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Your data matches 67 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1]
=> 1
[[3],[]]
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1]
=> 1
[[2,2],[1]]
=> [1]
=> [1]
=> 1
[[3,2],[2]]
=> [2]
=> [1,1]
=> 2
[[1,1,1],[]]
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [2]
=> 1
[[2,1,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[4],[]]
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1]
=> 1
[[2,2],[]]
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1]
=> 1
[[4,2],[2]]
=> [2]
=> [1,1]
=> 2
[[2,1,1],[]]
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [2]
=> 1
[[3,1,1],[1]]
=> [1]
=> [1]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[3,3],[2]]
=> [2]
=> [1,1]
=> 2
[[4,3],[3]]
=> [3]
=> [1,1,1]
=> 3
[[2,2,1],[1]]
=> [1]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,1]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> [2,1,1]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> [2,2]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [2,2,1]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [3]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [3,2]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [3,1]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [3,2,1]
=> 3
[[5],[]]
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2],[]]
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1]
=> 1
[[5,2],[2]]
=> [2]
=> [1,1]
=> 2
[[3,1,1],[]]
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [2]
=> 1
[[4,1,1],[1]]
=> [1]
=> [1]
=> 1
Description
The length of the partition.
Matching statistic: St000147
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> 0
[[2],[]]
=> []
=> 0
[[1,1],[]]
=> []
=> 0
[[2,1],[1]]
=> [1]
=> 1
[[3],[]]
=> []
=> 0
[[2,1],[]]
=> []
=> 0
[[3,1],[1]]
=> [1]
=> 1
[[2,2],[1]]
=> [1]
=> 1
[[3,2],[2]]
=> [2]
=> 2
[[1,1,1],[]]
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> 1
[[2,1,1],[1]]
=> [1]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> 2
[[4],[]]
=> []
=> 0
[[3,1],[]]
=> []
=> 0
[[4,1],[1]]
=> [1]
=> 1
[[2,2],[]]
=> []
=> 0
[[3,2],[1]]
=> [1]
=> 1
[[4,2],[2]]
=> [2]
=> 2
[[2,1,1],[]]
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> 1
[[3,1,1],[1]]
=> [1]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> 2
[[3,3],[2]]
=> [2]
=> 2
[[4,3],[3]]
=> [3]
=> 3
[[2,2,1],[1]]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> 2
[[3,2,1],[2]]
=> [2]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> 3
[[1,1,1,1],[]]
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 3
[[5],[]]
=> []
=> 0
[[4,1],[]]
=> []
=> 0
[[5,1],[1]]
=> [1]
=> 1
[[3,2],[]]
=> []
=> 0
[[4,2],[1]]
=> [1]
=> 1
[[5,2],[2]]
=> [2]
=> 2
[[3,1,1],[]]
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> 1
[[4,1,1],[1]]
=> [1]
=> 1
[[6,5,5,4,3,1],[5,4,4,3,1]]
=> [5,4,4,3,1]
=> ? = 5
[[5,5,5,4,3,2],[4,4,4,3,2]]
=> [4,4,4,3,2]
=> ? = 4
[[6,5,4,4,3,2],[5,4,3,3,2]]
=> [5,4,3,3,2]
=> ? = 5
[[5,5,4,4,3,2,1],[4,4,3,3,2,1]]
=> [4,4,3,3,2,1]
=> ? = 4
[[6,5,4,3,3,2,1],[5,4,3,2,2,1]]
=> [5,4,3,2,2,1]
=> ? = 5
Description
The largest part of an integer partition.
Matching statistic: St000676
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Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3],[]]
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[1,1,1],[]]
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4],[]]
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,2],[]]
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[2,1,1],[]]
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[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]
=> 2
[[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]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[[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]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[5],[]]
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2],[]]
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[3,1,1],[]]
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[5,5,4,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[[6,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,4,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[[7,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 6
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 6
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 6
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 6
[[6,6,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,4,3,2,1],[5,3,3,2,1]]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,3,3,2,1],[5,3,2,2,1]]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,3,2,2,1],[5,3,2,1,1]]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5
[[7,6,4,3,2,1],[6,4,3,2,1]]
=> [6,4,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 6
[[6,6,5,2,1],[5,5,2,1]]
=> [5,5,2,1]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 5
[[7,6,5,2,1],[6,5,2,1]]
=> [6,5,2,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 6
[[6,6,5,3,1],[5,5,3,1]]
=> [5,5,3,1]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 5
[[7,6,5,3,1],[6,5,3,1]]
=> [6,5,3,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 6
[[6,6,5,3,2],[5,5,3,2]]
=> [5,5,3,2]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 5
[[7,6,5,3,2],[6,5,3,2]]
=> [6,5,3,2]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 6
[[5,5,5,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[[5,5,4,3,2,1],[4,4,2,2,1]]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[[5,5,4,2,2,1],[4,4,2,1,1]]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4
[[6,6,5,3,2,1],[5,5,3,2,1]]
=> [5,5,3,2,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,5,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,4,3,2,1],[5,4,2,2,1]]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,4,2,2,1],[5,4,2,1,1]]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5
[[7,6,5,3,2,1],[6,5,3,2,1]]
=> [6,5,3,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 6
[[6,6,5,4,1],[5,5,4,1]]
=> [5,5,4,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 5
[[7,6,5,4,1],[6,5,4,1]]
=> [6,5,4,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 6
[[6,6,5,4,2],[5,5,4,2]]
=> [5,5,4,2]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 5
[[7,6,5,4,2],[6,5,4,2]]
=> [6,5,4,2]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 6
[[5,5,5,4,2,1],[4,4,4,2,1]]
=> [4,4,4,2,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[[5,5,4,4,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[[5,5,4,3,2,1],[4,4,3,1,1]]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[[6,6,5,4,2,1],[5,5,4,2,1]]
=> [5,5,4,2,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,5,4,2,1],[5,4,4,2,1]]
=> [5,4,4,2,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,4,4,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[6,5,4,3,2,1],[5,4,3,1,1]]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 5
[[6,6,5,4,3],[5,5,4,3]]
=> [5,5,4,3]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 5
[[5,5,5,4,3,1],[4,4,4,3,1]]
=> [4,4,4,3,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 4
[[5,5,4,4,3,1],[4,4,3,3,1]]
=> [4,4,3,3,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 4
[[5,5,4,3,3,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[[6,5,5,4,3,1],[5,4,4,3,1]]
=> [5,4,4,3,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 5
[[6,5,4,4,3,1],[5,4,3,3,1]]
=> [5,4,3,3,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 5
[[6,5,4,3,3,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
[[5,5,5,4,3,2],[4,4,4,3,2]]
=> [4,4,4,3,2]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
[[5,5,4,4,3,2],[4,4,3,3,2]]
=> [4,4,3,3,2]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
[[5,5,4,3,3,2],[4,4,3,2,2]]
=> [4,4,3,2,2]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4
[[5,5,4,3,2,2],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[[6,5,4,4,3,2],[5,4,3,3,2]]
=> [5,4,3,3,2]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 5
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: St000378
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[3],[]]
=> []
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[2,2],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[3,2],[2]]
=> [2]
=> [1,1]
=> [2]
=> 2
[[1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[[2,1,1],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> [3]
=> 2
[[4],[]]
=> []
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[2,2],[]]
=> []
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[4,2],[2]]
=> [2]
=> [1,1]
=> [2]
=> 2
[[2,1,1],[]]
=> []
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[[3,1,1],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> [3]
=> 2
[[3,3],[2]]
=> [2]
=> [1,1]
=> [2]
=> 2
[[4,3],[3]]
=> [3]
=> [1,1,1]
=> [2,1]
=> 3
[[2,2,1],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [2,1]
=> [3]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,1]
=> [2]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> [2,1,1]
=> [2,2]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> [2,2]
=> [4]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [2,1]
=> [3]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [2,2,1]
=> [2,2,1]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [3,2]
=> [5]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [2,1]
=> [3]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [3,2,1]
=> [5,1]
=> 3
[[5],[]]
=> []
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[3,2],[]]
=> []
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[5,2],[2]]
=> [2]
=> [1,1]
=> [2]
=> 2
[[3,1,1],[]]
=> []
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[[4,1,1],[1]]
=> [1]
=> [1]
=> [1]
=> 1
[[6,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[5,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[7,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[6,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[6,6,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[6,5,4,3,2],[5,3,3,2]]
=> [5,3,3,2]
=> [4,4,3,1,1]
=> [4,2,2,2,2,1]
=> ? = 5
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> [4,4,3,2,1,1]
=> [4,4,2,2,2,1]
=> ? = 6
[[5,5,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[6,5,4,3,2,1],[5,3,3,2,1]]
=> [5,3,3,2,1]
=> [5,4,3,1,1]
=> [9,4,1]
=> ? = 5
[[7,6,4,3,2,1],[6,4,3,2,1]]
=> [6,4,3,2,1]
=> [5,4,3,2,1,1]
=> [9,5,2]
=> ? = 6
[[7,6,5,2,1],[6,5,2,1]]
=> [6,5,2,1]
=> [4,3,2,2,2,1]
=> [4,3,3,2,2]
=> ? = 6
[[7,6,5,3],[6,5,3]]
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [7,6,1]
=> ? = 6
[[6,6,5,3,1],[5,5,3,1]]
=> [5,5,3,1]
=> [4,3,3,2,2]
=> [7,2,2,1,1,1]
=> ? = 5
[[6,5,5,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[7,6,5,3,1],[6,5,3,1]]
=> [6,5,3,1]
=> [4,3,3,2,2,1]
=> [7,4,1,1,1,1]
=> ? = 6
[[6,6,5,3,2],[5,5,3,2]]
=> [5,5,3,2]
=> [4,4,3,2,2]
=> [7,2,2,2,1,1]
=> ? = 5
[[7,6,5,3,2],[6,5,3,2]]
=> [6,5,3,2]
=> [4,4,3,2,2,1]
=> [4,4,3,2,2,1]
=> ? = 6
[[5,5,4,3,2,1],[4,4,2,2,1]]
=> [4,4,2,2,1]
=> [5,4,2,2]
=> [9,4]
=> ? = 4
[[6,6,5,3,2,1],[5,5,3,2,1]]
=> [5,5,3,2,1]
=> [5,4,3,2,2]
=> [7,2,2,2,1,1,1]
=> ? = 5
[[6,5,4,3,2,1],[5,4,2,2,1]]
=> [5,4,2,2,1]
=> [5,4,2,2,1]
=> [9,2,2,1]
=> ? = 5
[[7,6,5,3,2,1],[6,5,3,2,1]]
=> [6,5,3,2,1]
=> [5,4,3,2,2,1]
=> [7,3,2,2,1,1,1]
=> ? = 6
[[6,5,5,4],[5,4,4]]
=> [5,4,4]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> ? = 5
[[7,6,5,4],[6,5,4]]
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [8,6,1]
=> ? = 6
[[5,5,5,4,1],[4,4,4,1]]
=> [4,4,4,1]
=> [4,3,3,3]
=> [7,1,1,1,1,1,1]
=> ? = 4
[[6,6,5,4,1],[5,5,4,1]]
=> [5,5,4,1]
=> [4,3,3,3,2]
=> [7,2,2,1,1,1,1]
=> ? = 5
[[6,5,5,4,1],[5,4,4,1]]
=> [5,4,4,1]
=> [4,3,3,3,1]
=> [7,2,1,1,1,1,1]
=> ? = 5
[[6,5,4,4,1],[5,4,3,1]]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[7,6,5,4,1],[6,5,4,1]]
=> [6,5,4,1]
=> [4,3,3,3,2,1]
=> [7,4,1,1,1,1,1]
=> ? = 6
[[5,5,5,4,2],[4,4,4,2]]
=> [4,4,4,2]
=> [4,4,3,3]
=> [8,1,1,1,1,1,1]
=> ? = 4
[[6,6,5,4,2],[5,5,4,2]]
=> [5,5,4,2]
=> [4,4,3,3,2]
=> [8,2,2,1,1,1,1]
=> ? = 5
[[6,5,5,4,2],[5,4,4,2]]
=> [5,4,4,2]
=> [4,4,3,3,1]
=> [8,2,1,1,1,1,1]
=> ? = 5
[[6,5,4,3,2],[5,4,3,1]]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[7,6,5,4,2],[6,5,4,2]]
=> [6,5,4,2]
=> [4,4,3,3,2,1]
=> [8,4,1,1,1,1,1]
=> ? = 6
[[6,6,5,4,2,1],[5,5,4,2,1]]
=> [5,5,4,2,1]
=> [5,4,3,3,2]
=> [9,2,2,2,2]
=> ? = 5
[[5,4,4,4,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[6,5,5,4,2,1],[5,4,4,2,1]]
=> [5,4,4,2,1]
=> [5,4,3,3,1]
=> [4,3,3,2,2,2]
=> ? = 5
[[6,5,4,3,1,1],[5,4,3,1]]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 5
[[5,5,5,4,3],[4,4,4,3]]
=> [4,4,4,3]
=> [4,4,4,3]
=> [8,1,1,1,1,1,1,1]
=> ? = 4
[[6,6,5,4,3],[5,5,4,3]]
=> [5,5,4,3]
=> [4,4,4,3,2]
=> [8,2,2,1,1,1,1,1]
=> ? = 5
[[6,5,5,4,3],[5,4,4,3]]
=> [5,4,4,3]
=> [4,4,4,3,1]
=> [8,2,1,1,1,1,1,1]
=> ? = 5
[[6,5,4,4,3],[5,4,3,3]]
=> [5,4,3,3]
=> [4,4,4,2,1]
=> [5,2,2,2,2,2]
=> ? = 5
[[5,5,5,4,3,1],[4,4,4,3,1]]
=> [4,4,4,3,1]
=> [5,4,4,3]
=> [3,3,3,2,2,2,1]
=> ? = 4
[[5,5,4,4,3,1],[4,4,3,3,1]]
=> [4,4,3,3,1]
=> [5,4,4,2]
=> [3,3,2,2,2,2,1]
=> ? = 4
[[5,4,4,4,3,1],[4,3,3,3,1]]
=> [4,3,3,3,1]
=> [5,4,4,1]
=> [3,2,2,2,2,2,1]
=> ? = 4
[[5,4,4,3,3,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 4
[[6,5,5,4,3,1],[5,4,4,3,1]]
=> [5,4,4,3,1]
=> [5,4,4,3,1]
=> [4,3,3,2,2,2,1]
=> ? = 5
[[6,5,4,4,3,1],[5,4,3,3,1]]
=> [5,4,3,3,1]
=> [5,4,4,2,1]
=> [5,2,2,2,2,2,1]
=> ? = 5
[[4,4,4,3,3,2],[3,3,3,2,2]]
=> [3,3,3,2,2]
=> [5,5,3]
=> [10,1,1,1]
=> ? = 3
[[5,5,5,4,3,2],[4,4,4,3,2]]
=> [4,4,4,3,2]
=> [5,5,4,3]
=> [3,3,3,2,2,2,2]
=> ? = 4
[[5,5,4,3,3,2],[4,4,3,2,2]]
=> [4,4,3,2,2]
=> [5,5,3,2]
=> [10,5]
=> ? = 4
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells c in the diagram of an integer partition \lambda for which \operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000013
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3],[]]
=> []
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> []
=> 0
[[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
[[1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[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]
=> 2
[[4],[]]
=> []
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[2,2],[]]
=> []
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[2,1,1],[]]
=> []
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[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]
=> 2
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 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]
=> 2
[[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]
=> [1,1,1,0,0,0,1,0]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[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]
=> [1,1,0,0,1,0,1,0]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[5],[]]
=> []
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[]]
=> []
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[3,1,1],[]]
=> []
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 5
[[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]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[[6,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 5
[[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]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[[7,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 6
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 6
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ?
=> ? = 6
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 6
[[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]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[[6,6,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[6,5,4,3,2,1],[5,3,3,2,1]]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[6,5,3,3,2,1],[5,3,2,2,1]]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 5
[[6,5,3,2,2,1],[5,3,2,1,1]]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[7,6,4,3,2,1],[6,4,3,2,1]]
=> [6,4,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 6
[[6,6,5,2,1],[5,5,2,1]]
=> [5,5,2,1]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 5
[[7,6,5,2,1],[6,5,2,1]]
=> [6,5,2,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 6
[[6,6,5,3,1],[5,5,3,1]]
=> [5,5,3,1]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[7,6,5,3,1],[6,5,3,1]]
=> [6,5,3,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ?
=> ? = 6
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
[[7,6,5,3,2],[6,5,3,2]]
=> [6,5,3,2]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 6
[[5,5,4,3,2,1],[4,4,2,2,1]]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 4
[[5,5,4,2,2,1],[4,4,2,1,1]]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 4
[[6,6,5,3,2,1],[5,5,3,2,1]]
=> [5,5,3,2,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 5
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[[6,5,5,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
Description
The height of a Dyck path.
The height of a Dyck path D of semilength n is defined as the maximal height of a peak of D. The height of D at position i is the number of up-steps minus the number of down-steps before position i.
Matching statistic: St000025
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 91%●distinct values known / distinct values provided: 89%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 91%●distinct values known / distinct values provided: 89%
Values
[[1],[]]
=> []
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3],[]]
=> []
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> []
=> 0
[[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
[[1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[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]
=> 2
[[4],[]]
=> []
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[2,2],[]]
=> []
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[2,1,1],[]]
=> []
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[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]
=> 2
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 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]
=> 2
[[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]
=> [1,1,1,0,0,0,1,0]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[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]
=> [1,1,0,0,1,0,1,0]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[5],[]]
=> []
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[]]
=> []
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[3,1,1],[]]
=> []
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 5
[[5,5,4,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[[6,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 5
[[6,5,4,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[[7,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[7,6,2,1],[6,2,1]]
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 6
[[7,6,3,1],[6,3,1]]
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6
[[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]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 6
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[6,5,3,2,1],[5,2,2,1]]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 5
[[6,5,2,2,1],[5,2,1,1]]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 5
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 6
[[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]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 6
[[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]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 6
[[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]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[6,5,3,2,1],[5,3,1,1]]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 5
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 6
[[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]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 6
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ?
=> ? = 6
[[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5
[[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 6
[[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]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 4
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[[6,6,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[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]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[[5,4,3,2,2,1],[4,2,2,1,1]]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 4
[[6,5,4,3,2,1],[5,3,3,2,1]]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[5,4,2,2,2,1],[4,2,1,1,1]]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[6,5,3,3,2,1],[5,3,2,2,1]]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 5
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of D.
Matching statistic: St000476
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Values
[[1],[]]
=> []
=> []
=> []
=> ? = 0
[[2],[]]
=> []
=> []
=> []
=> ? = 0
[[1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[3],[]]
=> []
=> []
=> []
=> ? = 0
[[2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[4],[]]
=> []
=> []
=> []
=> ? = 0
[[3,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[2,2],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[2,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[[5],[]]
=> []
=> []
=> []
=> ? = 0
[[4,1],[]]
=> []
=> []
=> []
=> ? = 0
[[5,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[3,2],[]]
=> []
=> []
=> []
=> ? = 0
[[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[3,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[4,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[5,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[2,2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,3,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[4,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[5,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[5,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[6],[]]
=> []
=> []
=> []
=> ? = 0
[[5,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,2],[]]
=> []
=> []
=> []
=> ? = 0
[[4,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,3],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,2],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[1,1,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[7],[]]
=> []
=> []
=> []
=> ? = 0
[[6,1],[]]
=> []
=> []
=> []
=> ? = 0
[[5,2],[]]
=> []
=> []
=> []
=> ? = 0
[[5,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,3],[]]
=> []
=> []
=> []
=> ? = 0
[[4,2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[4,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,3,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,2],[]]
=> []
=> []
=> []
=> ? = 0
[[3,2,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[3,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,2,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,2,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[2,1,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[1,1,1,1,1,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[8],[]]
=> []
=> []
=> []
=> ? = 0
[[7,1],[]]
=> []
=> []
=> []
=> ? = 0
[[6,2],[]]
=> []
=> []
=> []
=> ? = 0
[[6,1,1],[]]
=> []
=> []
=> []
=> ? = 0
[[5,3],[]]
=> []
=> []
=> []
=> ? = 0
[[5,2,1],[]]
=> []
=> []
=> []
=> ? = 0
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley v in a Dyck path D there is a corresponding tunnel, which
is the factor T_v = s_i\dots s_j of D where s_i is the step after the first intersection of D with the line y = ht(v) to the left of s_j. This statistic is
\sum_v (j_v-i_v)/2.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 89%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 89%
Values
[[1],[]]
=> []
=> []
=> => ? = 0
[[2],[]]
=> []
=> []
=> => ? = 0
[[1,1],[]]
=> []
=> []
=> => ? = 0
[[2,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[3],[]]
=> []
=> []
=> => ? = 0
[[2,1],[]]
=> []
=> []
=> => ? = 0
[[3,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[2,2],[1]]
=> [1]
=> [1]
=> 10 => 1
[[3,2],[2]]
=> [2]
=> [1,1]
=> 110 => 2
[[1,1,1],[]]
=> []
=> []
=> => ? = 0
[[2,2,1],[1,1]]
=> [1,1]
=> [2]
=> 100 => 1
[[2,1,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[3,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> 1010 => 2
[[4],[]]
=> []
=> []
=> => ? = 0
[[3,1],[]]
=> []
=> []
=> => ? = 0
[[4,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[2,2],[]]
=> []
=> []
=> => ? = 0
[[3,2],[1]]
=> [1]
=> [1]
=> 10 => 1
[[4,2],[2]]
=> [2]
=> [1,1]
=> 110 => 2
[[2,1,1],[]]
=> []
=> []
=> => ? = 0
[[3,2,1],[1,1]]
=> [1,1]
=> [2]
=> 100 => 1
[[3,1,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[4,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> 1010 => 2
[[3,3],[2]]
=> [2]
=> [1,1]
=> 110 => 2
[[4,3],[3]]
=> [3]
=> [1,1,1]
=> 1110 => 3
[[2,2,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[3,3,1],[2,1]]
=> [2,1]
=> [2,1]
=> 1010 => 2
[[3,2,1],[2]]
=> [2]
=> [1,1]
=> 110 => 2
[[4,3,1],[3,1]]
=> [3,1]
=> [2,1,1]
=> 10110 => 3
[[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> 100 => 1
[[3,3,2],[2,2]]
=> [2,2]
=> [2,2]
=> 1100 => 2
[[3,2,2],[2,1]]
=> [2,1]
=> [2,1]
=> 1010 => 2
[[4,3,2],[3,2]]
=> [3,2]
=> [2,2,1]
=> 11010 => 3
[[1,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [3]
=> 1000 => 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> 100 => 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [3,2]
=> 10100 => 2
[[2,1,1,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [3,1]
=> 10010 => 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [2,1]
=> 1010 => 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [3,2,1]
=> 101010 => 3
[[5],[]]
=> []
=> []
=> => ? = 0
[[4,1],[]]
=> []
=> []
=> => ? = 0
[[5,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[3,2],[]]
=> []
=> []
=> => ? = 0
[[4,2],[1]]
=> [1]
=> [1]
=> 10 => 1
[[5,2],[2]]
=> [2]
=> [1,1]
=> 110 => 2
[[3,1,1],[]]
=> []
=> []
=> => ? = 0
[[4,2,1],[1,1]]
=> [1,1]
=> [2]
=> 100 => 1
[[4,1,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[5,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> 1010 => 2
[[3,3],[1]]
=> [1]
=> [1]
=> 10 => 1
[[4,3],[2]]
=> [2]
=> [1,1]
=> 110 => 2
[[5,3],[3]]
=> [3]
=> [1,1,1]
=> 1110 => 3
[[2,2,1],[]]
=> []
=> []
=> => ? = 0
[[3,3,1],[1,1]]
=> [1,1]
=> [2]
=> 100 => 1
[[3,2,1],[1]]
=> [1]
=> [1]
=> 10 => 1
[[4,3,1],[2,1]]
=> [2,1]
=> [2,1]
=> 1010 => 2
[[4,2,1],[2]]
=> [2]
=> [1,1]
=> 110 => 2
[[5,3,1],[3,1]]
=> [3,1]
=> [2,1,1]
=> 10110 => 3
[[3,2,2],[1,1]]
=> [1,1]
=> [2]
=> 100 => 1
[[4,3,2],[2,2]]
=> [2,2]
=> [2,2]
=> 1100 => 2
[[4,2,2],[2,1]]
=> [2,1]
=> [2,1]
=> 1010 => 2
[[5,3,2],[3,2]]
=> [3,2]
=> [2,2,1]
=> 11010 => 3
[[2,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [3]
=> 1000 => 1
[[3,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> 100 => 1
[[1,1,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[6],[]]
=> []
=> []
=> => ? = 0
[[5,1],[]]
=> []
=> []
=> => ? = 0
[[4,2],[]]
=> []
=> []
=> => ? = 0
[[4,1,1],[]]
=> []
=> []
=> => ? = 0
[[3,3],[]]
=> []
=> []
=> => ? = 0
[[3,2,1],[]]
=> []
=> []
=> => ? = 0
[[3,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[2,2,2],[]]
=> []
=> []
=> => ? = 0
[[2,2,1,1],[]]
=> []
=> []
=> => ? = 0
[[2,1,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[1,1,1,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[7],[]]
=> []
=> []
=> => ? = 0
[[6,1],[]]
=> []
=> []
=> => ? = 0
[[5,2],[]]
=> []
=> []
=> => ? = 0
[[5,1,1],[]]
=> []
=> []
=> => ? = 0
[[4,3],[]]
=> []
=> []
=> => ? = 0
[[4,2,1],[]]
=> []
=> []
=> => ? = 0
[[4,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[3,3,1],[]]
=> []
=> []
=> => ? = 0
[[3,2,2],[]]
=> []
=> []
=> => ? = 0
[[3,2,1,1],[]]
=> []
=> []
=> => ? = 0
[[3,1,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[2,2,2,1],[]]
=> []
=> []
=> => ? = 0
[[2,2,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[2,1,1,1,1,1],[]]
=> []
=> []
=> => ? = 0
[[7,6,3,2],[6,3,2]]
=> [6,3,2]
=> [3,3,2,1,1,1]
=> 110101110 => ? = 6
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [4,3,2,1,1,1]
=> 1010101110 => ? = 6
[[7,6,4,1],[6,4,1]]
=> [6,4,1]
=> [3,2,2,2,1,1]
=> 101110110 => ? = 6
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> [4,3,2,2,1,1]
=> 1010110110 => ? = 6
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> [4,3,3,2,1,1]
=> 1011010110 => ? = 6
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> [4,4,3,2,1,1]
=> 1101010110 => ? = 6
[[7,6,4,3,2,1],[6,4,3,2,1]]
=> [6,4,3,2,1]
=> [5,4,3,2,1,1]
=> 10101010110 => ? = 6
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000691
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 89%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 89%
Values
[[1],[]]
=> []
=> => => ? = 0
[[2],[]]
=> []
=> => => ? = 0
[[1,1],[]]
=> []
=> => => ? = 0
[[2,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3],[]]
=> []
=> => => ? = 0
[[2,1],[]]
=> []
=> => => ? = 0
[[3,1],[1]]
=> [1]
=> 10 => 01 => 1
[[2,2],[1]]
=> [1]
=> 10 => 01 => 1
[[3,2],[2]]
=> [2]
=> 100 => 101 => 2
[[1,1,1],[]]
=> []
=> => => ? = 0
[[2,2,1],[1,1]]
=> [1,1]
=> 110 => 011 => 1
[[2,1,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3,2,1],[2,1]]
=> [2,1]
=> 1010 => 1001 => 2
[[4],[]]
=> []
=> => => ? = 0
[[3,1],[]]
=> []
=> => => ? = 0
[[4,1],[1]]
=> [1]
=> 10 => 01 => 1
[[2,2],[]]
=> []
=> => => ? = 0
[[3,2],[1]]
=> [1]
=> 10 => 01 => 1
[[4,2],[2]]
=> [2]
=> 100 => 101 => 2
[[2,1,1],[]]
=> []
=> => => ? = 0
[[3,2,1],[1,1]]
=> [1,1]
=> 110 => 011 => 1
[[3,1,1],[1]]
=> [1]
=> 10 => 01 => 1
[[4,2,1],[2,1]]
=> [2,1]
=> 1010 => 1001 => 2
[[3,3],[2]]
=> [2]
=> 100 => 101 => 2
[[4,3],[3]]
=> [3]
=> 1000 => 0101 => 3
[[2,2,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3,3,1],[2,1]]
=> [2,1]
=> 1010 => 1001 => 2
[[3,2,1],[2]]
=> [2]
=> 100 => 101 => 2
[[4,3,1],[3,1]]
=> [3,1]
=> 10010 => 01101 => 3
[[2,2,2],[1,1]]
=> [1,1]
=> 110 => 011 => 1
[[3,3,2],[2,2]]
=> [2,2]
=> 1100 => 1011 => 2
[[3,2,2],[2,1]]
=> [2,1]
=> 1010 => 1001 => 2
[[4,3,2],[3,2]]
=> [3,2]
=> 10100 => 01001 => 3
[[1,1,1,1],[]]
=> []
=> => => ? = 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0111 => 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 011 => 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 11010 => 10011 => 2
[[2,1,1,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 10110 => 10001 => 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> 1010 => 1001 => 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 101010 => 011001 => 3
[[5],[]]
=> []
=> => => ? = 0
[[4,1],[]]
=> []
=> => => ? = 0
[[5,1],[1]]
=> [1]
=> 10 => 01 => 1
[[3,2],[]]
=> []
=> => => ? = 0
[[4,2],[1]]
=> [1]
=> 10 => 01 => 1
[[5,2],[2]]
=> [2]
=> 100 => 101 => 2
[[3,1,1],[]]
=> []
=> => => ? = 0
[[4,2,1],[1,1]]
=> [1,1]
=> 110 => 011 => 1
[[4,1,1],[1]]
=> [1]
=> 10 => 01 => 1
[[5,2,1],[2,1]]
=> [2,1]
=> 1010 => 1001 => 2
[[3,3],[1]]
=> [1]
=> 10 => 01 => 1
[[4,3],[2]]
=> [2]
=> 100 => 101 => 2
[[5,3],[3]]
=> [3]
=> 1000 => 0101 => 3
[[2,2,1],[]]
=> []
=> => => ? = 0
[[3,3,1],[1,1]]
=> [1,1]
=> 110 => 011 => 1
[[3,2,1],[1]]
=> [1]
=> 10 => 01 => 1
[[4,3,1],[2,1]]
=> [2,1]
=> 1010 => 1001 => 2
[[4,2,1],[2]]
=> [2]
=> 100 => 101 => 2
[[5,3,1],[3,1]]
=> [3,1]
=> 10010 => 01101 => 3
[[3,2,2],[1,1]]
=> [1,1]
=> 110 => 011 => 1
[[4,3,2],[2,2]]
=> [2,2]
=> 1100 => 1011 => 2
[[4,2,2],[2,1]]
=> [2,1]
=> 1010 => 1001 => 2
[[5,3,2],[3,2]]
=> [3,2]
=> 10100 => 01001 => 3
[[2,1,1,1],[]]
=> []
=> => => ? = 0
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0111 => 1
[[3,2,1,1],[1,1]]
=> [1,1]
=> 110 => 011 => 1
[[1,1,1,1,1],[]]
=> []
=> => => ? = 0
[[6],[]]
=> []
=> => => ? = 0
[[5,1],[]]
=> []
=> => => ? = 0
[[4,2],[]]
=> []
=> => => ? = 0
[[4,1,1],[]]
=> []
=> => => ? = 0
[[3,3],[]]
=> []
=> => => ? = 0
[[3,2,1],[]]
=> []
=> => => ? = 0
[[3,1,1,1],[]]
=> []
=> => => ? = 0
[[2,2,2],[]]
=> []
=> => => ? = 0
[[2,2,1,1],[]]
=> []
=> => => ? = 0
[[2,1,1,1,1],[]]
=> []
=> => => ? = 0
[[1,1,1,1,1,1],[]]
=> []
=> => => ? = 0
[[6,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> 1010101010 => 0110011001 => ? = 5
[[7],[]]
=> []
=> => => ? = 0
[[6,1],[]]
=> []
=> => => ? = 0
[[5,2],[]]
=> []
=> => => ? = 0
[[5,1,1],[]]
=> []
=> => => ? = 0
[[4,3],[]]
=> []
=> => => ? = 0
[[4,2,1],[]]
=> []
=> => => ? = 0
[[4,1,1,1],[]]
=> []
=> => => ? = 0
[[3,3,1],[]]
=> []
=> => => ? = 0
[[3,2,2],[]]
=> []
=> => => ? = 0
[[3,2,1,1],[]]
=> []
=> => => ? = 0
[[3,1,1,1,1],[]]
=> []
=> => => ? = 0
[[2,2,2,1],[]]
=> []
=> => => ? = 0
[[2,2,1,1,1],[]]
=> []
=> => => ? = 0
[[2,1,1,1,1,1],[]]
=> []
=> => => ? = 0
[[7,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> 1010101010 => 0110011001 => ? = 5
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> 1000101010 => ? => ? = 6
[[7,6,4,2,1],[6,4,2,1]]
=> [6,4,2,1]
=> 1001001010 => ? => ? = 6
[[7,6,4,3,1],[6,4,3,1]]
=> [6,4,3,1]
=> 1001010010 => ? => ? = 6
[[7,6,4,3,2],[6,4,3,2]]
=> [6,4,3,2]
=> 1001010100 => ? => ? = 6
[[6,6,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> 1010101010 => 0110011001 => ? = 5
Description
The number of changes of a binary word.
This is the number of indices i such that w_i \neq w_{i+1}.
Matching statistic: St000653
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Values
[[1],[]]
=> []
=> []
=> [] => ? = 0
[[2],[]]
=> []
=> []
=> [] => ? = 0
[[1,1],[]]
=> []
=> []
=> [] => ? = 0
[[2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3],[]]
=> []
=> []
=> [] => ? = 0
[[2,1],[]]
=> []
=> []
=> [] => ? = 0
[[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[[1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[[4],[]]
=> []
=> []
=> [] => ? = 0
[[3,1],[]]
=> []
=> []
=> [] => ? = 0
[[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[2,2],[]]
=> []
=> []
=> [] => ? = 0
[[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[[2,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 3
[[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 3
[[1,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[[5],[]]
=> []
=> []
=> [] => ? = 0
[[4,1],[]]
=> []
=> []
=> [] => ? = 0
[[5,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,2],[]]
=> []
=> []
=> [] => ? = 0
[[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[[3,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[4,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[[5,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[[2,2,1],[]]
=> []
=> []
=> [] => ? = 0
[[3,3,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[[4,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[[5,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 3
[[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[[5,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 3
[[2,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[1,1,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[6],[]]
=> []
=> []
=> [] => ? = 0
[[5,1],[]]
=> []
=> []
=> [] => ? = 0
[[4,2],[]]
=> []
=> []
=> [] => ? = 0
[[4,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[3,3],[]]
=> []
=> []
=> [] => ? = 0
[[3,2,1],[]]
=> []
=> []
=> [] => ? = 0
[[3,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[2,2,2],[]]
=> []
=> []
=> [] => ? = 0
[[2,2,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[2,1,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[1,1,1,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[7],[]]
=> []
=> []
=> [] => ? = 0
[[6,1],[]]
=> []
=> []
=> [] => ? = 0
[[5,2],[]]
=> []
=> []
=> [] => ? = 0
[[5,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[4,3],[]]
=> []
=> []
=> [] => ? = 0
[[4,2,1],[]]
=> []
=> []
=> [] => ? = 0
[[4,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[3,3,1],[]]
=> []
=> []
=> [] => ? = 0
[[3,2,2],[]]
=> []
=> []
=> [] => ? = 0
[[3,2,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[3,1,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[2,2,2,1],[]]
=> []
=> []
=> [] => ? = 0
[[2,2,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[2,1,1,1,1,1],[]]
=> []
=> []
=> [] => ? = 0
[[7,6,1],[6,1]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ? = 6
[[7,6,2],[6,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => ? = 6
[[7,6,2,1],[6,2,1]]
=> [6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => ? = 6
[[7,6,3,1],[6,3,1]]
=> [6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,6,2,3,7,4] => ? = 6
[[7,6,3,2],[6,3,2]]
=> [6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,6,2,3,7,4] => ? = 6
[[7,6,3,2,1],[6,3,2,1]]
=> [6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ? = 6
[[7,6,4,1],[6,4,1]]
=> [6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,7,4] => ? = 6
Description
The last descent of a permutation.
For a permutation \pi of \{1,\ldots,n\}, this is the largest index 0 \leq i < n such that \pi(i) > \pi(i+1) where one considers \pi(0) = n+1.
The following 57 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000734The last entry in the first row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000733The row containing the largest entry of a standard tableau. St001497The position of the largest weak excedence of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000031The number of cycles in the cycle decomposition of a permutation. St000439The position of the first down step of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000382The first part of an integer composition. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000740The last entry of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000381The largest part of an integer composition. St000808The number of up steps of the associated bargraph. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000840The number of closers smaller than the largest opener in a perfect matching. St000702The number of weak deficiencies of a permutation. St000738The first entry in the last row of a standard tableau. St000083The number of left oriented leafs of a binary tree except the first one. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000383The last part of an integer composition. St000505The biggest entry in the block containing the 1. St000746The number of pairs with odd minimum in a perfect matching. St000971The smallest closer of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000308The height of the tree associated to a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000989The number of final rises of a permutation. St000225Difference between largest and smallest parts in a partition. St001435The number of missing boxes in the first row. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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