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Your data matches 72 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,1] => [2] => [2]
=> 2
[2] => [1,1] => [1,1]
=> 1
[1,1,1] => [3] => [3]
=> 3
[1,2] => [2,1] => [2,1]
=> 2
[2,1] => [1,2] => [2,1]
=> 2
[3] => [1,1,1] => [1,1,1]
=> 1
[1,1,1,1] => [4] => [4]
=> 4
[1,1,2] => [3,1] => [3,1]
=> 3
[1,2,1] => [2,2] => [2,2]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> 2
[2,1,1] => [1,3] => [3,1]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> 1
[1,1,1,1,1] => [5] => [5]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 1
[1,1] => [2] => [2]
=> [1,1]
=> 2
[2] => [1,1] => [1,1]
=> [2]
=> 1
[1,1,1] => [3] => [3]
=> [1,1,1]
=> 3
[1,2] => [2,1] => [2,1]
=> [2,1]
=> 2
[2,1] => [1,2] => [2,1]
=> [2,1]
=> 2
[3] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,1,1,1] => [4] => [4]
=> [1,1,1,1]
=> 4
[1,1,2] => [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,2,1] => [2,2] => [2,2]
=> [2,2]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[2,1,1] => [1,3] => [3,1]
=> [2,1,1]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [1,1,1,1,1]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> [2,1,1,1]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> [2,2,1]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> [2,2,1]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> [3,2]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> [3,2]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> [2,1,1,1]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> [3,2]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,1,1,1,1,1]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> [2,1,1,1,1]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> [2,2,1,1]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [3,1,1,1]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> [2,2,2]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [3,2,1]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [3,2,1]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [4,1,1]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> [2,2,1,1]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [3,2,1]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [3,3]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [4,2]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [3,2,1]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> [2,1,1,1,1]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [3,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000734
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 92%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 92%
Values
[1] => [1] => [1]
=> [[1]]
=> 1
[1,1] => [2] => [2]
=> [[1,2]]
=> 2
[2] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[1,1,1] => [3] => [3]
=> [[1,2,3]]
=> 3
[1,2] => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
[2,1] => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[3] => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,1,1,1] => [4] => [4]
=> [[1,2,3,4]]
=> 4
[1,1,2] => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,2,1] => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,1] => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [[1,2,3,4,5]]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [[1,2,3,4,5,6]]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,10] => [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
[1,8,1,1] => [2,1,1,1,1,1,1,3] => [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[1,7,2,1] => [2,1,1,1,1,1,2,2] => [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,6,3,1] => [2,1,1,1,1,2,1,2] => [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,5,4,1] => [2,1,1,1,2,1,1,2] => [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,4,5,1] => [2,1,1,2,1,1,1,2] => [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,3,6,1] => [2,1,2,1,1,1,1,2] => [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,2,7,1] => [2,2,1,1,1,1,1,2] => [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,1,8,1] => [3,1,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[10,1] => [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]]
=> ? = 2
[1,1,1,1,1,1,1,1,2,2] => [9,2,1] => [9,2,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11],[12]]
=> ? = 9
[1,1,1,1,1,1,2,2,1,1] => [7,2,3] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? = 7
[1,1,1,1,1,1,2,1,1,2] => [7,4,1] => [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ? = 7
[1,1,1,1,1,1,3,3] => [7,1,2,1,1] => [7,2,1,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11],[12]]
=> ? = 7
[1,1,1,1,2,2,1,1,1,1] => [5,2,5] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? = 5
[1,1,1,1,2,1,1,1,1,2] => [5,6,1] => [6,5,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12]]
=> ? = 6
[1,1,1,1,4,4] => [5,1,1,2,1,1,1] => [5,2,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 5
[1,1,2,2,1,1,1,1,1,1] => [3,2,7] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? = 7
[1,1,2,2,3,3] => [3,2,2,1,2,1,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,1,2,1,1,1,1,2,1,1] => [3,6,3] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? = 6
[1,1,2,1,1,1,1,1,1,2] => [3,8,1] => [8,3,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12]]
=> ? = 8
[1,1,2,1,3,4] => [3,3,1,2,1,1,1] => [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,1,3,3,2,2] => [3,1,2,1,2,2,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,1,3,2,2,3] => [3,1,2,2,2,1,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,1,3,1,1,1,1,3] => [3,1,6,1,1] => [6,3,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11],[12]]
=> ? = 6
[1,1,3,1,2,4] => [3,1,3,2,1,1,1] => [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,1,4,4,1,1] => [3,1,1,2,1,1,3] => [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,1,4,3,1,2] => [3,1,1,2,1,3,1] => [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,1,4,2,1,3] => [3,1,1,2,3,1,1] => [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,1,4,1,1,4] => [3,1,1,4,1,1,1] => [4,3,1,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11],[12]]
=> ? = 4
[1,1,5,5] => [3,1,1,1,2,1,1,1,1] => [3,2,1,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 3
[2,2,1,1,1,1,1,1,1,1] => [1,2,9] => [9,2,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11],[12]]
=> ? = 9
[2,2,1,1,1,1,2,2] => [1,2,6,2,1] => [6,2,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11],[12]]
=> ? = 6
[2,2,1,1,3,3] => [1,2,4,1,2,1,1] => [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[2,2,2,2,2,2] => [1,2,2,2,2,2,1] => [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 2
[2,2,2,1,1,1,1,2] => [1,2,2,6,1] => [6,2,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11],[12]]
=> ? = 6
[2,2,2,1,2,3] => [1,2,2,3,2,1,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[2,2,3,3,1,1] => [1,2,2,1,2,1,3] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[2,2,3,2,1,2] => [1,2,2,1,2,3,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[2,2,3,1,1,3] => [1,2,2,1,4,1,1] => [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[2,2,4,4] => [1,2,2,1,1,2,1,1,1] => [2,2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,1,1,2,1,1,1,1,1,1] => [1,4,7] => [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ? = 7
[2,1,1,2,3,3] => [1,4,2,1,2,1,1] => [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[2,1,1,1,1,2,1,1,1,1] => [1,6,5] => [6,5,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12]]
=> ? = 6
[2,1,1,1,1,2,2,2] => [1,6,2,2,1] => [6,2,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11],[12]]
=> ? = 6
[2,1,1,1,1,1,1,2,1,1] => [1,8,3] => [8,3,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12]]
=> ? = 8
[2,1,1,1,1,1,1,1,1,2] => [1,10,1] => [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ? = 10
[2,1,1,1,1,1,2,3] => [1,7,2,1,1] => [7,2,1,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11],[12]]
=> ? = 7
[2,1,1,1,3,4] => [1,5,1,2,1,1,1] => [5,2,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 5
[2,1,2,3,2,2] => [1,3,2,1,2,2,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000676
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1]
=> [1,0]
=> 1
[1,1] => [2] => [2]
=> [1,0,1,0]
=> 2
[2] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[1,1,1] => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,1] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[8] => [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
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,2] => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,1,1,1,3] => [7,1,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,4] => [6,1,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[1,1,1,1,5] => [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]
=> ? = 5
[1,1,1,6] => [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
[1,1,7] => [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
[1,2,6] => [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
[1,3,5] => [2,1,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,4,4] => [2,1,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,5,3] => [2,1,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,6,2] => [2,1,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,7,1] => [2,1,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,8] => [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
[2,1,1,1,1,1,1,1] => [1,8] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[2,1,1,1,1,1,2] => [1,7,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[2,1,1,1,1,3] => [1,6,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[2,1,1,1,4] => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[2,1,1,5] => [1,4,1,1,1,1] => [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
[2,1,6] => [1,3,1,1,1,1,1] => [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
[2,2,5] => [1,2,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,3,4] => [1,2,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,4,3] => [1,2,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,5,2] => [1,2,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,6,1] => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,7] => [1,2,1,1,1,1,1,1] => [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
[3,1,1,1,1,1,1] => [1,1,7] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[3,1,1,1,1,2] => [1,1,6,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[3,1,1,1,3] => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[3,1,1,4] => [1,1,4,1,1,1] => [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
[3,1,5] => [1,1,3,1,1,1,1] => [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
[3,2,4] => [1,1,2,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,3,3] => [1,1,2,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,4,2] => [1,1,2,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,5,1] => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,6] => [1,1,2,1,1,1,1,1] => [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
[4,1,1,1,1,1] => [1,1,1,6] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[4,1,1,1,2] => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[4,1,1,3] => [1,1,1,4,1,1] => [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
[4,1,4] => [1,1,1,3,1,1,1] => [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
[4,2,3] => [1,1,1,2,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,3,2] => [1,1,1,2,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,4,1] => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,5] => [1,1,1,2,1,1,1,1] => [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
[5,1,1,1,1] => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[5,1,1,2] => [1,1,1,1,4,1] => [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
[5,1,3] => [1,1,1,1,3,1,1] => [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
[5,2,2] => [1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[5,3,1] => [1,1,1,1,2,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 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 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1]
=> [1,0]
=> ? = 1
[1,1] => [2] => [2]
=> [1,0,1,0]
=> 2
[2] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[1,1,1] => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,1] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[2,1,3] => [1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[8] => [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
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,2] => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,1,1,1,3] => [7,1,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,4] => [6,1,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[1,1,1,1,5] => [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]
=> ? = 5
[1,1,1,6] => [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
[1,1,7] => [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
[1,2,6] => [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
[1,3,5] => [2,1,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,4,4] => [2,1,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,5,3] => [2,1,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,6,2] => [2,1,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,7,1] => [2,1,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,8] => [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
[2,1,1,1,1,1,1,1] => [1,8] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[2,1,1,1,1,1,2] => [1,7,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[2,1,1,1,1,3] => [1,6,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[2,1,1,1,4] => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[2,1,1,5] => [1,4,1,1,1,1] => [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
[2,1,6] => [1,3,1,1,1,1,1] => [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
[2,2,5] => [1,2,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,3,4] => [1,2,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,4,3] => [1,2,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,5,2] => [1,2,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,6,1] => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,7] => [1,2,1,1,1,1,1,1] => [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
[3,1,1,1,1,1,1] => [1,1,7] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[3,1,1,1,1,2] => [1,1,6,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[3,1,1,1,3] => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[3,1,1,4] => [1,1,4,1,1,1] => [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
[3,1,5] => [1,1,3,1,1,1,1] => [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
[3,2,4] => [1,1,2,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,3,3] => [1,1,2,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,4,2] => [1,1,2,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,5,1] => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,6] => [1,1,2,1,1,1,1,1] => [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
[4,1,1,1,1,1] => [1,1,1,6] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[4,1,1,1,2] => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[4,1,1,3] => [1,1,1,4,1,1] => [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
[4,1,4] => [1,1,1,3,1,1,1] => [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
[4,2,3] => [1,1,1,2,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,3,2] => [1,1,1,2,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,4,1] => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,5] => [1,1,1,2,1,1,1,1] => [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
[5,1,1,1,1] => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[5,1,1,2] => [1,1,1,1,4,1] => [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
[5,1,3] => [1,1,1,1,3,1,1] => [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
[5,2,2] => [1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000392
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 92%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 92%
Values
[1] => [1] => 1 => 0 => 0 = 1 - 1
[1,1] => [2] => 10 => 01 => 1 = 2 - 1
[2] => [1,1] => 11 => 00 => 0 = 1 - 1
[1,1,1] => [3] => 100 => 011 => 2 = 3 - 1
[1,2] => [2,1] => 101 => 010 => 1 = 2 - 1
[2,1] => [1,2] => 110 => 001 => 1 = 2 - 1
[3] => [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,1,1,1] => [4] => 1000 => 0111 => 3 = 4 - 1
[1,1,2] => [3,1] => 1001 => 0110 => 2 = 3 - 1
[1,2,1] => [2,2] => 1010 => 0101 => 1 = 2 - 1
[1,3] => [2,1,1] => 1011 => 0100 => 1 = 2 - 1
[2,1,1] => [1,3] => 1100 => 0011 => 2 = 3 - 1
[2,2] => [1,2,1] => 1101 => 0010 => 1 = 2 - 1
[3,1] => [1,1,2] => 1110 => 0001 => 1 = 2 - 1
[4] => [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
[1,1,1,1,1] => [5] => 10000 => 01111 => 4 = 5 - 1
[1,1,1,2] => [4,1] => 10001 => 01110 => 3 = 4 - 1
[1,1,2,1] => [3,2] => 10010 => 01101 => 2 = 3 - 1
[1,1,3] => [3,1,1] => 10011 => 01100 => 2 = 3 - 1
[1,2,1,1] => [2,3] => 10100 => 01011 => 2 = 3 - 1
[1,2,2] => [2,2,1] => 10101 => 01010 => 1 = 2 - 1
[1,3,1] => [2,1,2] => 10110 => 01001 => 1 = 2 - 1
[1,4] => [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
[2,1,1,1] => [1,4] => 11000 => 00111 => 3 = 4 - 1
[2,1,2] => [1,3,1] => 11001 => 00110 => 2 = 3 - 1
[2,2,1] => [1,2,2] => 11010 => 00101 => 1 = 2 - 1
[2,3] => [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
[3,1,1] => [1,1,3] => 11100 => 00011 => 2 = 3 - 1
[3,2] => [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
[4,1] => [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
[5] => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
[1,1,1,1,1,1] => [6] => 100000 => 011111 => 5 = 6 - 1
[1,1,1,1,2] => [5,1] => 100001 => 011110 => 4 = 5 - 1
[1,1,1,2,1] => [4,2] => 100010 => 011101 => 3 = 4 - 1
[1,1,1,3] => [4,1,1] => 100011 => 011100 => 3 = 4 - 1
[1,1,2,1,1] => [3,3] => 100100 => 011011 => 2 = 3 - 1
[1,1,2,2] => [3,2,1] => 100101 => 011010 => 2 = 3 - 1
[1,1,3,1] => [3,1,2] => 100110 => 011001 => 2 = 3 - 1
[1,1,4] => [3,1,1,1] => 100111 => 011000 => 2 = 3 - 1
[1,2,1,1,1] => [2,4] => 101000 => 010111 => 3 = 4 - 1
[1,2,1,2] => [2,3,1] => 101001 => 010110 => 2 = 3 - 1
[1,2,2,1] => [2,2,2] => 101010 => 010101 => 1 = 2 - 1
[1,2,3] => [2,2,1,1] => 101011 => 010100 => 1 = 2 - 1
[1,3,1,1] => [2,1,3] => 101100 => 010011 => 2 = 3 - 1
[1,3,2] => [2,1,2,1] => 101101 => 010010 => 1 = 2 - 1
[1,4,1] => [2,1,1,2] => 101110 => 010001 => 1 = 2 - 1
[1,5] => [2,1,1,1,1] => 101111 => 010000 => 1 = 2 - 1
[2,1,1,1,1] => [1,5] => 110000 => 001111 => 4 = 5 - 1
[2,1,1,2] => [1,4,1] => 110001 => 001110 => 3 = 4 - 1
[2,1,2,1] => [1,3,2] => 110010 => 001101 => 2 = 3 - 1
[1,1,1,1,1,1,1,1,2] => [9,1] => 1000000001 => 0111111110 => ? = 9 - 1
[1,1,1,1,1,1,1,3] => [8,1,1] => 1000000011 => 0111111100 => ? = 8 - 1
[1,1,1,1,1,1,2,2] => [7,2,1] => 1000000101 => 0111111010 => ? = 7 - 1
[1,1,1,1,1,1,4] => [7,1,1,1] => 1000000111 => 0111111000 => ? = 7 - 1
[1,1,1,1,1,2,1,1,1] => [6,4] => 1000001000 => 0111110111 => ? = 6 - 1
[1,1,1,1,1,2,1,2] => [6,3,1] => 1000001001 => 0111110110 => ? = 6 - 1
[1,1,1,1,1,2,3] => [6,2,1,1] => 1000001011 => 0111110100 => ? = 6 - 1
[1,1,1,1,1,3,1,1] => [6,1,3] => 1000001100 => 0111110011 => ? = 6 - 1
[1,1,1,1,1,3,2] => [6,1,2,1] => 1000001101 => 0111110010 => ? = 6 - 1
[1,1,1,1,2,1,1,1,1] => [5,5] => 1000010000 => 0111101111 => ? = 5 - 1
[1,1,1,1,2,1,1,2] => [5,4,1] => 1000010001 => 0111101110 => ? = 5 - 1
[1,1,1,1,2,1,2,1] => [5,3,2] => 1000010010 => 0111101101 => ? = 5 - 1
[1,1,1,1,2,1,3] => [5,3,1,1] => 1000010011 => 0111101100 => ? = 5 - 1
[1,1,1,1,2,2,1,1] => [5,2,3] => 1000010100 => 0111101011 => ? = 5 - 1
[1,1,1,1,2,2,2] => [5,2,2,1] => 1000010101 => 0111101010 => ? = 5 - 1
[1,1,1,1,3,1,1,1] => [5,1,4] => 1000011000 => 0111100111 => ? = 5 - 1
[1,1,1,1,3,1,2] => [5,1,3,1] => 1000011001 => ? => ? = 5 - 1
[1,1,1,1,3,2,1] => [5,1,2,2] => 1000011010 => 0111100101 => ? = 5 - 1
[1,1,1,1,4,1,1] => [5,1,1,3] => 1000011100 => 0111100011 => ? = 5 - 1
[1,1,1,1,6] => [5,1,1,1,1,1] => 1000011111 => 0111100000 => ? = 5 - 1
[1,1,1,2,1,1,1,1,1] => [4,6] => 1000100000 => 0111011111 => ? = 6 - 1
[1,1,1,2,1,1,1,2] => [4,5,1] => 1000100001 => 0111011110 => ? = 5 - 1
[1,1,1,2,1,1,2,1] => [4,4,2] => 1000100010 => 0111011101 => ? = 4 - 1
[1,1,1,2,1,1,3] => [4,4,1,1] => 1000100011 => 0111011100 => ? = 4 - 1
[1,1,1,2,1,2,1,1] => [4,3,3] => 1000100100 => 0111011011 => ? = 4 - 1
[1,1,1,2,1,2,2] => [4,3,2,1] => 1000100101 => 0111011010 => ? = 4 - 1
[1,1,1,2,1,3,1] => [4,3,1,2] => 1000100110 => 0111011001 => ? = 4 - 1
[1,1,1,2,2,1,1,1] => [4,2,4] => 1000101000 => 0111010111 => ? = 4 - 1
[1,1,1,2,2,1,2] => [4,2,3,1] => 1000101001 => 0111010110 => ? = 4 - 1
[1,1,1,2,2,2,1] => [4,2,2,2] => 1000101010 => 0111010101 => ? = 4 - 1
[1,1,1,2,3,1,1] => [4,2,1,3] => 1000101100 => 0111010011 => ? = 4 - 1
[1,1,1,2,5] => [4,2,1,1,1,1] => 1000101111 => 0111010000 => ? = 4 - 1
[1,1,1,3,1,1,1,1] => [4,1,5] => 1000110000 => 0111001111 => ? = 5 - 1
[1,1,1,3,1,1,2] => [4,1,4,1] => 1000110001 => 0111001110 => ? = 4 - 1
[1,1,1,3,1,2,1] => [4,1,3,2] => 1000110010 => 0111001101 => ? = 4 - 1
[1,1,1,3,2,1,1] => [4,1,2,3] => 1000110100 => 0111001011 => ? = 4 - 1
[1,1,1,3,4] => [4,1,2,1,1,1] => 1000110111 => 0111001000 => ? = 4 - 1
[1,1,1,4,1,1,1] => [4,1,1,4] => 1000111000 => 0111000111 => ? = 4 - 1
[1,1,1,4,3] => [4,1,1,2,1,1] => 1000111011 => 0111000100 => ? = 4 - 1
[1,1,1,5,1,1] => [4,1,1,1,3] => 1000111100 => 0111000011 => ? = 4 - 1
[1,1,1,5,2] => [4,1,1,1,2,1] => 1000111101 => 0111000010 => ? = 4 - 1
[1,1,1,7] => [4,1,1,1,1,1,1] => 1000111111 => 0111000000 => ? = 4 - 1
[1,1,2,1,1,1,1,2] => [3,6,1] => 1001000001 => 0110111110 => ? = 6 - 1
[1,1,2,1,1,1,2,1] => [3,5,2] => 1001000010 => 0110111101 => ? = 5 - 1
[1,1,2,1,1,1,3] => [3,5,1,1] => 1001000011 => 0110111100 => ? = 5 - 1
[1,1,2,1,1,2,1,1] => [3,4,3] => 1001000100 => 0110111011 => ? = 4 - 1
[1,1,2,1,1,2,2] => [3,4,2,1] => 1001000101 => 0110111010 => ? = 4 - 1
[1,1,2,1,1,3,1] => [3,4,1,2] => 1001000110 => 0110111001 => ? = 4 - 1
[1,1,2,1,2,1,1,1] => [3,3,4] => 1001001000 => 0110110111 => ? = 4 - 1
[1,1,2,1,2,1,2] => [3,3,3,1] => 1001001001 => 0110110110 => ? = 3 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001291
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 42%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 42%
Values
[1] => [1] => [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1] => [2] => [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2] => [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,2] => [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1] => [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3] => [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,2] => [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,2,1] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,1] => [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[2,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4] => [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,2] => [4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,2,1] => [3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,3] => [3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,2,1,1] => [2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,2] => [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1] => [2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[2,1,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[2,2,1] => [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[5] => [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
[1,1,1,1,1,1] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,2] => [5,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,2,1] => [4,2] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,2,1,1] => [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,2,1,1,1] => [2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,5] => [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
[2,1,1,1,1] => [1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 6 = 5 + 1
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,3] => [1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[6] => [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] => [7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,1,1,1,1,2] => [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,6] => [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
[2,1,1,1,1,1] => [1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[2,5] => [1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[3,4] => [1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[4,3] => [1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[5,2] => [1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[6,1] => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7] => [1,1,1,1,1,1,1] => [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
[1,1,1,1,1,1,1,1] => [8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,1,1,1,1,1,2] => [7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,1,1,1,1,2,1] => [6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,3] => [6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,6] => [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
[1,2,1,1,1,1,1] => [2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 6 + 1
[1,2,5] => [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
[1,3,4] => [2,1,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,4,3] => [2,1,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,5,2] => [2,1,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,1] => [2,1,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,7] => [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
[2,1,1,1,1,1,1] => [1,7] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[2,1,1,1,1,2] => [1,6,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[2,1,5] => [1,3,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[2,2,4] => [1,2,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,3,3] => [1,2,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,4,2] => [1,2,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,5,1] => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,6] => [1,2,1,1,1,1,1] => [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
[3,1,1,1,1,1] => [1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[3,1,4] => [1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[3,2,3] => [1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[3,3,2] => [1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[3,4,1] => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[3,5] => [1,1,2,1,1,1,1] => [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
[4,1,3] => [1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[4,2,2] => [1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[4,3,1] => [1,1,1,2,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[4,4] => [1,1,1,2,1,1,1] => [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
[5,1,2] => [1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[5,2,1] => [1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[5,3] => [1,1,1,1,2,1,1] => [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
[6,1,1] => [1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[6,2] => [1,1,1,1,1,2,1] => [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
[7,1] => [1,1,1,1,1,1,2] => [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] => [1,1,1,1,1,1,1,1] => [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
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 + 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: St000381
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 83%
St000381: Integer compositions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => 1
[1,1] => [2] => 2
[2] => [1,1] => 1
[1,1,1] => [3] => 3
[1,2] => [1,2] => 2
[2,1] => [2,1] => 2
[3] => [1,1,1] => 1
[1,1,1,1] => [4] => 4
[1,1,2] => [1,3] => 3
[1,2,1] => [2,2] => 2
[1,3] => [1,1,2] => 2
[2,1,1] => [3,1] => 3
[2,2] => [1,2,1] => 2
[3,1] => [2,1,1] => 2
[4] => [1,1,1,1] => 1
[1,1,1,1,1] => [5] => 5
[1,1,1,2] => [1,4] => 4
[1,1,2,1] => [2,3] => 3
[1,1,3] => [1,1,3] => 3
[1,2,1,1] => [3,2] => 3
[1,2,2] => [1,2,2] => 2
[1,3,1] => [2,1,2] => 2
[1,4] => [1,1,1,2] => 2
[2,1,1,1] => [4,1] => 4
[2,1,2] => [1,3,1] => 3
[2,2,1] => [2,2,1] => 2
[2,3] => [1,1,2,1] => 2
[3,1,1] => [3,1,1] => 3
[3,2] => [1,2,1,1] => 2
[4,1] => [2,1,1,1] => 2
[5] => [1,1,1,1,1] => 1
[1,1,1,1,1,1] => [6] => 6
[1,1,1,1,2] => [1,5] => 5
[1,1,1,2,1] => [2,4] => 4
[1,1,1,3] => [1,1,4] => 4
[1,1,2,1,1] => [3,3] => 3
[1,1,2,2] => [1,2,3] => 3
[1,1,3,1] => [2,1,3] => 3
[1,1,4] => [1,1,1,3] => 3
[1,2,1,1,1] => [4,2] => 4
[1,2,1,2] => [1,3,2] => 3
[1,2,2,1] => [2,2,2] => 2
[1,2,3] => [1,1,2,2] => 2
[1,3,1,1] => [3,1,2] => 3
[1,3,2] => [1,2,1,2] => 2
[1,4,1] => [2,1,1,2] => 2
[1,5] => [1,1,1,1,2] => 2
[2,1,1,1,1] => [5,1] => 5
[2,1,1,2] => [1,4,1] => 4
[2,1,2,1] => [2,3,1] => 3
[1,1,1,1,1,1,1,1,1,1] => [10] => ? = 10
[1,1,1,1,1,1,1,2,1] => [2,8] => ? = 8
[1,1,1,1,1,1,1,3] => [1,1,8] => ? = 8
[1,1,1,1,1,1,2,1,1] => [3,7] => ? = 7
[1,1,1,1,1,1,2,2] => [1,2,7] => ? = 7
[1,1,1,1,1,1,3,1] => [2,1,7] => ? = 7
[1,1,1,1,1,1,4] => [1,1,1,7] => ? = 7
[1,1,1,1,1,2,1,1,1] => [4,6] => ? = 6
[1,1,1,1,1,2,1,2] => [1,3,6] => ? = 6
[1,1,1,1,1,2,2,1] => [2,2,6] => ? = 6
[1,1,1,1,1,2,3] => [1,1,2,6] => ? = 6
[1,1,1,1,1,3,1,1] => [3,1,6] => ? = 6
[1,1,1,1,1,3,2] => [1,2,1,6] => ? = 6
[1,1,1,1,1,4,1] => [2,1,1,6] => ? = 6
[1,1,1,1,1,5] => [1,1,1,1,6] => ? = 6
[1,1,1,1,2,1,1,2] => [1,4,5] => ? = 5
[1,1,1,1,2,1,2,1] => [2,3,5] => ? = 5
[1,1,1,1,2,1,3] => [1,1,3,5] => ? = 5
[1,1,1,1,2,2,1,1] => [3,2,5] => ? = 5
[1,1,1,1,2,2,2] => [1,2,2,5] => ? = 5
[1,1,1,1,2,3,1] => [2,1,2,5] => ? = 5
[1,1,1,1,2,4] => [1,1,1,2,5] => ? = 5
[1,1,1,1,3,1,1,1] => [4,1,5] => ? = 5
[1,1,1,1,3,1,2] => [1,3,1,5] => ? = 5
[1,1,1,1,3,2,1] => [2,2,1,5] => ? = 5
[1,1,1,1,3,3] => [1,1,2,1,5] => ? = 5
[1,1,1,1,4,1,1] => [3,1,1,5] => ? = 5
[1,1,1,1,4,2] => [1,2,1,1,5] => ? = 5
[1,1,1,1,5,1] => [2,1,1,1,5] => ? = 5
[1,1,1,1,6] => [1,1,1,1,1,5] => ? = 5
[1,1,1,2,1,1,1,1,1] => [6,4] => ? = 6
[1,1,1,2,1,1,1,2] => [1,5,4] => ? = 5
[1,1,1,2,1,1,2,1] => [2,4,4] => ? = 4
[1,1,1,2,1,2,1,1] => [3,3,4] => ? = 4
[1,1,1,2,1,2,2] => [1,2,3,4] => ? = 4
[1,1,1,2,1,4] => [1,1,1,3,4] => ? = 4
[1,1,1,2,2,1,1,1] => [4,2,4] => ? = 4
[1,1,1,2,2,1,2] => [1,3,2,4] => ? = 4
[1,1,1,2,2,2,1] => [2,2,2,4] => ? = 4
[1,1,1,2,2,3] => [1,1,2,2,4] => ? = 4
[1,1,1,2,3,2] => [1,2,1,2,4] => ? = 4
[1,1,1,2,4,1] => [2,1,1,2,4] => ? = 4
[1,1,1,2,5] => [1,1,1,1,2,4] => ? = 4
[1,1,1,3,1,1,1,1] => [5,1,4] => ? = 5
[1,1,1,3,1,1,2] => [1,4,1,4] => ? = 4
[1,1,1,3,1,2,1] => [2,3,1,4] => ? = 4
[1,1,1,3,1,3] => [1,1,3,1,4] => ? = 4
[1,1,1,3,2,1,1] => [3,2,1,4] => ? = 4
[1,1,1,3,2,2] => [1,2,2,1,4] => ? = 4
[1,1,1,3,3,1] => [2,1,2,1,4] => ? = 4
Description
The largest part of an integer composition.
Matching statistic: St000982
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 83%
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 83%
Values
[1] => 1 => 1 => 0 => 1
[1,1] => 11 => 11 => 00 => 2
[2] => 10 => 01 => 10 => 1
[1,1,1] => 111 => 111 => 000 => 3
[1,2] => 110 => 011 => 100 => 2
[2,1] => 101 => 001 => 110 => 2
[3] => 100 => 101 => 010 => 1
[1,1,1,1] => 1111 => 1111 => 0000 => 4
[1,1,2] => 1110 => 0111 => 1000 => 3
[1,2,1] => 1101 => 0011 => 1100 => 2
[1,3] => 1100 => 1011 => 0100 => 2
[2,1,1] => 1011 => 0001 => 1110 => 3
[2,2] => 1010 => 1001 => 0110 => 2
[3,1] => 1001 => 1101 => 0010 => 2
[4] => 1000 => 0101 => 1010 => 1
[1,1,1,1,1] => 11111 => 11111 => 00000 => 5
[1,1,1,2] => 11110 => 01111 => 10000 => 4
[1,1,2,1] => 11101 => 00111 => 11000 => 3
[1,1,3] => 11100 => 10111 => 01000 => 3
[1,2,1,1] => 11011 => 00011 => 11100 => 3
[1,2,2] => 11010 => 10011 => 01100 => 2
[1,3,1] => 11001 => 11011 => 00100 => 2
[1,4] => 11000 => 01011 => 10100 => 2
[2,1,1,1] => 10111 => 00001 => 11110 => 4
[2,1,2] => 10110 => 10001 => 01110 => 3
[2,2,1] => 10101 => 11001 => 00110 => 2
[2,3] => 10100 => 01001 => 10110 => 2
[3,1,1] => 10011 => 11101 => 00010 => 3
[3,2] => 10010 => 01101 => 10010 => 2
[4,1] => 10001 => 00101 => 11010 => 2
[5] => 10000 => 10101 => 01010 => 1
[1,1,1,1,1,1] => 111111 => 111111 => 000000 => 6
[1,1,1,1,2] => 111110 => 011111 => 100000 => 5
[1,1,1,2,1] => 111101 => 001111 => 110000 => 4
[1,1,1,3] => 111100 => 101111 => 010000 => 4
[1,1,2,1,1] => 111011 => 000111 => 111000 => 3
[1,1,2,2] => 111010 => 100111 => 011000 => 3
[1,1,3,1] => 111001 => 110111 => 001000 => 3
[1,1,4] => 111000 => 010111 => 101000 => 3
[1,2,1,1,1] => 110111 => 000011 => 111100 => 4
[1,2,1,2] => 110110 => 100011 => 011100 => 3
[1,2,2,1] => 110101 => 110011 => 001100 => 2
[1,2,3] => 110100 => 010011 => 101100 => 2
[1,3,1,1] => 110011 => 111011 => 000100 => 3
[1,3,2] => 110010 => 011011 => 100100 => 2
[1,4,1] => 110001 => 001011 => 110100 => 2
[1,5] => 110000 => 101011 => 010100 => 2
[2,1,1,1,1] => 101111 => 000001 => 111110 => 5
[2,1,1,2] => 101110 => 100001 => 011110 => 4
[2,1,2,1] => 101101 => 110001 => 001110 => 3
[1,1,1,1,1,1,1,2,1] => 1111111101 => 0011111111 => 1100000000 => ? = 8
[1,1,1,1,1,1,1,3] => 1111111100 => ? => ? => ? = 8
[1,1,1,1,1,1,2,1,1] => 1111111011 => ? => ? => ? = 7
[1,1,1,1,1,1,2,2] => 1111111010 => ? => ? => ? = 7
[1,1,1,1,1,1,4] => 1111111000 => ? => ? => ? = 7
[1,1,1,1,1,2,1,1,1] => 1111110111 => ? => ? => ? = 6
[1,1,1,1,1,2,1,2] => 1111110110 => ? => ? => ? = 6
[1,1,1,1,1,2,2,1] => 1111110101 => ? => ? => ? = 6
[1,1,1,1,1,2,3] => 1111110100 => ? => ? => ? = 6
[1,1,1,1,1,3,1,1] => 1111110011 => ? => ? => ? = 6
[1,1,1,1,1,3,2] => 1111110010 => ? => ? => ? = 6
[1,1,1,1,1,4,1] => 1111110001 => ? => ? => ? = 6
[1,1,1,1,1,5] => 1111110000 => ? => ? => ? = 6
[1,1,1,1,2,1,1,1,1] => 1111101111 => ? => ? => ? = 5
[1,1,1,1,2,1,1,2] => 1111101110 => ? => ? => ? = 5
[1,1,1,1,2,1,2,1] => 1111101101 => 1100011111 => 0011100000 => ? = 5
[1,1,1,1,2,1,3] => 1111101100 => ? => ? => ? = 5
[1,1,1,1,2,2,2] => 1111101010 => ? => ? => ? = 5
[1,1,1,1,2,3,1] => 1111101001 => ? => ? => ? = 5
[1,1,1,1,2,4] => 1111101000 => ? => ? => ? = 5
[1,1,1,1,3,1,2] => 1111100110 => ? => ? => ? = 5
[1,1,1,1,3,2,1] => 1111100101 => ? => ? => ? = 5
[1,1,1,1,3,3] => 1111100100 => ? => ? => ? = 5
[1,1,1,1,4,1,1] => 1111100011 => ? => ? => ? = 5
[1,1,1,1,4,2] => 1111100010 => ? => ? => ? = 5
[1,1,1,1,6] => 1111100000 => 0101011111 => 1010100000 => ? = 5
[1,1,1,2,1,1,1,1,1] => 1111011111 => ? => ? => ? = 6
[1,1,1,2,1,1,1,2] => 1111011110 => ? => ? => ? = 5
[1,1,1,2,1,1,2,1] => 1111011101 => ? => ? => ? = 4
[1,1,1,2,1,1,3] => 1111011100 => ? => ? => ? = 4
[1,1,1,2,1,2,1,1] => 1111011011 => ? => ? => ? = 4
[1,1,1,2,1,2,2] => 1111011010 => ? => ? => ? = 4
[1,1,1,2,1,3,1] => 1111011001 => ? => ? => ? = 4
[1,1,1,2,1,4] => 1111011000 => ? => ? => ? = 4
[1,1,1,2,2,1,1,1] => 1111010111 => ? => ? => ? = 4
[1,1,1,2,2,1,2] => 1111010110 => ? => ? => ? = 4
[1,1,1,2,2,2,1] => 1111010101 => ? => ? => ? = 4
[1,1,1,2,2,3] => 1111010100 => ? => ? => ? = 4
[1,1,1,2,3,1,1] => 1111010011 => ? => ? => ? = 4
[1,1,1,2,3,2] => 1111010010 => ? => ? => ? = 4
[1,1,1,2,4,1] => 1111010001 => ? => ? => ? = 4
[1,1,1,2,5] => 1111010000 => 0101001111 => 1010110000 => ? = 4
[1,1,1,3,1,1,1,1] => 1111001111 => ? => ? => ? = 5
[1,1,1,3,1,1,2] => 1111001110 => ? => ? => ? = 4
[1,1,1,3,1,2,1] => 1111001101 => ? => ? => ? = 4
[1,1,1,3,1,3] => 1111001100 => ? => ? => ? = 4
[1,1,1,3,2,1,1] => 1111001011 => ? => ? => ? = 4
[1,1,1,3,2,2] => 1111001010 => ? => ? => ? = 4
[1,1,1,3,3,1] => 1111001001 => ? => ? => ? = 4
[1,1,1,3,4] => 1111001000 => 0101101111 => 1010010000 => ? = 4
Description
The length of the longest constant subword.
Matching statistic: St000983
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 75%
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 75%
Values
[1] => 1 => 1 => 1 => 1
[1,1] => 11 => 11 => 10 => 2
[2] => 10 => 01 => 00 => 1
[1,1,1] => 111 => 111 => 101 => 3
[1,2] => 110 => 011 => 001 => 2
[2,1] => 101 => 001 => 011 => 2
[3] => 100 => 101 => 111 => 1
[1,1,1,1] => 1111 => 1111 => 1010 => 4
[1,1,2] => 1110 => 0111 => 0010 => 3
[1,2,1] => 1101 => 0011 => 0110 => 2
[1,3] => 1100 => 1011 => 1110 => 2
[2,1,1] => 1011 => 0001 => 0100 => 3
[2,2] => 1010 => 1001 => 1100 => 2
[3,1] => 1001 => 1101 => 1000 => 2
[4] => 1000 => 0101 => 0000 => 1
[1,1,1,1,1] => 11111 => 11111 => 10101 => 5
[1,1,1,2] => 11110 => 01111 => 00101 => 4
[1,1,2,1] => 11101 => 00111 => 01101 => 3
[1,1,3] => 11100 => 10111 => 11101 => 3
[1,2,1,1] => 11011 => 00011 => 01001 => 3
[1,2,2] => 11010 => 10011 => 11001 => 2
[1,3,1] => 11001 => 11011 => 10001 => 2
[1,4] => 11000 => 01011 => 00001 => 2
[2,1,1,1] => 10111 => 00001 => 01011 => 4
[2,1,2] => 10110 => 10001 => 11011 => 3
[2,2,1] => 10101 => 11001 => 10011 => 2
[2,3] => 10100 => 01001 => 00011 => 2
[3,1,1] => 10011 => 11101 => 10111 => 3
[3,2] => 10010 => 01101 => 00111 => 2
[4,1] => 10001 => 00101 => 01111 => 2
[5] => 10000 => 10101 => 11111 => 1
[1,1,1,1,1,1] => 111111 => 111111 => 101010 => 6
[1,1,1,1,2] => 111110 => 011111 => 001010 => 5
[1,1,1,2,1] => 111101 => 001111 => 011010 => 4
[1,1,1,3] => 111100 => 101111 => 111010 => 4
[1,1,2,1,1] => 111011 => 000111 => 010010 => 3
[1,1,2,2] => 111010 => 100111 => 110010 => 3
[1,1,3,1] => 111001 => 110111 => 100010 => 3
[1,1,4] => 111000 => 010111 => 000010 => 3
[1,2,1,1,1] => 110111 => 000011 => 010110 => 4
[1,2,1,2] => 110110 => 100011 => 110110 => 3
[1,2,2,1] => 110101 => 110011 => 100110 => 2
[1,2,3] => 110100 => 010011 => 000110 => 2
[1,3,1,1] => 110011 => 111011 => 101110 => 3
[1,3,2] => 110010 => 011011 => 001110 => 2
[1,4,1] => 110001 => 001011 => 011110 => 2
[1,5] => 110000 => 101011 => 111110 => 2
[2,1,1,1,1] => 101111 => 000001 => 010100 => 5
[2,1,1,2] => 101110 => 100001 => 110100 => 4
[2,1,2,1] => 101101 => 110001 => 100100 => 3
[1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => 1010101010 => ? = 10
[1,1,1,1,1,1,1,2,1] => 1111111101 => 0011111111 => 0110101010 => ? = 8
[1,1,1,1,1,1,1,3] => 1111111100 => ? => ? => ? = 8
[1,1,1,1,1,1,2,1,1] => 1111111011 => ? => ? => ? = 7
[1,1,1,1,1,1,2,2] => 1111111010 => ? => ? => ? = 7
[1,1,1,1,1,1,3,1] => 1111111001 => 1101111111 => 1000101010 => ? = 7
[1,1,1,1,1,1,4] => 1111111000 => ? => ? => ? = 7
[1,1,1,1,1,2,1,1,1] => 1111110111 => ? => ? => ? = 6
[1,1,1,1,1,2,1,2] => 1111110110 => ? => ? => ? = 6
[1,1,1,1,1,2,2,1] => 1111110101 => ? => ? => ? = 6
[1,1,1,1,1,2,3] => 1111110100 => ? => ? => ? = 6
[1,1,1,1,1,3,1,1] => 1111110011 => ? => ? => ? = 6
[1,1,1,1,1,3,2] => 1111110010 => ? => ? => ? = 6
[1,1,1,1,1,4,1] => 1111110001 => ? => ? => ? = 6
[1,1,1,1,1,5] => 1111110000 => ? => ? => ? = 6
[1,1,1,1,2,1,1,1,1] => 1111101111 => ? => ? => ? = 5
[1,1,1,1,2,1,1,2] => 1111101110 => ? => ? => ? = 5
[1,1,1,1,2,1,2,1] => 1111101101 => 1100011111 => 1001001010 => ? = 5
[1,1,1,1,2,1,3] => 1111101100 => ? => ? => ? = 5
[1,1,1,1,2,2,1,1] => 1111101011 => 1110011111 => 1011001010 => ? = 5
[1,1,1,1,2,2,2] => 1111101010 => ? => ? => ? = 5
[1,1,1,1,2,3,1] => 1111101001 => ? => ? => ? = 5
[1,1,1,1,2,4] => 1111101000 => ? => ? => ? = 5
[1,1,1,1,3,1,1,1] => 1111100111 => 1111011111 => 1010001010 => ? = 5
[1,1,1,1,3,1,2] => 1111100110 => ? => ? => ? = 5
[1,1,1,1,3,2,1] => 1111100101 => ? => ? => ? = 5
[1,1,1,1,3,3] => 1111100100 => ? => ? => ? = 5
[1,1,1,1,4,1,1] => 1111100011 => ? => ? => ? = 5
[1,1,1,1,4,2] => 1111100010 => ? => ? => ? = 5
[1,1,1,1,5,1] => 1111100001 => 1101011111 => 1000001010 => ? = 5
[1,1,1,2,1,1,1,1,1] => 1111011111 => ? => ? => ? = 6
[1,1,1,2,1,1,1,2] => 1111011110 => ? => ? => ? = 5
[1,1,1,2,1,1,2,1] => 1111011101 => ? => ? => ? = 4
[1,1,1,2,1,1,3] => 1111011100 => ? => ? => ? = 4
[1,1,1,2,1,2,1,1] => 1111011011 => ? => ? => ? = 4
[1,1,1,2,1,2,2] => 1111011010 => ? => ? => ? = 4
[1,1,1,2,1,3,1] => 1111011001 => ? => ? => ? = 4
[1,1,1,2,1,4] => 1111011000 => ? => ? => ? = 4
[1,1,1,2,2,1,1,1] => 1111010111 => ? => ? => ? = 4
[1,1,1,2,2,1,2] => 1111010110 => ? => ? => ? = 4
[1,1,1,2,2,2,1] => 1111010101 => ? => ? => ? = 4
[1,1,1,2,2,3] => 1111010100 => ? => ? => ? = 4
[1,1,1,2,3,1,1] => 1111010011 => ? => ? => ? = 4
[1,1,1,2,3,2] => 1111010010 => ? => ? => ? = 4
[1,1,1,2,4,1] => 1111010001 => ? => ? => ? = 4
[1,1,1,3,1,1,1,1] => 1111001111 => ? => ? => ? = 5
[1,1,1,3,1,1,2] => 1111001110 => ? => ? => ? = 4
[1,1,1,3,1,2,1] => 1111001101 => ? => ? => ? = 4
[1,1,1,3,1,3] => 1111001100 => ? => ? => ? = 4
[1,1,1,3,2,1,1] => 1111001011 => ? => ? => ? = 4
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St001933The largest multiplicity of a part in an integer partition. St000521The number of distinct subtrees of an ordered tree. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000439The position of the first down step of a Dyck path. St000306The bounce count of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000662The staircase size of the code of a permutation. St000141The maximum drop size of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St001058The breadth of the ordered tree. St000686The finitistic dominant dimension of a Dyck path. St000025The number of initial rises of a Dyck path. St001203We associate to 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 Dyck path as follows:
St001674The number of vertices of the largest induced star graph in the graph. St000171The degree of the graph. St001826The maximal number of leaves on a vertex of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000730The maximal arc length of a set partition. St000028The number of stack-sorts needed to sort a permutation. St001118The acyclic chromatic index of a graph. St000651The maximal size of a rise in a permutation. St000308The height of the tree associated to a permutation. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St001268The size of the largest ordinal summand in the poset. St001372The length of a longest cyclic run of ones of a binary word. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001652The length of a longest interval of consecutive numbers. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000299The number of nonisomorphic vertex-induced subtrees. St000328The maximum number of child nodes in a tree. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000094The depth of an ordered tree. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001117The game chromatic index of a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000652The maximal difference between successive positions of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001651The Frankl number of a lattice.
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