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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000395
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Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 4
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[[5,1],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[5,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[[5,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 4
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001034
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 4
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6
[[5,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[5,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[5,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 4
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[6,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 11
[[6,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 11
[[6,5,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 13
[[5,4,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 14
[[5,4,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12
[[7,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 11
[[7,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 11
[[7,5,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 13
[[6,4,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 14
[[6,4,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12
[[6,6,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 11
[[7,6,4,1],[6,4,1]]
=> [6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 13
[[6,5,4,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 11
[[6,6,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 11
[[6,5,3,3,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 11
[[6,6,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 13
[[6,5,4,3,2],[5,3,3,2]]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 11
[[6,5,3,2,2],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 11
[[5,5,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 14
[[5,5,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12
[[5,4,3,3,2,1],[4,2,2,2,1]]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 12
[[6,5,3,2,1,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 11
[[6,5,4,2,1],[5,4,1,1]]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 14
[[7,6,5,3],[6,5,3]]
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 12
[[6,5,5,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 11
[[6,5,5,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 13
[[5,4,4,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 14
[[5,4,4,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12
[[6,5,4,4,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 11
[[6,5,4,4,2],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 13
[[6,5,4,3,2],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 11
[[5,4,3,3,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12
[[6,5,4,3,1,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 11
[[6,5,4,3,3],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 13
[[5,4,3,3,3,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 14
[[6,5,4,3,2,1],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 13
[[5,4,3,3,2,2],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 14
[[5,4,3,2,2,2],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12
[[4,3,3,3,2,2,1],[3,2,2,2,1,1]]
=> [3,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 13
[[5,4,3,3,2,1,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 14
[[5,4,3,2,2,1,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000394
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 6
[[5,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[5,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[5,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[6,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 11
[[6,5,4,2],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 11
[[6,5,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 13
[[6,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 11
[[6,5,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 13
[[5,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 12
[[5,4,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 14
[[5,4,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 12
[[7,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 11
[[7,5,4,2],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 11
[[7,5,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 13
[[7,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 11
[[7,5,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 13
[[6,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 12
[[6,4,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 14
[[6,4,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 12
[[7,6,3,2],[6,3,2]]
=> [6,3,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 10
[[6,6,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 11
[[7,6,4,1],[6,4,1]]
=> [6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 13
[[6,6,4,2],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 11
[[7,6,4,2],[6,4,2]]
=> [6,4,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 12
[[6,6,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 13
[[6,5,4,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 11
[[7,6,4,3],[6,4,3]]
=> [6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 10
[[6,6,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 11
[[6,5,4,3,1],[5,3,3,1]]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 9
[[6,5,3,3,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 11
[[6,6,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 13
[[6,5,4,3,2],[5,3,3,2]]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 11
[[6,5,3,3,2],[5,3,2,2]]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 13
[[6,5,3,2,2],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 11
[[5,5,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 12
[[5,5,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 14
[[5,5,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 12
[[5,4,3,3,2,1],[4,2,2,2,1]]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 12
[[6,5,3,2,1,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 11
[[7,6,5,1],[6,5,1]]
=> [6,5,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 15
[[6,5,5,2],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 11
[[7,6,5,2],[6,5,2]]
=> [6,5,2]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 14
[[6,5,5,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 13
[[6,5,4,2,1],[5,4,1,1]]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 14
[[6,5,4,3],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 11
[[7,6,5,3],[6,5,3]]
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 12
[[6,5,5,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 11
[[6,5,4,3,1],[5,4,2,1]]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 13
[[6,5,4,2,1],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 11
[[6,5,5,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 13
[[6,5,4,3,2],[5,4,2,2]]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 15
[[6,5,4,2,2],[5,4,2,1]]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 13
[[5,4,4,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 14
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000029
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 61%●distinct values known / distinct values provided: 56%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 61%●distinct values known / distinct values provided: 56%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 4
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 4
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 4
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 5
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 5
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 6
[[5,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [2,1] => 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2
[[5,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 4
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[[5,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 4
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 4
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2
[[5,4,2,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 7
[[5,4,3,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 9
[[5,4,3,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 8
[[4,4,3,2,1],[3,3,2,1]]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 9
[[4,3,3,2,1],[3,2,2,1]]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 7
[[4,3,2,2,1],[3,2,1,1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 8
[[5,4,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 10
[[6,4,2,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 7
[[6,4,3,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 9
[[6,4,3,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 8
[[5,4,3,2,1],[3,3,2,1]]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 9
[[5,3,3,2,1],[3,2,2,1]]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 7
[[5,3,2,2,1],[3,2,1,1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 8
[[6,4,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 10
[[6,5,1],[5,1]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 6
[[6,5,2],[5,2]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? = 6
[[5,5,2,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 7
[[5,4,2,1],[4,1,1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 7
[[6,5,2,1],[5,2,1]]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => ? = 8
[[6,5,3],[5,3]]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? = 8
[[5,5,3,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 9
[[5,4,3,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 7
[[6,5,3,1],[5,3,1]]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [8,1,2,5,7,3,4,6] => ? = 10
[[5,5,3,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 8
[[5,4,3,2],[4,2,2]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 6
[[5,4,2,2],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 7
[[6,5,3,2],[5,3,2]]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => ? = 9
[[4,4,3,2,1],[3,2,2,1]]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 7
[[4,4,2,2,1],[3,2,1,1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 8
[[5,5,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 10
[[4,3,2,2,1],[3,1,1,1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 8
[[5,4,3,2,1],[4,2,2,1]]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ? = 8
[[5,4,2,2,1],[4,2,1,1]]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => ? = 9
[[5,4,2,1,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 7
[[6,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [9,1,2,5,6,3,8,4,7] => ? = 11
[[6,5,4],[5,4]]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? = 10
[[5,5,4,1],[4,4,1]]
=> [4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 11
[[5,4,4,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 9
[[6,5,4,1],[5,4,1]]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => ? = 12
[[5,5,4,2],[4,4,2]]
=> [4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ? = 10
[[5,4,4,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 8
[[5,4,3,2],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 9
[[6,5,4,2],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => ? = 11
[[4,4,4,2,1],[3,3,2,1]]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 9
[[4,4,3,2,1],[3,3,1,1]]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 10
[[5,5,4,2,1],[4,4,2,1]]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [8,5,4,1,2,7,3,6] => ? = 12
[[4,3,3,2,1],[3,2,1,1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 8
[[5,4,4,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 10
[[5,4,3,2,1],[4,3,1,1]]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => ? = 11
[[5,4,3,1,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 9
Description
The depth of a permutation.
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Matching statistic: St000197
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000197: Alternating sign matrices ⟶ ℤResult quality: 56% ●values known / values provided: 61%●distinct values known / distinct values provided: 56%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000197: Alternating sign matrices ⟶ ℤResult quality: 56% ●values known / values provided: 61%●distinct values known / distinct values provided: 56%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 4
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 4
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 4
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 5
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 5
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 6
[[5,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [[1]]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[[5,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 4
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[[5,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 4
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 4
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[5,4,2,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[5,4,3,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 9
[[5,4,3,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 8
[[4,4,3,2,1],[3,3,2,1]]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
=> ? = 9
[[4,3,3,2,1],[3,2,2,1]]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[4,3,2,2,1],[3,2,1,1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 8
[[5,4,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 10
[[6,4,2,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[6,4,3,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 9
[[6,4,3,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 8
[[5,4,3,2,1],[3,3,2,1]]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
=> ? = 9
[[5,3,3,2,1],[3,2,2,1]]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[5,3,2,2,1],[3,2,1,1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 8
[[6,4,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 10
[[6,5,1],[5,1]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 6
[[6,5,2],[5,2]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 6
[[5,5,2,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[5,4,2,1],[4,1,1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[6,5,2,1],[5,2,1]]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 8
[[6,5,3],[5,3]]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? = 8
[[5,5,3,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 9
[[5,4,3,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[6,5,3,1],[5,3,1]]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 10
[[5,5,3,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 8
[[5,4,3,2],[4,2,2]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 6
[[5,4,2,2],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[6,5,3,2],[5,3,2]]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 9
[[4,4,3,2,1],[3,2,2,1]]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[4,4,2,2,1],[3,2,1,1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 8
[[5,5,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 10
[[4,3,2,2,1],[3,1,1,1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 8
[[5,4,3,2,1],[4,2,2,1]]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 8
[[5,4,2,2,1],[4,2,1,1]]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 9
[[5,4,2,1,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 7
[[6,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,1,-1,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 11
[[6,5,4],[5,4]]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? = 10
[[5,5,4,1],[4,4,1]]
=> [4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 11
[[5,4,4,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 9
[[6,5,4,1],[5,4,1]]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 12
[[5,5,4,2],[4,4,2]]
=> [4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 10
[[5,4,4,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 8
[[5,4,3,2],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 9
[[6,5,4,2],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,0,1],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 11
[[4,4,4,2,1],[3,3,2,1]]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
=> ? = 9
[[4,4,3,2,1],[3,3,1,1]]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 10
[[5,5,4,2,1],[4,4,2,1]]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,-1,0,1,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 12
[[4,3,3,2,1],[3,2,1,1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 8
[[5,4,4,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 10
[[5,4,3,2,1],[4,3,1,1]]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 11
[[5,4,3,1,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> ? = 9
Description
The number of entries equal to positive one in the alternating sign matrix.
Matching statistic: St001582
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 21%●distinct values known / distinct values provided: 19%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 21%●distinct values known / distinct values provided: 19%
Values
[[2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4
[[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 4
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 4
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 5
[[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 5
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 6
[[5,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[4,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[5,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[[3,3,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[5,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4
[[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[5,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 4
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 4
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[4,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 5
[[3,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[4,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 5
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[5,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 6
[[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[[5,4],[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 4
[[3,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[4,4,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4
[[4,3,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3
[[5,4,1],[4,1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 5
[[2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4,4,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 4
[[3,2,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[4,3,2],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4
[[5,4,2],[4,2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 5
[[2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[3,3,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 5
[[3,3,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4,4,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 6
[[3,2,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[[4,3,2,1],[3,1,1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ? = 6
[[4,3,1,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4
[[5,4,2,1],[4,2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ? = 7
[[3,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3
[[4,4,3],[3,3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 6
[[4,3,3],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 4
[[5,4,3],[4,3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 7
[[2,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[3,3,3,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 5
[[3,3,2,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3
[[4,4,3,1],[3,3,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ? = 8
[[3,2,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[[4,3,3,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 6
[[4,3,2,1],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 4
[[5,4,3,1],[4,3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ? = 9
[[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 4
[[3,3,3,2],[2,2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 4
[[3,3,2,2],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 5
[[4,4,3,2],[3,3,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? = 7
[[3,2,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 5
[[4,3,3,2],[3,2,2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ? = 5
[[4,3,2,2],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 6
[[5,4,3,2],[4,3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? = 8
[[2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 6
[[2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 4
[[3,3,3,2,1],[2,2,2,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 6
[[2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[[2,1,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001491
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Values
[[2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[[4,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[4,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 4 - 1
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 4 - 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 5 - 1
[[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 5 - 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 6 - 1
[[5,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[[4,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[[4,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[[5,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[[3,3,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[4,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[[5,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[5,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 4 - 1
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 4 - 1
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[[4,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 5 - 1
[[3,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 5 - 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[5,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 6 - 1
[[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[[5,4],[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 4 - 1
[[3,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[[4,4,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[[4,3,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[[5,4,1],[4,1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 5 - 1
[[2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 3 - 1
[[4,4,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 4 - 1
[[3,2,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 2 - 1
[[4,3,2],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[[2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[2,1,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[6,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[5,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[5,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,1,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[2,2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[2,2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[2,1,1,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[7,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[6,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[6,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[5,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[5,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[5,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,4],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[4,1,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[3,1,1,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[2,2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[2,2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
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