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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000929
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Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [4]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[4,3,2]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[3,3,3]
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [4]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St000658
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[3,1,1,1]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,2,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
[2,1,1,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[4,1,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 0
[3,2,2]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[3,2,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0
[3,1,1,1,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 0
[2,2,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[2,1,1,1,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,1,1,1,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 0
[5,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> 0
[4,2,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[4,2,1,1]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[4,1,1,1,1]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> 0
[3,3,2]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[3,3,1,1]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 0
[3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 0
[3,2,1,1,1]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 0
[3,1,1,1,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> 0
[2,2,2,2]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[2,2,2,1,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0
[2,2,1,1,1,1]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[2,1,1,1,1,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 0
[6,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,2,2]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[5,2,1,1]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> 0
[5,1,1,1,1]
=> [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]
=> ? = 0
[4,3,2]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[4,3,1,1]
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0
[4,2,2,1]
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 0
[4,2,1,1,1]
=> [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]
=> ? = 0
[4,1,1,1,1,1]
=> [4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 0
[3,3,3]
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[3,3,2,1]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 0
[3,3,1,1,1]
=> [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]
=> ? = 0
[3,2,2,2]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,2,2,1,1]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[3,2,1,1,1,1]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 0
[3,1,1,1,1,1,1]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 0
[2,2,2,2,1]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0
[2,2,2,1,1,1]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 0
[2,2,1,1,1,1,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> 0
[2,1,1,1,1,1,1,1]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 0
[7,1,1,1]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,2,2]
=> [5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[6,2,1,1]
=> [3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,1,1,1,1]
=> [4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,3,2]
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[5,3,1,1]
=> [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 0
[5,2,2,1]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> 0
[5,2,1,1,1]
=> [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]
=> ? = 0
[5,1,1,1,1,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
[4,4,2]
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[4,4,1,1]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 0
[4,3,3]
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[4,3,2,1]
=> [7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 0
[4,3,1,1,1]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> ? = 0
[4,2,2,2]
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[4,2,2,1,1]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0
[4,2,1,1,1,1]
=> [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]
=> ? = 0
[4,1,1,1,1,1,1]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 0
[3,3,3,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> 0
[3,3,2,2]
=> [6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,3,2,1,1]
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 0
[3,3,1,1,1,1]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 0
[3,2,2,2,1]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[3,2,2,1,1,1]
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 0
[3,2,1,1,1,1,1]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> 0
[2,2,2,1,1,1,1]
=> [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]
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [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]
=> ? = 0
[8,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,2,2]
=> [5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7,2,1,1]
=> [3,3,1,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,3,2]
=> [7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[6,3,1,1]
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,2,2,1]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[6,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[6,1,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,4,2]
=> [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,4,1,1]
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> ? = 0
[5,3,3]
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[5,3,2,1]
=> [7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 0
[5,2,2,2]
=> [7,4]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0
[5,2,2,1,1]
=> [6,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 0
[5,2,1,1,1,1]
=> [6,3,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 0
[5,1,1,1,1,1,1]
=> [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]
=> ? = 0
[4,4,2,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? = 0
[4,4,1,1,1]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
=> ? = 0
[4,3,2,1,1]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 0
[4,3,1,1,1,1]
=> [7,3,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> ? = 0
[4,2,2,1,1,1]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
Description
The number of rises of length 2 of a Dyck path.
This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1].
A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St000661
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
[2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 0
[3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[3,2,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 0
[3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 0
[2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,2,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1
[4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 0
[3,2,2,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 0
[3,2,1,1,1]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> 0
[3,1,1,1,1,1]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[2,2,2,1,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0
[2,2,1,1,1,1]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 0
[2,1,1,1,1,1,1]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,1,1,1,1]
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[6,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,2,2]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> 1
[5,2,1,1]
=> [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]
=> ? = 0
[5,1,1,1,1]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,3,2]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1
[4,3,1,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> 0
[4,2,2,1]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> 0
[4,2,1,1,1]
=> [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]
=> ? = 0
[4,1,1,1,1,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,3,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[3,3,2,1]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 0
[3,3,1,1,1]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 0
[3,2,2,2]
=> [4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 0
[3,2,2,1,1]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 0
[3,2,1,1,1,1]
=> [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]
=> ? = 0
[3,1,1,1,1,1,1]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[2,2,2,2,1]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0
[2,2,2,1,1,1]
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 0
[2,2,1,1,1,1,1]
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[7,1,1,1]
=> [4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,2,2]
=> [3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[6,2,1,1]
=> [4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,1,1,1,1]
=> [5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,3,2]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> 1
[5,3,1,1]
=> [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]
=> ? = 0
[5,2,2,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
[5,2,1,1,1]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[5,1,1,1,1,1]
=> [6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,4,2]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[4,4,1,1]
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 0
[4,3,3]
=> [3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1
[4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> 0
[4,3,1,1,1]
=> [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]
=> ? = 0
[4,2,2,2]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> 0
[4,2,2,1,1]
=> [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]
=> ? = 0
[4,2,1,1,1,1]
=> [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 0
[4,1,1,1,1,1,1]
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,3,3,1]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 0
[3,3,2,2]
=> [4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 0
[3,3,2,1,1]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> 0
[3,3,1,1,1,1]
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 0
[3,2,2,2,1]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 0
[3,2,2,1,1,1]
=> [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]
=> ? = 0
[3,2,1,1,1,1,1]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> ? = 0
[3,1,1,1,1,1,1,1]
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[2,2,2,2,2]
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[2,2,2,2,1,1]
=> [6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 0
[2,2,2,1,1,1,1]
=> [7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,2,2]
=> [3,3,1,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,3,2]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[6,3,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[6,2,2,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,4,2]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> 1
[5,4,1,1]
=> [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]
=> ? = 0
[5,3,3]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> 1
[5,3,2,1]
=> [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]
=> ? = 0
[5,3,1,1,1]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[5,2,2,2]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[5,2,2,1,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
[5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
Description
The number of rises of length 3 of a Dyck path.
Matching statistic: St001133
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001133: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001133: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10)]
=> 2 = 0 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10)]
=> 2 = 0 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [(1,2),(3,8),(4,9),(5,10),(6,11),(7,12)]
=> ? = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10)]
=> 2 = 0 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10)]
=> 3 = 1 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10)]
=> 2 = 0 + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [(1,2),(3,7),(4,9),(5,10),(6,11),(8,12)]
=> ? = 0 + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [(1,2),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14)]
=> ? = 0 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10)]
=> 2 = 0 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10)]
=> 3 = 1 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10)]
=> 2 = 0 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [(1,2),(3,7),(4,8),(5,10),(6,11),(9,12)]
=> ? = 0 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10)]
=> 2 = 0 + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [(1,2),(3,6),(4,9),(5,10),(7,11),(8,12)]
=> ? = 0 + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [(1,2),(3,8),(4,10),(5,11),(6,12),(7,13),(9,14)]
=> ? = 0 + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [(1,2),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16)]
=> ? = 0 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> [(1,3),(2,7),(4,8),(5,9),(6,10),(11,12)]
=> ? = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10)]
=> 3 = 1 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10)]
=> 2 = 0 + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [(1,2),(3,7),(4,8),(5,9),(6,11),(10,12)]
=> ? = 0 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10)]
=> 3 = 1 + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10)]
=> 2 = 0 + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10)]
=> 2 = 0 + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> [(1,2),(3,6),(4,8),(5,10),(7,11),(9,12)]
=> ? = 0 + 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> [(1,2),(3,8),(4,9),(5,11),(6,12),(7,13),(10,14)]
=> ? = 0 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12)]
=> ? = 0 + 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [(1,2),(3,5),(4,9),(6,10),(7,11),(8,12)]
=> ? = 0 + 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> [(1,2),(3,7),(4,10),(5,11),(6,12),(8,13),(9,14)]
=> ? = 0 + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> [(1,2),(3,9),(4,11),(5,12),(6,13),(7,14),(8,15),(10,16)]
=> ? = 0 + 2
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,2),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> [(1,2),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18)]
=> ? = 0 + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> [(1,4),(2,8),(3,9),(5,10),(6,11),(7,12),(13,14)]
=> ? = 0 + 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8),(11,12)]
=> [(1,4),(2,5),(3,8),(6,9),(7,10),(11,12)]
=> ? = 1 + 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]
=> [(1,3),(2,6),(4,8),(5,9),(7,10),(11,12)]
=> ? = 0 + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12)]
=> ? = 0 + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10)]
=> 3 = 1 + 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10)]
=> 2 = 0 + 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10)]
=> 2 = 0 + 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> [(1,2),(3,6),(4,8),(5,9),(7,11),(10,12)]
=> ? = 0 + 2
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> [(1,2),(3,8),(4,9),(5,10),(6,12),(7,13),(11,14)]
=> ? = 0 + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)]
=> ? = 1 + 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10)]
=> 2 = 0 + 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)]
=> [(1,2),(3,6),(4,7),(5,10),(8,11),(9,12)]
=> ? = 0 + 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [(1,3),(2,4),(5,8),(6,10),(7,11),(9,12)]
=> ? = 0 + 2
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,10),(7,11),(9,12)]
=> ? = 0 + 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> [(1,2),(3,7),(4,9),(5,11),(6,12),(8,13),(10,14)]
=> ? = 0 + 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> [(1,2),(3,9),(4,10),(5,12),(6,13),(7,14),(8,15),(11,16)]
=> ? = 0 + 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> 2 = 0 + 2
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> [(1,2),(3,6),(4,10),(5,11),(7,12),(8,13),(9,14)]
=> ? = 0 + 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> [(1,2),(3,8),(4,11),(5,12),(6,13),(7,14),(9,15),(10,16)]
=> ? = 0 + 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [(1,2),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,10),(11,12)]
=> [(1,2),(3,10),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(11,18)]
=> ? = 0 + 2
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [(1,2),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(11,12)]
=> [(1,2),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20)]
=> ? = 0 + 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> [(1,5),(2,9),(3,10),(4,11),(6,12),(7,13),(8,14),(15,16)]
=> ? = 0 + 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> [(1,5),(2,6),(3,9),(4,10),(7,11),(8,12),(13,14)]
=> ? = 1 + 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> [(1,4),(2,7),(3,9),(5,10),(6,11),(8,12),(13,14)]
=> ? = 0 + 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> [(1,3),(2,8),(4,9),(5,10),(6,11),(7,12),(13,14)]
=> ? = 0 + 2
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)]
=> [(1,4),(2,5),(3,7),(6,9),(8,10),(11,12)]
=> ? = 1 + 2
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [(1,3),(2,6),(4,7),(5,9),(8,10),(11,12)]
=> ? = 0 + 2
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [(1,3),(2,5),(4,8),(6,9),(7,10),(11,12)]
=> ? = 0 + 2
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10),(11,12)]
=> ? = 0 + 2
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> [(1,2),(3,8),(4,9),(5,10),(6,11),(7,13),(12,14)]
=> ? = 0 + 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,12),(10,11)]
=> [(1,4),(2,5),(3,7),(6,8),(9,11),(10,12)]
=> ? = 1 + 2
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [(1,3),(2,6),(4,7),(5,8),(9,11),(10,12)]
=> ? = 0 + 2
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,9),(10,11)]
=> [(1,4),(2,5),(3,6),(7,9),(8,11),(10,12)]
=> ? = 1 + 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 2 = 0 + 2
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [(1,2),(3,6),(4,7),(5,9),(8,11),(10,12)]
=> ? = 0 + 2
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [(1,3),(2,4),(5,8),(6,9),(7,11),(10,12)]
=> ? = 0 + 2
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [(1,2),(3,5),(4,8),(6,9),(7,11),(10,12)]
=> ? = 0 + 2
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> [(1,2),(3,7),(4,9),(5,10),(6,12),(8,13),(11,14)]
=> ? = 0 + 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> [(1,2),(3,9),(4,10),(5,11),(6,13),(7,14),(8,15),(12,16)]
=> ? = 0 + 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [(1,3),(2,5),(4,6),(7,10),(8,11),(9,12)]
=> ? = 0 + 2
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> 2 = 0 + 2
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> 2 = 0 + 2
[4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> 2 = 0 + 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> 2 = 0 + 2
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> 2 = 0 + 2
[3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> 2 = 0 + 2
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> 2 = 0 + 2
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> 2 = 0 + 2
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> 2 = 0 + 2
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> 2 = 0 + 2
[4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> 2 = 0 + 2
[3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> 2 = 0 + 2
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> 2 = 0 + 2
[5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> 2 = 0 + 2
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> 2 = 0 + 2
[4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> 2 = 0 + 2
[4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> 2 = 0 + 2
[4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> 2 = 0 + 2
[5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> 2 = 0 + 2
[5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> 2 = 0 + 2
[5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> 2 = 0 + 2
[4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> 2 = 0 + 2
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 2 = 0 + 2
Description
The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001134
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001134: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001134: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10)]
=> 2 = 0 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10)]
=> 2 = 0 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [(1,2),(3,8),(4,9),(5,10),(6,11),(7,12)]
=> ? = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10)]
=> 2 = 0 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10)]
=> 3 = 1 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10)]
=> 2 = 0 + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [(1,2),(3,7),(4,9),(5,10),(6,11),(8,12)]
=> ? = 0 + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [(1,2),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14)]
=> ? = 0 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10)]
=> 2 = 0 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10)]
=> 3 = 1 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10)]
=> 2 = 0 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [(1,2),(3,7),(4,8),(5,10),(6,11),(9,12)]
=> ? = 0 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10)]
=> 2 = 0 + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [(1,2),(3,6),(4,9),(5,10),(7,11),(8,12)]
=> ? = 0 + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [(1,2),(3,8),(4,10),(5,11),(6,12),(7,13),(9,14)]
=> ? = 0 + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [(1,2),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16)]
=> ? = 0 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> [(1,3),(2,7),(4,8),(5,9),(6,10),(11,12)]
=> ? = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10)]
=> 3 = 1 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10)]
=> 2 = 0 + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [(1,2),(3,7),(4,8),(5,9),(6,11),(10,12)]
=> ? = 0 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10)]
=> 3 = 1 + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10)]
=> 2 = 0 + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10)]
=> 2 = 0 + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> [(1,2),(3,6),(4,8),(5,10),(7,11),(9,12)]
=> ? = 0 + 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> [(1,2),(3,8),(4,9),(5,11),(6,12),(7,13),(10,14)]
=> ? = 0 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12)]
=> ? = 0 + 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [(1,2),(3,5),(4,9),(6,10),(7,11),(8,12)]
=> ? = 0 + 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> [(1,2),(3,7),(4,10),(5,11),(6,12),(8,13),(9,14)]
=> ? = 0 + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> [(1,2),(3,9),(4,11),(5,12),(6,13),(7,14),(8,15),(10,16)]
=> ? = 0 + 2
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,2),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> [(1,2),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18)]
=> ? = 0 + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> [(1,4),(2,8),(3,9),(5,10),(6,11),(7,12),(13,14)]
=> ? = 0 + 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8),(11,12)]
=> [(1,4),(2,5),(3,8),(6,9),(7,10),(11,12)]
=> ? = 1 + 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]
=> [(1,3),(2,6),(4,8),(5,9),(7,10),(11,12)]
=> ? = 0 + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12)]
=> ? = 0 + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10)]
=> 3 = 1 + 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10)]
=> 2 = 0 + 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10)]
=> 2 = 0 + 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> [(1,2),(3,6),(4,8),(5,9),(7,11),(10,12)]
=> ? = 0 + 2
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> [(1,2),(3,8),(4,9),(5,10),(6,12),(7,13),(11,14)]
=> ? = 0 + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)]
=> ? = 1 + 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10)]
=> 2 = 0 + 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)]
=> [(1,2),(3,6),(4,7),(5,10),(8,11),(9,12)]
=> ? = 0 + 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [(1,3),(2,4),(5,8),(6,10),(7,11),(9,12)]
=> ? = 0 + 2
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,10),(7,11),(9,12)]
=> ? = 0 + 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> [(1,2),(3,7),(4,9),(5,11),(6,12),(8,13),(10,14)]
=> ? = 0 + 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> [(1,2),(3,9),(4,10),(5,12),(6,13),(7,14),(8,15),(11,16)]
=> ? = 0 + 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> 2 = 0 + 2
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> [(1,2),(3,6),(4,10),(5,11),(7,12),(8,13),(9,14)]
=> ? = 0 + 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> [(1,2),(3,8),(4,11),(5,12),(6,13),(7,14),(9,15),(10,16)]
=> ? = 0 + 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [(1,2),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,10),(11,12)]
=> [(1,2),(3,10),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(11,18)]
=> ? = 0 + 2
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [(1,2),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(11,12)]
=> [(1,2),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20)]
=> ? = 0 + 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> [(1,5),(2,9),(3,10),(4,11),(6,12),(7,13),(8,14),(15,16)]
=> ? = 0 + 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> [(1,5),(2,6),(3,9),(4,10),(7,11),(8,12),(13,14)]
=> ? = 1 + 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> [(1,4),(2,7),(3,9),(5,10),(6,11),(8,12),(13,14)]
=> ? = 0 + 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> [(1,3),(2,8),(4,9),(5,10),(6,11),(7,12),(13,14)]
=> ? = 0 + 2
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)]
=> [(1,4),(2,5),(3,7),(6,9),(8,10),(11,12)]
=> ? = 1 + 2
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [(1,3),(2,6),(4,7),(5,9),(8,10),(11,12)]
=> ? = 0 + 2
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [(1,3),(2,5),(4,8),(6,9),(7,10),(11,12)]
=> ? = 0 + 2
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10),(11,12)]
=> ? = 0 + 2
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> [(1,2),(3,8),(4,9),(5,10),(6,11),(7,13),(12,14)]
=> ? = 0 + 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,12),(10,11)]
=> [(1,4),(2,5),(3,7),(6,8),(9,11),(10,12)]
=> ? = 1 + 2
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [(1,3),(2,6),(4,7),(5,8),(9,11),(10,12)]
=> ? = 0 + 2
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,9),(10,11)]
=> [(1,4),(2,5),(3,6),(7,9),(8,11),(10,12)]
=> ? = 1 + 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 2 = 0 + 2
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [(1,2),(3,6),(4,7),(5,9),(8,11),(10,12)]
=> ? = 0 + 2
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [(1,3),(2,4),(5,8),(6,9),(7,11),(10,12)]
=> ? = 0 + 2
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [(1,2),(3,5),(4,8),(6,9),(7,11),(10,12)]
=> ? = 0 + 2
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> [(1,2),(3,7),(4,9),(5,10),(6,12),(8,13),(11,14)]
=> ? = 0 + 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> [(1,2),(3,9),(4,10),(5,11),(6,13),(7,14),(8,15),(12,16)]
=> ? = 0 + 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [(1,3),(2,5),(4,6),(7,10),(8,11),(9,12)]
=> ? = 0 + 2
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> 2 = 0 + 2
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> 2 = 0 + 2
[4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> 2 = 0 + 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> 2 = 0 + 2
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> 2 = 0 + 2
[3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> 2 = 0 + 2
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> 2 = 0 + 2
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> 2 = 0 + 2
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> 2 = 0 + 2
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> 2 = 0 + 2
[4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> 2 = 0 + 2
[3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> 2 = 0 + 2
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> 2 = 0 + 2
[5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> 2 = 0 + 2
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> 2 = 0 + 2
[4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> 2 = 0 + 2
[4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> 2 = 0 + 2
[4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> 2 = 0 + 2
[5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> 2 = 0 + 2
[5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> 2 = 0 + 2
[5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> 2 = 0 + 2
[4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> 2 = 0 + 2
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 2 = 0 + 2
Description
The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St000715
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000715: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000715: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 0
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 0
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 0
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 0
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 1
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 0
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 0
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> 0
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 0
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 1
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> 0
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> ? = 0
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 0
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [2,2,2,2,2]
=> 0
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> 0
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1,1]
=> 0
[5,1,1,1]
=> [[5,1,1,1],[]]
=> [[5,5,5,5],[4,4,4]]
=> [5,5,5,5]
=> ? = 0
[4,2,2]
=> [[4,2,2],[]]
=> [[4,4,4],[2,2]]
=> [4,4,4]
=> 1
[4,2,1,1]
=> [[4,2,1,1],[]]
=> [[4,4,4,4],[3,3,2]]
=> [4,4,4,4]
=> ? = 0
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [4,4,4,4,4]
=> ? = 0
[3,3,2]
=> [[3,3,2],[]]
=> [[3,3,3],[1]]
=> [3,3,3]
=> 1
[3,3,1,1]
=> [[3,3,1,1],[]]
=> [[3,3,3,3],[2,2]]
=> [3,3,3,3]
=> 0
[3,2,2,1]
=> [[3,2,2,1],[]]
=> [[3,3,3,3],[2,1,1]]
=> [3,3,3,3]
=> 0
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,1]]
=> [3,3,3,3,3]
=> ? = 0
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [3,3,3,3,3,3]
=> ? = 0
[2,2,2,2]
=> [[2,2,2,2],[]]
=> [[2,2,2,2],[]]
=> [2,2,2,2]
=> 0
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> [[2,2,2,2,2],[1,1]]
=> [2,2,2,2,2]
=> 0
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2,2]
=> 0
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? = 0
[1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1,1,1]
=> 0
[6,1,1,1]
=> [[6,1,1,1],[]]
=> [[6,6,6,6],[5,5,5]]
=> [6,6,6,6]
=> ? = 0
[5,2,2]
=> [[5,2,2],[]]
=> [[5,5,5],[3,3]]
=> [5,5,5]
=> ? = 1
[5,2,1,1]
=> [[5,2,1,1],[]]
=> [[5,5,5,5],[4,4,3]]
=> [5,5,5,5]
=> ? = 0
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> [[5,5,5,5,5],[4,4,4,4]]
=> [5,5,5,5,5]
=> ? = 0
[4,3,2]
=> [[4,3,2],[]]
=> [[4,4,4],[2,1]]
=> [4,4,4]
=> 1
[4,3,1,1]
=> [[4,3,1,1],[]]
=> [[4,4,4,4],[3,3,1]]
=> [4,4,4,4]
=> ? = 0
[4,2,2,1]
=> [[4,2,2,1],[]]
=> [[4,4,4,4],[3,2,2]]
=> [4,4,4,4]
=> ? = 0
[4,2,1,1,1]
=> [[4,2,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,2]]
=> [4,4,4,4,4]
=> ? = 0
[4,1,1,1,1,1]
=> [[4,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [4,4,4,4,4,4]
=> ? = 0
[3,3,3]
=> [[3,3,3],[]]
=> [[3,3,3],[]]
=> [3,3,3]
=> 1
[3,3,2,1]
=> [[3,3,2,1],[]]
=> [[3,3,3,3],[2,1]]
=> [3,3,3,3]
=> 0
[3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2]]
=> [3,3,3,3,3]
=> ? = 0
[3,2,2,2]
=> [[3,2,2,2],[]]
=> [[3,3,3,3],[1,1,1]]
=> [3,3,3,3]
=> 0
[3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> [[3,3,3,3,3],[2,2,1,1]]
=> [3,3,3,3,3]
=> ? = 0
[3,2,1,1,1,1]
=> [[3,2,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,1]]
=> [3,3,3,3,3,3]
=> ? = 0
[3,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> [3,3,3,3,3,3,3]
=> ? = 0
[2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> [[2,2,2,2,2],[1]]
=> [2,2,2,2,2]
=> 0
[2,2,2,1,1,1]
=> [[2,2,2,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1]]
=> [2,2,2,2,2,2]
=> 0
[2,2,1,1,1,1,1]
=> [[2,2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? = 0
[2,1,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2]
=> ? = 0
[1,1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1,1,1,1]
=> 0
[7,1,1,1]
=> [[7,1,1,1],[]]
=> [[7,7,7,7],[6,6,6]]
=> [7,7,7,7]
=> ? = 0
[6,2,2]
=> [[6,2,2],[]]
=> [[6,6,6],[4,4]]
=> [6,6,6]
=> ? = 1
[6,2,1,1]
=> [[6,2,1,1],[]]
=> [[6,6,6,6],[5,5,4]]
=> [6,6,6,6]
=> ? = 0
[6,1,1,1,1]
=> [[6,1,1,1,1],[]]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [6,6,6,6,6]
=> ? = 0
[5,3,2]
=> [[5,3,2],[]]
=> [[5,5,5],[3,2]]
=> [5,5,5]
=> ? = 1
[5,3,1,1]
=> [[5,3,1,1],[]]
=> [[5,5,5,5],[4,4,2]]
=> [5,5,5,5]
=> ? = 0
[5,2,2,1]
=> [[5,2,2,1],[]]
=> [[5,5,5,5],[4,3,3]]
=> [5,5,5,5]
=> ? = 0
[5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> [[5,5,5,5,5],[4,4,4,3]]
=> [5,5,5,5,5]
=> ? = 0
[5,1,1,1,1,1]
=> [[5,1,1,1,1,1],[]]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [5,5,5,5,5,5]
=> ? = 0
[4,4,2]
=> [[4,4,2],[]]
=> [[4,4,4],[2]]
=> [4,4,4]
=> 1
[4,4,1,1]
=> [[4,4,1,1],[]]
=> [[4,4,4,4],[3,3]]
=> [4,4,4,4]
=> ? = 0
[4,3,3]
=> [[4,3,3],[]]
=> [[4,4,4],[1,1]]
=> [4,4,4]
=> 1
[4,3,2,1]
=> [[4,3,2,1],[]]
=> [[4,4,4,4],[3,2,1]]
=> [4,4,4,4]
=> ? = 0
[4,3,1,1,1]
=> [[4,3,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,1]]
=> [4,4,4,4,4]
=> ? = 0
[4,2,2,2]
=> [[4,2,2,2],[]]
=> [[4,4,4,4],[2,2,2]]
=> [4,4,4,4]
=> ? = 0
[4,2,2,1,1]
=> [[4,2,2,1,1],[]]
=> [[4,4,4,4,4],[3,3,2,2]]
=> [4,4,4,4,4]
=> ? = 0
[4,2,1,1,1,1]
=> [[4,2,1,1,1,1],[]]
=> [[4,4,4,4,4,4],[3,3,3,3,2]]
=> [4,4,4,4,4,4]
=> ? = 0
[4,1,1,1,1,1,1]
=> [[4,1,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [4,4,4,4,4,4,4]
=> ? = 0
[3,3,3,1]
=> [[3,3,3,1],[]]
=> [[3,3,3,3],[2]]
=> [3,3,3,3]
=> 0
[3,3,2,2]
=> [[3,3,2,2],[]]
=> [[3,3,3,3],[1,1]]
=> [3,3,3,3]
=> 0
[3,3,2,1,1]
=> [[3,3,2,1,1],[]]
=> [[3,3,3,3,3],[2,2,1]]
=> [3,3,3,3,3]
=> ? = 0
[3,3,1,1,1,1]
=> [[3,3,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3,3]
=> ? = 0
[3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> [[3,3,3,3,3],[2,1,1,1]]
=> [3,3,3,3,3]
=> ? = 0
[3,2,2,1,1,1]
=> [[3,2,2,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,1,1]]
=> [3,3,3,3,3,3]
=> ? = 0
[3,2,1,1,1,1,1]
=> [[3,2,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,1]]
=> [3,3,3,3,3,3,3]
=> ? = 0
[3,1,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [3,3,3,3,3,3,3,3]
=> ? = 0
[2,2,2,2,2]
=> [[2,2,2,2,2],[]]
=> [[2,2,2,2,2],[]]
=> [2,2,2,2,2]
=> 0
[2,2,2,2,1,1]
=> [[2,2,2,2,1,1],[]]
=> [[2,2,2,2,2,2],[1,1]]
=> [2,2,2,2,2,2]
=> 0
[2,2,2,1,1,1,1]
=> [[2,2,2,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [[2,2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2]
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2,2]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1,1,1,1,1]
=> 0
[8,1,1,1]
=> [[8,1,1,1],[]]
=> ?
=> ?
=> ? = 0
[7,2,2]
=> [[7,2,2],[]]
=> [[7,7,7],[5,5]]
=> ?
=> ? = 1
[7,2,1,1]
=> [[7,2,1,1],[]]
=> ?
=> ?
=> ? = 0
[4,4,3]
=> [[4,4,3],[]]
=> [[4,4,4],[1]]
=> [4,4,4]
=> 1
[3,3,3,2]
=> [[3,3,3,2],[]]
=> [[3,3,3,3],[1]]
=> [3,3,3,3]
=> 0
[4,4,4]
=> [[4,4,4],[]]
=> [[4,4,4],[]]
=> [4,4,4]
=> 1
[3,3,3,3]
=> [[3,3,3,3],[]]
=> [[3,3,3,3],[]]
=> [3,3,3,3]
=> 0
[2,2,2,2,2,2]
=> [[2,2,2,2,2,2],[]]
=> [[2,2,2,2,2,2],[]]
=> [2,2,2,2,2,2]
=> 0
Description
The number of semistandard Young tableaux of given shape and entries at most 3.
This is also the dimension of the corresponding irreducible representation of $GL_3$.
Matching statistic: St000359
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => ? = 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,8,7,6,5,4,2] => ? = 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 0
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [8,1,7,6,5,4,3,2] => ? = 0
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,9,8,7,6,5,4,2] => ? = 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => ? = 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 0
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,1,8,7,6,5,4,2] => ? = 0
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 0
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 0
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 0
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [9,1,8,7,6,5,4,3,2] => ? = 0
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,10,9,8,7,6,5,4,2] => ? = 0
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [5,3,8,1,7,6,4,2] => ? = 0
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => ? = 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [7,4,1,6,5,3,2] => ? = 0
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 0
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 0
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,1,8,7,6,5,4,2] => ? = 0
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 0
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 0
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 0
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,6,5,4,3,2] => ? = 0
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [8,1,7,6,5,4,3,2] => ? = 0
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [3,1,9,8,7,6,5,4,2] => ? = 0
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 0
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [6,1,8,7,5,4,3,2] => ? = 0
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => [8,1,9,7,6,5,4,3,2] => ? = 0
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [10,1,4,5,6,7,8,9,2,3] => [10,1,9,8,7,6,5,4,3,2] => ? = 0
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => [3,1,11,10,9,8,7,6,5,4,2] => ? = 0
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [6,3,4,5,1,7,9,2,8] => [6,3,9,8,1,7,5,4,2] => ? = 0
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,6,4,5,1,8,3,7] => [2,8,7,6,1,5,4,3] => ? = 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,5,4,1,6,2,3,7] => [8,5,4,1,7,6,3,2] => ? = 0
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [4,3,1,5,6,8,2,7] => [4,3,1,8,7,6,5,2] => ? = 0
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [2,7,6,1,5,4,3] => ? = 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 0
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,4,1,7,6,3,2] => ? = 0
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => ? = 0
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => [3,1,8,7,6,5,4,2] => ? = 0
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => ? = 0
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => ? = 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => ? = 0
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => ? = 0
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => [8,1,7,6,5,4,3,2] => ? = 0
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3,1,4,9,6,7,8,2,5] => [3,1,9,8,7,6,5,4,2] => ? = 0
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => ? = 0
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 0
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 0
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 0
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 0
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 0
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 0
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => 0
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 0
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 0
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 0
[4,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [7,1,8,6,5,4,3,2] => 0
[5,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 0
[5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [7,1,8,6,5,4,3,2] => 0
[4,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [7,1,2,3,6,4,8,5] => [7,1,8,6,5,4,3,2] => 0
[5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => 0
[5,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 0
[5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 0
Description
The number of occurrences of the pattern 23-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St000245
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => ? = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [8,1,7,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,9,8,7,6,5,4,2] => ? = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => ? = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 0 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 0 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 0 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [9,1,8,7,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,10,9,8,7,6,5,4,2] => ? = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [5,3,8,1,7,6,4,2] => ? = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => ? = 1 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [7,4,1,6,5,3,2] => ? = 0 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 1 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 0 + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 1 = 0 + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,6,5,4,3,2] => ? = 0 + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [8,1,7,6,5,4,3,2] => ? = 0 + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [3,1,9,8,7,6,5,4,2] => ? = 0 + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 0 + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [6,1,8,7,5,4,3,2] => ? = 0 + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => [8,1,9,7,6,5,4,3,2] => ? = 0 + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [10,1,4,5,6,7,8,9,2,3] => [10,1,9,8,7,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => [3,1,11,10,9,8,7,6,5,4,2] => ? = 0 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [6,3,4,5,1,7,9,2,8] => [6,3,9,8,1,7,5,4,2] => ? = 0 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,6,4,5,1,8,3,7] => [2,8,7,6,1,5,4,3] => ? = 1 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,5,4,1,6,2,3,7] => [8,5,4,1,7,6,3,2] => ? = 0 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [4,3,1,5,6,8,2,7] => [4,3,1,8,7,6,5,2] => ? = 0 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [2,7,6,1,5,4,3] => ? = 1 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 0 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,4,1,7,6,3,2] => ? = 0 + 1
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => ? = 0 + 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 1 + 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => ? = 0 + 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => ? = 1 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 0 + 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => ? = 0 + 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => [8,1,7,6,5,4,3,2] => ? = 0 + 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3,1,4,9,6,7,8,2,5] => [3,1,9,8,7,6,5,4,2] => ? = 0 + 1
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => ? = 0 + 1
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [7,1,2,3,6,4,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
Description
The number of ascents of a permutation.
Matching statistic: St000834
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => ? = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [8,1,7,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,9,8,7,6,5,4,2] => ? = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => ? = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 0 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 0 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 0 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [9,1,8,7,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,10,9,8,7,6,5,4,2] => ? = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [5,3,8,1,7,6,4,2] => ? = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => ? = 1 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [7,4,1,6,5,3,2] => ? = 0 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 1 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 0 + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 1 = 0 + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,6,5,4,3,2] => ? = 0 + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [8,1,7,6,5,4,3,2] => ? = 0 + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [3,1,9,8,7,6,5,4,2] => ? = 0 + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 0 + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [6,1,8,7,5,4,3,2] => ? = 0 + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => [8,1,9,7,6,5,4,3,2] => ? = 0 + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [10,1,4,5,6,7,8,9,2,3] => [10,1,9,8,7,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => [3,1,11,10,9,8,7,6,5,4,2] => ? = 0 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [6,3,4,5,1,7,9,2,8] => [6,3,9,8,1,7,5,4,2] => ? = 0 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,6,4,5,1,8,3,7] => [2,8,7,6,1,5,4,3] => ? = 1 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,5,4,1,6,2,3,7] => [8,5,4,1,7,6,3,2] => ? = 0 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [4,3,1,5,6,8,2,7] => [4,3,1,8,7,6,5,2] => ? = 0 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [2,7,6,1,5,4,3] => ? = 1 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 0 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,4,1,7,6,3,2] => ? = 0 + 1
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => ? = 0 + 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 1 + 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => ? = 0 + 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => ? = 1 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 0 + 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => ? = 0 + 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => [8,1,7,6,5,4,3,2] => ? = 0 + 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3,1,4,9,6,7,8,2,5] => [3,1,9,8,7,6,5,4,2] => ? = 0 + 1
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => ? = 0 + 1
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [7,1,2,3,6,4,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000871
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => ? = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [8,1,7,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,9,8,7,6,5,4,2] => ? = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => ? = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 0 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 0 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 0 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [9,1,8,7,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,10,9,8,7,6,5,4,2] => ? = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [5,3,8,1,7,6,4,2] => ? = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => ? = 1 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [7,4,1,6,5,3,2] => ? = 0 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 1 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 0 + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 1 = 0 + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,6,5,4,3,2] => ? = 0 + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [8,1,7,6,5,4,3,2] => ? = 0 + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [3,1,9,8,7,6,5,4,2] => ? = 0 + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 0 + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [6,1,8,7,5,4,3,2] => ? = 0 + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => [8,1,9,7,6,5,4,3,2] => ? = 0 + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [10,1,4,5,6,7,8,9,2,3] => [10,1,9,8,7,6,5,4,3,2] => ? = 0 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => [3,1,11,10,9,8,7,6,5,4,2] => ? = 0 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [6,3,4,5,1,7,9,2,8] => [6,3,9,8,1,7,5,4,2] => ? = 0 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,6,4,5,1,8,3,7] => [2,8,7,6,1,5,4,3] => ? = 1 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,5,4,1,6,2,3,7] => [8,5,4,1,7,6,3,2] => ? = 0 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [4,3,1,5,6,8,2,7] => [4,3,1,8,7,6,5,2] => ? = 0 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [2,7,6,1,5,4,3] => ? = 1 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 0 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,4,1,7,6,3,2] => ? = 0 + 1
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => ? = 0 + 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => [3,1,8,7,6,5,4,2] => ? = 0 + 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 1 + 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [4,3,1,7,6,5,2] => ? = 0 + 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => ? = 1 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 0 + 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => ? = 0 + 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => [8,1,7,6,5,4,3,2] => ? = 0 + 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3,1,4,9,6,7,8,2,5] => [3,1,9,8,7,6,5,4,2] => ? = 0 + 1
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => ? = 0 + 1
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [7,1,2,3,6,4,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
Description
The number of very big ascents of a permutation.
A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$.
For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000742The number of big ascents of a permutation after prepending zero. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000383The last part of an integer composition. St001256Number of simple reflexive modules that are 2-stable reflexive. St000842The breadth of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000534The number of 2-rises of a permutation. St001964The interval resolution global dimension of a poset. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000454The largest eigenvalue of a graph if it is integral. St000352The Elizalde-Pak rank of a permutation. St000546The number of global descents of a permutation. St000648The number of 2-excedences of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000563The number of overlapping pairs of blocks of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition.
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