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Your data matches 65 different statistics following compositions of up to 3 maps.
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Matching statistic: St001280
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 2
[3,1,1]
=> 1
[2,2,1]
=> 2
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 2
[3,2,1]
=> 2
[3,1,1,1]
=> 1
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 1
[4,3]
=> 2
[4,2,1]
=> 2
[4,1,1,1]
=> 1
[3,3,1]
=> 2
[3,2,2]
=> 3
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 1
[5,3]
=> 2
[5,2,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1 = 0 + 1
[2]
=> [1,1]
=> 2 = 1 + 1
[1,1]
=> [2]
=> 1 = 0 + 1
[3]
=> [2,1]
=> 2 = 1 + 1
[2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,1]
=> [3]
=> 1 = 0 + 1
[4]
=> [3,1]
=> 2 = 1 + 1
[3,1]
=> [2,2]
=> 2 = 1 + 1
[2,2]
=> [2,1,1]
=> 3 = 2 + 1
[2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,1,1,1]
=> [4]
=> 1 = 0 + 1
[5]
=> [4,1]
=> 2 = 1 + 1
[4,1]
=> [3,2]
=> 2 = 1 + 1
[3,2]
=> [2,2,1]
=> 3 = 2 + 1
[3,1,1]
=> [3,2]
=> 2 = 1 + 1
[2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
[2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [5]
=> 1 = 0 + 1
[6]
=> [5,1]
=> 2 = 1 + 1
[5,1]
=> [4,2]
=> 2 = 1 + 1
[4,2]
=> [3,2,1]
=> 3 = 2 + 1
[4,1,1]
=> [3,3]
=> 2 = 1 + 1
[3,3]
=> [2,2,2]
=> 3 = 2 + 1
[3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[3,1,1,1]
=> [4,2]
=> 2 = 1 + 1
[2,2,2]
=> [3,1,1,1]
=> 4 = 3 + 1
[2,2,1,1]
=> [4,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [5,1]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [6]
=> 1 = 0 + 1
[7]
=> [6,1]
=> 2 = 1 + 1
[6,1]
=> [5,2]
=> 2 = 1 + 1
[5,2]
=> [4,2,1]
=> 3 = 2 + 1
[5,1,1]
=> [4,3]
=> 2 = 1 + 1
[4,3]
=> [3,2,2]
=> 3 = 2 + 1
[4,2,1]
=> [3,3,1]
=> 3 = 2 + 1
[4,1,1,1]
=> [4,3]
=> 2 = 1 + 1
[3,3,1]
=> [3,2,2]
=> 3 = 2 + 1
[3,2,2]
=> [3,2,1,1]
=> 4 = 3 + 1
[3,2,1,1]
=> [4,2,1]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [5,2]
=> 2 = 1 + 1
[2,2,2,1]
=> [4,1,1,1]
=> 4 = 3 + 1
[2,2,1,1,1]
=> [5,1,1]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [6,1]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [7]
=> 1 = 0 + 1
[8]
=> [7,1]
=> 2 = 1 + 1
[7,1]
=> [6,2]
=> 2 = 1 + 1
[6,2]
=> [5,2,1]
=> 3 = 2 + 1
[6,1,1]
=> [5,3]
=> 2 = 1 + 1
[5,3]
=> [4,2,2]
=> 3 = 2 + 1
[5,2,1]
=> [4,3,1]
=> 3 = 2 + 1
Description
The length of the partition.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: 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
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> []
=> 0
[2]
=> [1,1]
=> [1]
=> 1
[1,1]
=> [2]
=> []
=> 0
[3]
=> [1,1,1]
=> [1,1]
=> 1
[2,1]
=> [2,1]
=> [1]
=> 1
[1,1,1]
=> [3]
=> []
=> 0
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2]
=> [2,2]
=> [2]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> 1
[1,1,1,1]
=> [4]
=> []
=> 0
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> 2
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [5]
=> []
=> 0
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
[1,1,1,1,1,1]
=> [6]
=> []
=> 0
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000734
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 89%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 89%
Values
[1]
=> [1]
=> []
=> []
=> ? = 0
[2]
=> [1,1]
=> [1]
=> [[1]]
=> 1
[1,1]
=> [2]
=> []
=> []
=> ? = 0
[3]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 1
[1,1,1]
=> [3]
=> []
=> []
=> ? = 0
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [[1]]
=> 1
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 0
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [[1]]
=> 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 0
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> [[1,2,3]]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [[1]]
=> 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 0
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [[1,2,3]]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [[1]]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 0
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 0
[13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 1
[12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[8,5]
=> [2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> []
=> []
=> ? = 0
[14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 1
[13,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 1
[12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[11,3]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[11,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[10,3,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[9,5]
=> [2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[9,4,1]
=> [3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[9,3,2]
=> [3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[8,6]
=> [2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 2
[8,5,1]
=> [3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 2
[8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? = 3
[8,3,3]
=> [3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? = 3
[7,5,2]
=> [3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 3
[7,4,3]
=> [3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? = 3
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [14]
=> []
=> []
=> ? = 0
[15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14]]
=> ? = 1
[14,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 1
[13,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 2
[13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 1
[12,3]
=> [2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 2
[12,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[12,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[11,4]
=> [2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 2
[11,3,1]
=> [3,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[11,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 3
[11,2,1,1]
=> [4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000676
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> []
=> []
=> 0
[2]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[1,1]
=> [2]
=> []
=> []
=> 0
[3]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[1,1,1]
=> [3]
=> []
=> []
=> 0
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1
[1,1,1,1]
=> [4]
=> []
=> []
=> 0
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> 0
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> 0
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,3]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4]
=> [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[13]
=> [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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[10,2,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,2,2]
=> [3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[9,2,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,5]
=> [2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[8,4,1]
=> [3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,3,2]
=> [3,3,2,1,1,1,1,1]
=> [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]
=> ? = 3
[8,3,1,1]
=> [4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,2,2,1]
=> [4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[7,3,3]
=> [3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[7,2,2,2]
=> [4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[14]
=> [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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[13,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[11,3]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[10,3,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[10,2,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[10,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,5]
=> [2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001039
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 89%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 89%
Values
[1]
=> [1]
=> []
=> []
=> ? = 0
[2]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1]
=> [2]
=> []
=> []
=> ? = 0
[3]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1]
=> [3]
=> []
=> []
=> ? = 0
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 0
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 0
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 0
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,3,1,1]
=> [4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,2,1]
=> [4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,1,1,1]
=> [5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1,1,1]
=> [6,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2,2]
=> [4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,2,2,1,1]
=> [5,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 0
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 0
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 0
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,3]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 0
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4]
=> [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[2,1,1,1,1,1,1,1,1,1,1]
=> [11,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 0
[13]
=> [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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[10,2,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 56%
Values
[1]
=> [1]
=> []
=> []
=> ? = 0 + 1
[2]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1]
=> [2]
=> []
=> []
=> ? = 0 + 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,1]
=> [3]
=> []
=> []
=> ? = 0 + 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 0 + 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 0 + 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 0 + 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[3,3,1,1]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,2,1]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,2,1,1,1]
=> [5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 0 + 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 1 + 1
[8,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,2]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7,1,1]
=> [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 0 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 1 + 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 1 + 1
[8,2]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,3]
=> [2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,2,1]
=> [3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7,1,1,1]
=> [4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 0 + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 1 + 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 2 + 1
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 1 + 1
[8,3]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,4]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,3,1]
=> [3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,2,2]
=> [3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 0 + 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 2 + 1
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 1 + 1
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [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]
=> ? = 2 + 1
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 1 + 1
[8,4]
=> [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [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
[8,2,1,1]
=> [4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8,1,1,1,1]
=> [5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,5]
=> [2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000378
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 89%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 89%
Values
[1]
=> [1]
=> [1]
=> 1 = 0 + 1
[2]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[1,1]
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3]
=> [2,1]
=> [3]
=> 2 = 1 + 1
[2,1]
=> [2,1]
=> [3]
=> 2 = 1 + 1
[1,1,1]
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4]
=> [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[3,1]
=> [2,2]
=> [4]
=> 2 = 1 + 1
[2,2]
=> [2,1,1]
=> [2,2]
=> 3 = 2 + 1
[2,1,1]
=> [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[5]
=> [4,1]
=> [2,1,1,1]
=> 2 = 1 + 1
[4,1]
=> [3,2]
=> [5]
=> 2 = 1 + 1
[3,2]
=> [2,2,1]
=> [2,2,1]
=> 3 = 2 + 1
[3,1,1]
=> [3,2]
=> [5]
=> 2 = 1 + 1
[2,2,1]
=> [3,1,1]
=> [4,1]
=> 3 = 2 + 1
[2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[6]
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 1 + 1
[5,1]
=> [4,2]
=> [2,2,1,1]
=> 2 = 1 + 1
[4,2]
=> [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[4,1,1]
=> [3,3]
=> [6]
=> 2 = 1 + 1
[3,3]
=> [2,2,2]
=> [2,2,2]
=> 3 = 2 + 1
[3,2,1]
=> [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[3,1,1,1]
=> [4,2]
=> [2,2,1,1]
=> 2 = 1 + 1
[2,2,2]
=> [3,1,1,1]
=> [3,3]
=> 4 = 3 + 1
[2,2,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[7]
=> [6,1]
=> [2,1,1,1,1,1]
=> 2 = 1 + 1
[6,1]
=> [5,2]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[5,2]
=> [4,2,1]
=> [5,1,1]
=> 3 = 2 + 1
[5,1,1]
=> [4,3]
=> [7]
=> 2 = 1 + 1
[4,3]
=> [3,2,2]
=> [2,2,2,1]
=> 3 = 2 + 1
[4,2,1]
=> [3,3,1]
=> [6,1]
=> 3 = 2 + 1
[4,1,1,1]
=> [4,3]
=> [7]
=> 2 = 1 + 1
[3,3,1]
=> [3,2,2]
=> [2,2,2,1]
=> 3 = 2 + 1
[3,2,2]
=> [3,2,1,1]
=> [5,2]
=> 4 = 3 + 1
[3,2,1,1]
=> [4,2,1]
=> [5,1,1]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[2,2,2,1]
=> [4,1,1,1]
=> [3,2,1,1]
=> 4 = 3 + 1
[2,2,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[8]
=> [7,1]
=> [2,1,1,1,1,1,1]
=> 2 = 1 + 1
[7,1]
=> [6,2]
=> [2,2,1,1,1,1]
=> 2 = 1 + 1
[6,2]
=> [5,2,1]
=> [4,1,1,1,1]
=> 3 = 2 + 1
[6,1,1]
=> [5,3]
=> [2,2,2,1,1]
=> 2 = 1 + 1
[5,3]
=> [4,2,2]
=> [6,1,1]
=> 3 = 2 + 1
[5,2,1]
=> [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[13]
=> [12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[12,1]
=> [11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[11,2]
=> [10,2,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
[11,1,1]
=> [10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> ? = 1 + 1
[10,3]
=> [9,2,2]
=> [5,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
[10,2,1]
=> [9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> ? = 2 + 1
[9,2,1,1]
=> [8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 2 + 1
[9,1,1,1,1]
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 1 + 1
[8,5]
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 2 + 1
[8,4,1]
=> [7,3,3]
=> [8,1,1,1,1,1]
=> ? = 2 + 1
[8,3,2]
=> [7,3,2,1]
=> [6,2,1,1,1,1,1]
=> ? = 3 + 1
[8,3,1,1]
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 2 + 1
[8,2,2,1]
=> [7,4,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 3 + 1
[7,6]
=> [6,5,2]
=> [11,1,1]
=> ? = 2 + 1
[7,5,1]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 2 + 1
[7,4,2]
=> [6,3,3,1]
=> [8,2,1,1,1]
=> ? = 3 + 1
[7,4,1,1]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 2 + 1
[7,3,3]
=> [6,3,2,2]
=> [3,3,3,2,1,1]
=> ? = 3 + 1
[7,3,2,1]
=> [6,4,2,1]
=> [7,2,2,1,1]
=> ? = 3 + 1
[7,3,1,1,1]
=> [6,5,2]
=> [11,1,1]
=> ? = 2 + 1
[7,2,2,2]
=> [6,4,1,1,1]
=> [4,3,2,2,1,1]
=> ? = 4 + 1
[7,2,2,1,1]
=> [6,5,1,1]
=> [11,2]
=> ? = 3 + 1
[6,6,1]
=> [5,5,3]
=> [10,1,1,1]
=> ? = 2 + 1
[6,5,2]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 3 + 1
[6,4,2,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 3 + 1
[6,4,1,1,1]
=> [5,5,3]
=> [10,1,1,1]
=> ? = 2 + 1
[6,3,3,1]
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
[6,3,1,1,1,1]
=> [6,5,2]
=> [11,1,1]
=> ? = 2 + 1
[6,2,2,2,1]
=> [5,5,1,1,1]
=> [10,2,1]
=> ? = 4 + 1
[6,2,2,1,1,1]
=> [6,5,1,1]
=> [11,2]
=> ? = 3 + 1
[6,1,1,1,1,1,1,1]
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 1 + 1
[5,4,4]
=> [4,3,3,3]
=> [7,1,1,1,1,1,1]
=> ? = 3 + 1
[5,4,2,2]
=> [4,4,3,1,1]
=> [4,2,2,2,2,1]
=> ? = 4 + 1
[5,4,2,1,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 3 + 1
[5,4,1,1,1,1]
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 2 + 1
[5,3,3,1,1]
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
[5,3,2,1,1,1]
=> [6,4,2,1]
=> [7,2,2,1,1]
=> ? = 3 + 1
[5,3,1,1,1,1,1]
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 2 + 1
[5,2,2,2,2]
=> [5,4,1,1,1,1]
=> [9,3,1]
=> ? = 5 + 1
[5,2,2,2,1,1]
=> [6,4,1,1,1]
=> [4,3,2,2,1,1]
=> ? = 4 + 1
[5,2,2,1,1,1,1]
=> [7,4,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 3 + 1
[5,2,1,1,1,1,1,1]
=> [8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 2 + 1
[4,4,4,1]
=> [4,3,3,3]
=> [7,1,1,1,1,1,1]
=> ? = 3 + 1
[4,4,3,2]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 4 + 1
[4,4,2,1,1,1]
=> [6,3,3,1]
=> [8,2,1,1,1]
=> ? = 3 + 1
[4,4,1,1,1,1,1]
=> [7,3,3]
=> [8,1,1,1,1,1]
=> ? = 2 + 1
[4,3,3,1,1,1]
=> [6,3,2,2]
=> [3,3,3,2,1,1]
=> ? = 3 + 1
[4,3,2,2,1,1]
=> [6,3,2,1,1]
=> [7,3,1,1,1]
=> ? = 4 + 1
[4,3,2,1,1,1,1]
=> [7,3,2,1]
=> [6,2,1,1,1,1,1]
=> ? = 3 + 1
[4,2,2,2,2,1]
=> [6,3,1,1,1,1]
=> [4,3,3,1,1,1]
=> ? = 5 + 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000288
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1]
=> 10 => 01 => 1 = 0 + 1
[2]
=> [1,1]
=> 110 => 011 => 2 = 1 + 1
[1,1]
=> [2]
=> 100 => 001 => 1 = 0 + 1
[3]
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[2,1]
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,1,1]
=> [3]
=> 1000 => 0001 => 1 = 0 + 1
[4]
=> [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[3,1]
=> [2,2]
=> 1100 => 0011 => 2 = 1 + 1
[2,2]
=> [2,1,1]
=> 10110 => 01101 => 3 = 2 + 1
[2,1,1]
=> [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[1,1,1,1]
=> [4]
=> 10000 => 00001 => 1 = 0 + 1
[5]
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
[4,1]
=> [3,2]
=> 10100 => 00101 => 2 = 1 + 1
[3,2]
=> [2,2,1]
=> 11010 => 01011 => 3 = 2 + 1
[3,1,1]
=> [3,2]
=> 10100 => 00101 => 2 = 1 + 1
[2,2,1]
=> [3,1,1]
=> 100110 => 011001 => 3 = 2 + 1
[2,1,1,1]
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
[1,1,1,1,1]
=> [5]
=> 100000 => 000001 => 1 = 0 + 1
[6]
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
[5,1]
=> [4,2]
=> 100100 => 001001 => 2 = 1 + 1
[4,2]
=> [3,2,1]
=> 101010 => 010101 => 3 = 2 + 1
[4,1,1]
=> [3,3]
=> 11000 => 00011 => 2 = 1 + 1
[3,3]
=> [2,2,2]
=> 11100 => 00111 => 3 = 2 + 1
[3,2,1]
=> [3,2,1]
=> 101010 => 010101 => 3 = 2 + 1
[3,1,1,1]
=> [4,2]
=> 100100 => 001001 => 2 = 1 + 1
[2,2,2]
=> [3,1,1,1]
=> 1001110 => 0111001 => 4 = 3 + 1
[2,2,1,1]
=> [4,1,1]
=> 1000110 => 0110001 => 3 = 2 + 1
[2,1,1,1,1]
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
[1,1,1,1,1,1]
=> [6]
=> 1000000 => 0000001 => 1 = 0 + 1
[7]
=> [6,1]
=> 10000010 => 01000001 => 2 = 1 + 1
[6,1]
=> [5,2]
=> 1000100 => 0010001 => 2 = 1 + 1
[5,2]
=> [4,2,1]
=> 1001010 => 0101001 => 3 = 2 + 1
[5,1,1]
=> [4,3]
=> 101000 => 000101 => 2 = 1 + 1
[4,3]
=> [3,2,2]
=> 101100 => 001101 => 3 = 2 + 1
[4,2,1]
=> [3,3,1]
=> 110010 => 010011 => 3 = 2 + 1
[4,1,1,1]
=> [4,3]
=> 101000 => 000101 => 2 = 1 + 1
[3,3,1]
=> [3,2,2]
=> 101100 => 001101 => 3 = 2 + 1
[3,2,2]
=> [3,2,1,1]
=> 1010110 => 0110101 => 4 = 3 + 1
[3,2,1,1]
=> [4,2,1]
=> 1001010 => 0101001 => 3 = 2 + 1
[3,1,1,1,1]
=> [5,2]
=> 1000100 => 0010001 => 2 = 1 + 1
[2,2,2,1]
=> [4,1,1,1]
=> 10001110 => 01110001 => 4 = 3 + 1
[2,2,1,1,1]
=> [5,1,1]
=> 10000110 => 01100001 => 3 = 2 + 1
[2,1,1,1,1,1]
=> [6,1]
=> 10000010 => 01000001 => 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [7]
=> 10000000 => 00000001 => 1 = 0 + 1
[8]
=> [7,1]
=> 100000010 => 010000001 => 2 = 1 + 1
[7,1]
=> [6,2]
=> 10000100 => 00100001 => 2 = 1 + 1
[6,2]
=> [5,2,1]
=> 10001010 => 01010001 => 3 = 2 + 1
[6,1,1]
=> [5,3]
=> 1001000 => 0001001 => 2 = 1 + 1
[5,3]
=> [4,2,2]
=> 1001100 => 0011001 => 3 = 2 + 1
[5,2,1]
=> [4,3,1]
=> 1010010 => 0100101 => 3 = 2 + 1
[11]
=> [10,1]
=> 100000000010 => 010000000001 => ? = 1 + 1
[10,1]
=> [9,2]
=> 10000000100 => 00100000001 => ? = 1 + 1
[9,2]
=> [8,2,1]
=> 10000001010 => 01010000001 => ? = 2 + 1
[9,1,1]
=> [8,3]
=> 1000001000 => 0001000001 => ? = 1 + 1
[8,3]
=> [7,2,2]
=> 1000001100 => 0011000001 => ? = 2 + 1
[8,2,1]
=> [7,3,1]
=> 1000010010 => 0100100001 => ? = 2 + 1
[7,2,2]
=> [6,3,1,1]
=> 1000100110 => 0110010001 => ? = 3 + 1
[6,2,2,1]
=> [5,4,1,1]
=> 101000110 => 011000101 => ? = 3 + 1
[5,2,2,2]
=> [4,4,1,1,1]
=> 110001110 => 011100011 => ? = 4 + 1
[5,2,2,1,1]
=> [5,4,1,1]
=> 101000110 => 011000101 => ? = 3 + 1
[4,3,2,2]
=> [4,3,2,1,1]
=> 101010110 => 011010101 => ? = 4 + 1
[4,2,2,1,1,1]
=> [6,3,1,1]
=> 1000100110 => 0110010001 => ? = 3 + 1
[4,2,1,1,1,1,1]
=> [7,3,1]
=> 1000010010 => 0100100001 => ? = 2 + 1
[4,1,1,1,1,1,1,1]
=> [8,3]
=> 1000001000 => 0001000001 => ? = 1 + 1
[3,3,3,2]
=> [4,2,2,2,1]
=> 100111010 => 010111001 => ? = 4 + 1
[3,3,3,1,1]
=> [5,2,2,2]
=> 100011100 => 001110001 => ? = 3 + 1
[3,3,2,1,1,1]
=> [6,2,2,1]
=> 1000011010 => 0101100001 => ? = 3 + 1
[3,3,1,1,1,1,1]
=> [7,2,2]
=> 1000001100 => 0011000001 => ? = 2 + 1
[3,2,2,2,2]
=> [5,2,1,1,1,1]
=> 10001011110 => 01111010001 => ? = 5 + 1
[3,2,2,2,1,1]
=> [6,2,1,1,1]
=> 10000101110 => 01110100001 => ? = 4 + 1
[3,2,2,1,1,1,1]
=> [7,2,1,1]
=> 10000010110 => 01101000001 => ? = 3 + 1
[3,2,1,1,1,1,1,1]
=> [8,2,1]
=> 10000001010 => 01010000001 => ? = 2 + 1
[3,1,1,1,1,1,1,1,1]
=> [9,2]
=> 10000000100 => 00100000001 => ? = 1 + 1
[2,2,2,2,1,1,1]
=> [7,1,1,1,1]
=> 100000011110 => 011110000001 => ? = 4 + 1
[2,2,2,1,1,1,1,1]
=> [8,1,1,1]
=> 100000001110 => 011100000001 => ? = 3 + 1
[2,2,1,1,1,1,1,1,1]
=> [9,1,1]
=> 100000000110 => 011000000001 => ? = 2 + 1
[2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => 010000000001 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> 100000000000 => 000000000001 => ? = 0 + 1
[12]
=> [11,1]
=> 1000000000010 => ? => ? = 1 + 1
[11,1]
=> [10,2]
=> 100000000100 => 001000000001 => ? = 1 + 1
[10,2]
=> [9,2,1]
=> 100000001010 => ? => ? = 2 + 1
[10,1,1]
=> [9,3]
=> 10000001000 => 00010000001 => ? = 1 + 1
[9,3]
=> [8,2,2]
=> 10000001100 => 00110000001 => ? = 2 + 1
[9,2,1]
=> [8,3,1]
=> 10000010010 => 01001000001 => ? = 2 + 1
[9,1,1,1]
=> [8,4]
=> 1000010000 => 0000100001 => ? = 1 + 1
[8,4]
=> [7,3,2]
=> 1000010100 => 0010100001 => ? = 2 + 1
[8,3,1]
=> [7,3,2]
=> 1000010100 => 0010100001 => ? = 2 + 1
[8,2,2]
=> [7,3,1,1]
=> 10000100110 => 01100100001 => ? = 3 + 1
[8,2,1,1]
=> [7,4,1]
=> 1000100010 => 0100010001 => ? = 2 + 1
[7,4,1]
=> [6,3,3]
=> 100011000 => 000110001 => ? = 2 + 1
[7,3,2]
=> [6,3,2,1]
=> 1000101010 => 0101010001 => ? = 3 + 1
[7,2,2,1]
=> [6,4,1,1]
=> 1001000110 => 0110001001 => ? = 3 + 1
[6,4,2]
=> [5,3,3,1]
=> 100110010 => 010011001 => ? = 3 + 1
[6,3,3]
=> [5,3,2,2]
=> 100101100 => 001101001 => ? = 3 + 1
[6,2,2,1,1]
=> [5,5,1,1]
=> 110000110 => 011000011 => ? = 3 + 1
[5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 011010011 => ? = 4 + 1
[5,2,2,1,1,1]
=> [6,4,1,1]
=> 1001000110 => 0110001001 => ? = 3 + 1
[5,2,1,1,1,1,1]
=> [7,4,1]
=> 1000100010 => 0100010001 => ? = 2 + 1
[5,1,1,1,1,1,1,1]
=> [8,4]
=> 1000010000 => 0000100001 => ? = 1 + 1
[4,4,2,2]
=> [4,3,3,1,1]
=> 101100110 => 011001101 => ? = 4 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1]
=> [[1]]
=> 0
[2]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1]
=> [2]
=> [[1,2]]
=> 0
[3]
=> [2,1]
=> [[1,3],[2]]
=> 1
[2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> 0
[4]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[3,1]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[2,2]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0
[5]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[4,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[3,1,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[2,2,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0
[6]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[5,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[4,2]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[4,1,1]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1
[3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[3,1,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[2,2,2]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 3
[2,2,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 0
[7]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 1
[6,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1
[5,2]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 2
[5,1,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 1
[4,3]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 2
[4,2,1]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2
[4,1,1,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 1
[3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 2
[3,2,2]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 3
[3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 2
[3,1,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1
[2,2,2,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 3
[2,2,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0
[8]
=> [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> 1
[7,1]
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> 1
[6,2]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> 2
[6,1,1]
=> [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> 1
[5,3]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 2
[5,2,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 2
[11]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[10,1]
=> [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? = 1
[9,2]
=> [8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? = 2
[9,1,1]
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? = 1
[8,3]
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? = 2
[8,2,1]
=> [7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? = 2
[8,1,1,1]
=> [7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? = 1
[7,4]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 2
[7,3,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 2
[7,2,2]
=> [6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? = 3
[7,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 2
[7,1,1,1,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[6,2,1,1,1]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 2
[6,1,1,1,1,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[5,2,1,1,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 2
[5,1,1,1,1,1,1]
=> [7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? = 1
[4,3,1,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 2
[4,2,2,1,1,1]
=> [6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? = 3
[4,2,1,1,1,1,1]
=> [7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? = 2
[4,1,1,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? = 1
[3,3,2,1,1,1]
=> [6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? = 3
[3,3,1,1,1,1,1]
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? = 2
[3,2,2,2,2]
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? = 5
[3,2,2,2,1,1]
=> [6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? = 4
[3,2,2,1,1,1,1]
=> [7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ? = 3
[3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? = 2
[3,1,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? = 1
[2,2,2,2,2,1]
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? = 5
[2,2,2,2,1,1,1]
=> [7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? = 4
[2,2,2,1,1,1,1,1]
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? = 3
[2,2,1,1,1,1,1,1,1]
=> [9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? = 2
[2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 0
[12]
=> [11,1]
=> [[1,3,4,5,6,7,8,9,10,11,12],[2]]
=> ? = 1
[11,1]
=> [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 1
[10,2]
=> [9,2,1]
=> [[1,3,6,7,8,9,10,11,12],[2,5],[4]]
=> ? = 2
[10,1,1]
=> [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 1
[9,3]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 2
[9,2,1]
=> [8,3,1]
=> [[1,3,4,8,9,10,11,12],[2,6,7],[5]]
=> ? = 2
[9,1,1,1]
=> [8,4]
=> [[1,2,3,4,9,10,11,12],[5,6,7,8]]
=> ? = 1
[8,4]
=> [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? = 2
[8,3,1]
=> [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? = 2
[8,2,2]
=> [7,3,1,1]
=> [[1,4,5,9,10,11,12],[2,7,8],[3],[6]]
=> ? = 3
[8,2,1,1]
=> [7,4,1]
=> [[1,3,4,5,10,11,12],[2,7,8,9],[6]]
=> ? = 2
[8,1,1,1,1]
=> [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? = 1
[7,4,1]
=> [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? = 2
[7,3,2]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 3
[7,2,2,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 3
[7,2,1,1,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 2
[6,6]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000733The row containing the largest entry of a standard tableau. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000996The number of exclusive left-to-right maxima of a permutation. St000053The number of valleys of the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000678The number of up steps after the last double rise of a Dyck path. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000507The number of ascents of a standard tableau. St000291The number of descents of a binary word. St000806The semiperimeter of the associated bargraph. St000035The number of left outer peaks of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000245The number of ascents of a permutation. St000331The number of upper interactions of a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St001462The number of factors of a standard tableaux under concatenation. St000155The number of exceedances (also excedences) of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000672The number of minimal elements in Bruhat order not less than the permutation. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000730The maximal arc length of a set partition. St000884The number of isolated descents of a permutation. St000251The number of nonsingleton blocks of a set partition. St000007The number of saliances of the permutation. St001557The number of inversions of the second entry of a permutation. St001737The number of descents of type 2 in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000710The number of big deficiencies of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000021The number of descents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001298The number of repeated entries in the Lehmer code of a permutation. St000325The width of the tree associated to a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
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