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Your data matches 66 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00102: Dyck paths —rise composition⟶ 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
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,1] => [1,1]
=> 1
[1,1,0,0]
=> [2] => [2]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3] => [3]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00102: Dyck paths —rise composition⟶ 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
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [2]
=> 1
[1,1,0,0]
=> [2] => [2]
=> [1,1]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> [2,1]
=> 2
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> [2,2]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [3,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> [3,2]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [2,2,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> [3,2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> [2,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000734
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 91%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [1]
=> [[1]]
=> 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [[1],[2]]
=> 1
[1,1,0,0]
=> [2] => [2]
=> [[1,2]]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [3] => [3]
=> [[1,2,3]]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [[1,2,3,4]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => [11,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12]]
=> ? = 11
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? = 9
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? = 9
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,8,2] => [8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10,1] => [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ? = 10
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? = 9
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1,1] => [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,7,3] => [7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,6,4] => [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[1,0,1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,5,5] => [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 5
[1,0,1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,4,6] => [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[1,0,1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,3,7] => [7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,3,1] => [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,5,1] => [5,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 5
[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,1,1,1,1,1] => [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? = 6
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,1,1,1,1,2] => [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? = 5
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,1,1,1,1,1] => [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? = 6
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [9,2] => [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1] => [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [10,1] => [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [10,1] => [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,8,1] => [8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,1,1,1,1,1,2,1] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,1,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [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
[1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,2,2,2,3,1,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,2,2,3,1,2,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> [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
[1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0]
=> [1,2,2,4,1,1,1] => [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,2,3,1,2,2,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,2,3,1,3,1,1] => [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,2,3,2,1,2,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,2,4,1,1,2,1] => [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,2,3,2,2,1,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0]
=> [1,2,3,3,1,1,1] => [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,0]
=> [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
[1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0]
=> [1,2,4,2,1,1,1] => [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> [1,2,5,1,1,1,1] => [5,2,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 5
[1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,1,2,2,2,1] => [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,3,1,2,3,1,1] => [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,1,3,1,2,1] => [3,3,2,1,1,1,1]
=> [[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: St000392
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 0 => 0 = 1 - 1
[1,0,1,0]
=> [1,1] => 11 => 00 => 0 = 1 - 1
[1,1,0,0]
=> [2] => 10 => 01 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 001 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 010 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1] => 101 => 010 => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3] => 100 => 011 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0001 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 0010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 0010 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 0100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 0100 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 0100 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 0101 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 0110 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 0110 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 0110 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0111 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00011 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 00101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 00101 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 00110 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 00110 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 00110 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00111 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 01011 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 01001 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 01001 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 01011 => 2 = 3 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 9 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 01111111110 => ? = 10 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 0111111100 => ? = 8 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => 0011111110 => ? = 8 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00111111111 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => 100000000001 => 011111111110 => ? = 11 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 0111111100 => ? = 8 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,2,1] => 1000000101 => 0111111010 => ? = 7 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => 0011111110 => ? = 8 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => 11000000001 => 00111111110 => ? = 9 - 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 0111111100 => ? = 8 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => 1000000111 => 0111111000 => ? = 7 - 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,3,1] => 1000001001 => 0111110110 => ? = 6 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => 0011111110 => ? = 8 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => 0011111100 => ? = 7 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,9,1] => 11000000001 => 00111111110 => ? = 9 - 1
[1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,6,3] => 1100000100 => 0011111011 => ? = 6 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10,1] => 110000000001 => 001111111110 => ? = 10 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 0111111100 => ? = 8 - 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => 1000000111 => 0111111000 => ? = 7 - 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [6,2,1,1] => 1000001011 => 0111110100 => ? = 6 - 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [5,4,1] => 1000010001 => 0111101110 => ? = 5 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,8,1] => 1100000001 => 0011111110 => ? = 8 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => 0011111100 => ? = 7 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,9,1] => 11000000001 => 00111111110 => ? = 9 - 1
[1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,6,2,1] => 1100000101 => 0011111010 => ? = 6 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1,1] => 11000000011 => 00111111100 => ? = 8 - 1
[1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,5,4] => 1100001000 => 0011110111 => ? = 5 - 1
[1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,7,3] => 11000000100 => 00111111011 => ? = 7 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 0111111100 => ? = 8 - 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => 1000000111 => 0111111000 => ? = 7 - 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,1,2,1] => 1000001101 => 0111110010 => ? = 6 - 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [4,5,1] => 1000100001 => 0111011110 => ? = 5 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,8,1] => 1100000001 => 0011111110 => ? = 8 - 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => 0011111100 => ? = 7 - 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
=> [1,6,2,1] => 1100000101 => 0011111010 => ? = 6 - 1
[1,0,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,6,1,2] => 1100000110 => 0011111001 => ? = 6 - 1
[1,0,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,5] => 1100010000 => 0011101111 => ? = 5 - 1
[1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,6,4] => 11000001000 => 00111110111 => ? = 6 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 0111111100 => ? = 8 - 1
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0]
=> [7,2,1] => 1000000101 => 0111111010 => ? = 7 - 1
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [6,3,1] => 1000001001 => 0111110110 => ? = 6 - 1
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [3,6,1] => 1001000001 => 0110111110 => ? = 6 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,8,1] => 1100000001 => 0011111110 => ? = 8 - 1
[1,0,1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,5,2,2] => 1100001010 => 0011110101 => ? = 5 - 1
[1,0,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,3,6] => 1100100000 => 0011011111 => ? = 6 - 1
[1,0,1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,5,5] => 11000010000 => 00111101111 => ? = 5 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
=> [8,1,1] => 1000000011 => 0111111100 => ? = 8 - 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,7,1] => 1010000001 => 0101111110 => ? = 7 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000381
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 91%
St000381: Integer compositions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1] => 1
[1,1,0,0]
=> [2] => 2
[1,0,1,0,1,0]
=> [1,1,1] => 1
[1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,0,1,0]
=> [2,1] => 2
[1,1,0,1,0,0]
=> [2,1] => 2
[1,1,1,0,0,0]
=> [3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,1] => 3
[1,1,1,0,1,0,0,0]
=> [3,1] => 3
[1,1,1,1,0,0,0,0]
=> [4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 3
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => ? = 11
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,2,1] => ? = 7
[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,7,2] => ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => ? = 9
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => ? = 7
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,3,1] => ? = 6
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,9,1] => ? = 9
[1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,6,3] => ? = 6
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,8,2] => ? = 8
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10,1] => ? = 10
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => ? = 7
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [6,2,1,1] => ? = 6
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [5,4,1] => ? = 5
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,9,1] => ? = 9
[1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,7,2] => ? = 7
[1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,5,4] => ? = 5
[1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,7,3] => ? = 7
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => ? = 7
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,1,2,1] => ? = 6
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [4,5,1] => ? = 5
[1,0,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,5] => ? = 5
[1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,6,4] => ? = 6
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0]
=> [7,2,1] => ? = 7
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1,1,1] => ? = 6
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [6,3,1] => ? = 6
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [3,6,1] => ? = 6
[1,0,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> [1,7,2] => ? = 7
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,6,1,1,1] => ? = 6
[1,0,1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,5,2,2] => ? = 5
[1,0,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,3,6] => ? = 6
[1,0,1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,5,5] => ? = 5
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,7,1] => ? = 7
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,2] => ? = 8
[1,0,1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,5,1,3] => ? = 5
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,8] => ? = 8
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,7] => ? = 7
[1,0,1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,4,6] => ? = 6
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,1,0]
=> [7,1,1,1] => ? = 7
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,1,2] => ? = 7
[1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,7,1] => ? = 7
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,8] => ? = 8
[1,0,1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,3,7] => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,2,1] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,2,1,1] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,2,1,1,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1,1,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1,1,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1,1,1] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1,1,1] => ? = 2
Description
The largest part of an integer composition.
Matching statistic: St000676
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 73%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => [1]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [2] => [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,8] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 10
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => [11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 11
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,2,1] => [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,8,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,7,2] => [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,8,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1,1] => [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,8,2] => [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10,1] => [10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 10
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [8,1,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [6,2,1,1] => [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,8,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> [1,7,1,1] => [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9
[1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,7,2] => [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,6,2,1] => [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1,1] => [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,7,3] => [7,3,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 7
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)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 73%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => [1]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [2] => [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,8] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 10
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => [11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 11
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,2,1] => [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,8,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,7,2] => [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,8,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1,1] => [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,8,2] => [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10,1] => [10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 10
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [8,1,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [6,2,1,1] => [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,8,1] => [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [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
[1,0,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> [1,7,1,1] => [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9
[1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,7,2] => [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,6,2,1] => [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1,1] => [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 8
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000983
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 82%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 82%
Values
[1,0]
=> [1] => 1 => 1 => 1
[1,0,1,0]
=> [1,1] => 11 => 11 => 1
[1,1,0,0]
=> [2] => 10 => 01 => 2
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 111 => 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 011 => 2
[1,1,0,0,1,0]
=> [2,1] => 101 => 001 => 2
[1,1,0,1,0,0]
=> [2,1] => 101 => 001 => 2
[1,1,1,0,0,0]
=> [3] => 100 => 101 => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1111 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0111 => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 0011 => 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 0011 => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1011 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 0001 => 2
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 1001 => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 0001 => 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 0001 => 2
[1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 1001 => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1101 => 3
[1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1101 => 3
[1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1101 => 3
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0101 => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 11111 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 01111 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 00111 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 00111 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 10111 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 00011 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 10011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 00011 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 00011 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 10011 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 11011 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 11011 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 11011 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 01011 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 00001 => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 10001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 11001 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 11001 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 01001 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 00001 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 10001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 10111 => 00001 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 00001 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 10001 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 11001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 11001 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 11001 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 01001 => 3
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1101010101 => ? = 9
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? => ? = 9
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? => ? = 10
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 8
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => ? => ? = 8
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => 100000000001 => 110101010101 => ? = 11
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 8
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,2,1] => 1000000101 => ? => ? = 7
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => ? => ? = 8
[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,7,2] => 1100000010 => ? => ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => 11000000001 => ? => ? = 9
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 8
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => 1000000111 => 1111010101 => ? = 7
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,3,1] => 1000001001 => ? => ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => ? => ? = 8
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => ? => ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,9,1] => 11000000001 => ? => ? = 9
[1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,6,3] => 1100000100 => ? => ? = 6
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,8,2] => 11000000010 => ? => ? = 8
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10,1] => 110000000001 => ? => ? = 10
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 8
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => 1000000111 => 1111010101 => ? = 7
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [6,2,1,1] => 1000001011 => ? => ? = 6
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [5,4,1] => 1000010001 => ? => ? = 5
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,8,1] => 1100000001 => ? => ? = 8
[1,0,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => ? => ? = 7
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,9,1] => 11000000001 => ? => ? = 9
[1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,7,2] => 1100000010 => ? => ? = 7
[1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,6,2,1] => 1100000101 => ? => ? = 6
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1,1] => 11000000011 => ? => ? = 8
[1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,5,4] => 1100001000 => 0101101011 => ? = 5
[1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,7,3] => 11000000100 => ? => ? = 7
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 8
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1] => 1000000111 => 1111010101 => ? = 7
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,1,2,1] => 1000001101 => 1100010101 => ? = 6
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [4,5,1] => 1000100001 => ? => ? = 5
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,8,1] => 1100000001 => ? => ? = 8
[1,0,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => ? => ? = 7
[1,0,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
=> [1,6,2,1] => 1100000101 => ? => ? = 6
[1,0,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,6,1,2] => 1100000110 => ? => ? = 6
[1,0,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,5] => 1100010000 => ? => ? = 5
[1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,6,4] => 11000001000 => ? => ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 8
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0]
=> [7,2,1] => 1000000101 => ? => ? = 7
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1,1,1] => 1000001111 => ? => ? = 6
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [6,3,1] => 1000001001 => ? => ? = 6
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [3,6,1] => 1001000001 => ? => ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,8,1] => 1100000001 => ? => ? = 8
[1,0,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> [1,7,2] => 1100000010 => ? => ? = 7
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 $101...$.
Matching statistic: St001291
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 51%●distinct values known / distinct values provided: 45%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 51%●distinct values known / distinct values provided: 45%
Values
[1,0]
=> [1] => [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [2] => [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [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
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [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
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [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
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [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
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [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
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [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
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [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
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [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
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [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
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [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
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [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
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [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
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [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
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [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
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [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
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [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
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [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
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [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
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [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
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [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
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [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
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [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
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [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
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [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
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: St000013
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 91%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,2,2] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,2,1,2] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,2,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,2,1,2] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,2,1,2] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,2,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,2,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,2,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,2,3] => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,3,2] => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,3,2] => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,3,2] => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,4,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,4,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,2,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,2,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,2,3,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,2,3,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,2,3,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,2,1,1,2] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,2,1,1,2] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,2,1,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,2,1,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,2,1,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,2,1,3] => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,2,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,2,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,2,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,2,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,2,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,2,3,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St001058The breadth of the ordered tree. St000439The position of the first down step of a Dyck path. St000306The bounce count of a Dyck path. 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. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000025The number of initial rises of a Dyck path. St000686The finitistic dominant dimension 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:
St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000662The staircase size of the code of a permutation. St000651The maximal size of a rise in 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. St000141The maximum drop size of a permutation. St001933The largest multiplicity of a part in an integer partition. St000308The height of the tree associated to a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001330The hat guessing number of a graph. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000846The maximal number of elements covering an element of a poset. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000328The maximum number of child nodes in a tree. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000166The depth minus 1 of an ordered tree. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. St001530The depth of a Dyck path. St000094The depth of an ordered tree. 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. St000326The position of the first one in a binary word after appending a 1 at the end. St000028The number of stack-sorts needed to sort a permutation. St001589The nesting number of a perfect matching. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001624The breadth of a lattice.
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