1]]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [4,3,1,1] => ? = 3
[[5,4,2,1],[4,2,1]]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [5,4,2,1] => ? = 4
[[3,3,3],[2,2]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [3,3,3] => ? = 0
[[4,4,3],[3,3]]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [4,4,3] => ? = 1
[[4,3,3],[3,2]]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 1
[[5,4,3],[4,3]]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> [5,4,3] => ? = 2
[[3,3,3,1],[2,2,1]]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? = 2
[[3,3,2,1],[2,2]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => ? = 2
[[4,4,3,1],[3,3,1]]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [4,4,3,1] => ? = 3
[[3,2,2,1],[2,1]]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ? = 2
[[4,3,3,1],[3,2,1]]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [4,3,3,1] => ? = 3
[[4,3,2,1],[3,2]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 3
[[5,4,3,1],[4,3,1]]
=> [5,4,3,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
=> [5,4,3,1] => ? = 4
[[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ? = 0
[[3,3,3,2],[2,2,2]]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => ? = 1
[[3,3,2,2],[2,2,1]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 1
[[4,4,3,2],[3,3,2]]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [4,4,3,2] => ? = 2
[[3,2,2,2],[2,1,1]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [3,2,2,2] => ? = 1
[[4,3,3,2],[3,2,2]]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [4,3,3,2] => ? = 2
Description
The absolute variation of a composition.
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