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Your data matches 99 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
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Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ 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},{2}}
=> [1,1]
=> 1
[1,1,0,0]
=> {{1,2}}
=> [2]
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [2,1]
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1]
=> 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> 3
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1]
=> 3
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [4,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [3,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> 4
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> [1]
=> [1]
=> 1
[1,0,1,0]
=> {{1},{2}}
=> [1,1]
=> [2]
=> 1
[1,1,0,0]
=> {{1,2}}
=> [2]
=> [1,1]
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [2,1]
=> [2,1]
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1]
=> [2,1]
=> 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> [2,1]
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [2,2,1]
=> [3,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [4,1]
=> [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> [3,2]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000734
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> [1]
=> [[1]]
=> 1
[1,0,1,0]
=> {{1},{2}}
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,0,0]
=> {{1,2}}
=> [2]
=> [[1,2]]
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> [[1,2,3]]
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9,10},{11}}
=> [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},{2,3,4,5,6,7,8,9,10,11}}
=> [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> [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},{2},{3},{4},{5},{6},{7},{8},{9,11},{10}}
=> [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]
=> {{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,10,11},{5},{6},{7},{8},{9}}
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? = 6
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,11},{5,6},{7},{8},{9},{10}}
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [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]
=> {{1,2,3,4,5,6,7,8,10,11},{9}}
=> [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,11},{9,10}}
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> {{1,11},{2,3,4,5,6,7,8,9,10}}
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[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},{2,4,6,8,10,12},{3},{5},{7},{9},{11}}
=> [6,1,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 6
[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},{2,4,6,8,12},{3},{5},{7},{9,11},{10}}
=> [5,2,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 5
[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},{2,4,6,10,12},{3},{5},{7,9},{8},{11}}
=> [5,2,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 5
[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},{2,4,6,12},{3},{5},{7,9,11},{8},{10}}
=> [4,3,1,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11],[12]]
=> ? = 4
[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},{2,4,6,12},{3},{5},{7,11},{8,10},{9}}
=> [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},{2,4,8,10,12},{3},{5,7},{6},{9},{11}}
=> [5,2,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 5
[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},{2,4,8,12},{3},{5,7},{6},{9,11},{10}}
=> [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,0]
=> {{1},{2,4,10,12},{3},{5,7,9},{6},{8},{11}}
=> [4,3,1,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,0]
=> {{1},{2,4,10,12},{3},{5,9},{6,8},{7},{11}}
=> [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},{2,4,12},{3},{5,7,9,11},{6},{8},{10}}
=> [4,3,1,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0]
=> {{1},{2,4,12},{3},{5,7,11},{6},{8,10},{9}}
=> [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},{2,4,12},{3},{5,9,11},{6,8},{7},{10}}
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0]
=> {{1},{2,4,12},{3},{5,11},{6,8,10},{7},{9}}
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> {{1},{2,4,12},{3},{5,11},{6,10},{7,9},{8}}
=> [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> {{1},{2,6,8,10,12},{3,5},{4},{7},{9},{11}}
=> [5,2,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 5
[1,0,1,1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> {{1},{2,6,8,12},{3,5},{4},{7},{9,11},{10}}
=> [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[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},{2,6,10,12},{3,5},{4},{7,9},{8},{11}}
=> [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> {{1},{2,6,12},{3,5},{4},{7,9,11},{8},{10}}
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> {{1},{2,6,12},{3,5},{4},{7,11},{8,10},{9}}
=> [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,8,10,12},{3,5,7},{4},{6},{9},{11}}
=> [4,3,1,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> {{1},{2,8,12},{3,5,7},{4},{6},{9,11},{10}}
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,8,10,12},{3,7},{4,6},{5},{9},{11}}
=> [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> {{1},{2,8,12},{3,7},{4,6},{5},{9,11},{10}}
=> [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0,0]
=> {{1},{2,10,12},{3,5,7,9},{4},{6},{8},{11}}
=> [4,3,1,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0,0]
=> {{1},{2,10,12},{3,5,9},{4},{6,8},{7},{11}}
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> {{1},{2,10,12},{3,7,9},{4,6},{5},{8},{11}}
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0,0]
=> {{1},{2,10,12},{3,9},{4,6,8},{5},{7},{11}}
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> {{1},{2,10,12},{3,9},{4,8},{5,7},{6},{11}}
=> [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1},{2,12},{3,5,7,9,11},{4},{6},{8},{10}}
=> [5,2,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 5
[1,0,1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0]
=> {{1},{2,12},{3,5,7,11},{4},{6},{8,10},{9}}
=> [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,0]
=> {{1},{2,12},{3,5,9,11},{4},{6,8},{7},{10}}
=> [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0]
=> {{1},{2,12},{3,5,11},{4},{6,8,10},{7},{9}}
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0,0]
=> {{1},{2,12},{3,5,11},{4},{6,10},{7,9},{8}}
=> [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> {{1},{2,12},{3,7,9,11},{4,6},{5},{8},{10}}
=> [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> {{1},{2,12},{3,7,11},{4,6},{5},{8,10},{9}}
=> [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0,0]
=> {{1},{2,12},{3,9,11},{4,6,8},{5},{7},{10}}
=> [3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> {{1},{2,12},{3,9,11},{4,8},{5,7},{6},{10}}
=> [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 3
[1,0,1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0,0]
=> {{1},{2,12},{3,11},{4,6,8,10},{5},{7},{9}}
=> [4,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000381
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> [1] => 1
[1,0,1,0]
=> {{1},{2}}
=> [1,1] => 1
[1,1,0,0]
=> {{1,2}}
=> [2] => 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1] => 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [1,2] => 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1] => 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1] => 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3] => 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [1,1,2] => 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [1,2,1] => 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [1,2,1] => 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [1,3] => 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1] => 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1] => 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1] => 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1] => 3
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1] => 3
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2] => 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1] => 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [1,1,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [1,1,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [1,3,1] => 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [1,3,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [1,2,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [1,3,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [1,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,1,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1] => 4
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,10},{8,9}}
=> [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},{2,3,4,5,6,10},{7},{8},{9}}
=> [1,6,1,1,1] => ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6,7,8},{9,10}}
=> [8,2] => ? = 8
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8,9,10}}
=> [2,8] => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> [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},{2},{3},{4},{5},{6},{7},{8,10},{9}}
=> [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},{2},{3},{4},{5},{6},{7,10},{8},{9}}
=> [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},{2},{3},{4},{5},{6,10},{7},{8},{9}}
=> [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},{2},{3},{4},{5,10},{6},{7},{8},{9}}
=> [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},{2},{3},{4,10},{5},{6},{7},{8},{9}}
=> [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},{2},{3,10},{4},{5},{6},{7},{8},{9}}
=> [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},{2,10},{3},{4},{5},{6},{7},{8},{9}}
=> [1,2,1,1,1,1,1,1,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> [1,1,1,1,1,1,1,1,1,2] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9,11},{10}}
=> [1,1,1,1,1,1,1,1,2,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1] => ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1},{2,3,10},{4},{5},{6},{7},{8},{9}}
=> [1,3,1,1,1,1,1,1] => ? = 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1},{2,3,4,10},{5},{6},{7},{8},{9}}
=> [1,4,1,1,1,1,1] => ? = 4
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6,7},{8},{9},{10}}
=> [1,1,1,1,1,2,1,1,1] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8},{9},{10}}
=> [3,1,1,1,1,1,1,1] => ? = 3
[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8,9,10}}
=> [3,7] => ? = 7
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8},{9},{10}}
=> [4,1,1,1,1,1,1] => ? = 4
[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8,9,10}}
=> [4,6] => ? = 6
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8},{9},{10}}
=> [5,1,1,1,1,1] => ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8},{9},{10}}
=> [6,1,1,1,1] => ? = 6
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4,5,6},{7,8,9,10}}
=> [6,4] => ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5,6,7},{8},{9},{10}}
=> [7,1,1,1] => ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> {{1,2,3,4,5,6,7},{8,9},{10}}
=> [7,2,1] => ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5,6,7},{8,9,10}}
=> [7,3] => ? = 7
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> [10] => ? = 10
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,4},{2},{3},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,6},{2},{3},{4},{5},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,7},{2},{3},{4},{5},{6},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> {{1,8},{2},{3},{4},{5},{6},{7},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 2
[1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,9,10},{5},{6},{7},{8}}
=> [6,1,1,1,1] => ? = 6
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,10},{5,6},{7},{8},{9}}
=> [5,2,1,1,1] => ? = 5
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,10,11},{5},{6},{7},{8},{9}}
=> [6,1,1,1,1,1] => ? = 6
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,3,9,10},{4},{5},{6},{7},{8}}
=> [5,1,1,1,1,1] => ? = 5
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,10},{4,5},{6},{7},{8},{9}}
=> [4,2,1,1,1,1] => ? = 4
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,11},{5,6},{7},{8},{9},{10}}
=> [5,2,1,1,1,1] => ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,9,10},{2},{3},{4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1,1,1] => ? = 3
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,10},{8,9}}
=> [8,2] => ? = 8
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,10},{7,8},{9}}
=> [7,2,1] => ? = 7
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,11},{9,10}}
=> [9,2] => ? = 9
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7},{8,10},{9}}
=> [2,1,1,1,1,1,2,1] => ? = 2
Description
The largest part of an integer composition.
Matching statistic: St000676
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 81%●distinct values known / distinct values provided: 80%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 81%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> {{1}}
=> [1]
=> [1,0]
=> 1
[1,0,1,0]
=> {{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> {{1,2}}
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [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},{2},{3},{4,5}}
=> [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},{2},{3,4},{5}}
=> [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},{2},{3,5},{4}}
=> [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},{2},{3,4,5}}
=> [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},{2,3},{4},{5}}
=> [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},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [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},{2,5},{3},{4}}
=> [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},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [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},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [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},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [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]
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [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]
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [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]
=> {{1,5},{2},{3},{4}}
=> [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]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> [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]
=> {{1,2,3,4,5,6,7,8},{9}}
=> [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},{2,3,4,5,6,7,8,9}}
=> [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]
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> [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]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> [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},{2,3,4,5,6,7,9},{8}}
=> [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},{2,3,4,5,6,7,8,9,10}}
=> [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]
=> {{1,2,3,4,5,6,7,8,9,10},{11}}
=> [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]
=> {{1,2,3,4,5,6,7,9},{8},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7,8},{6},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8,9},{7}}
=> [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},{2,3,4,5,6,7,8,10},{9}}
=> [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},{2,3,4,5,6,7,8,9,10,11}}
=> [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
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,9},{7},{8}}
=> [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},{2,3,4,5,6,7,10},{8,9}}
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,10},{8},{9}}
=> [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
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7,8},{2},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,9},{6},{7},{8}}
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8,9},{3}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6,7},{8},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,10},{7},{8},{9}}
=> [6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8,9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7,8,9},{2},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6,7,8},{9,10}}
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8,9,10}}
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8,9,10},{3}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> [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
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> [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
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7,8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,7,8,9},{2},{3},{4},{5},{6}}
=> [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
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6,7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6,8},{7}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,6,9},{2},{3},{4},{5},{7},{8}}
=> [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
[1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> {{1,6,8,9},{2},{3},{4},{5},{7}}
=> [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
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> {{1,6,7,9},{2},{3},{4},{5},{8}}
=> [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
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> {{1,5,9},{2},{3},{4},{6},{7},{8}}
=> [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
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> {{1,5,6,9},{2},{3},{4},{7},{8}}
=> [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
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,4,5,9},{2},{3},{6},{7},{8}}
=> [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
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3,4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,3,9},{2},{4},{5},{6},{7},{8}}
=> [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
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,3,4,9},{2},{5},{6},{7},{8}}
=> [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
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,3,4,5,6,7,9},{2},{8}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,3,4,5,6,7,8,9},{2}}
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2,3},{4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2,4},{3},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> {{1,9},{2,8},{3},{4},{5},{6},{7}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,9},{3},{4},{5},{6},{7},{8}}
=> [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
[1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,4,8,9},{3},{5},{6},{7}}
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,4,5,6,7,8,9},{3}}
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
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: St000392
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => => ? = 1 - 1
[1,0,1,0]
=> [1,2] => 0 => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => 1 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 01 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => 01 => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => 11 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 011 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 101 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 011 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0011 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0101 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0011 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0011 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0110 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0101 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 0011 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0111 => 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,5,8,7,3] => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,2,5,6,4,8,7,3] => ? => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => ? => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,2,6,5,7,4,8,3] => ? => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,2,7,8,6,5,4] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,7,8,6,5,2] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,5,4,2,8,7,6] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,4,6,2,8,7] => ? => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,3,5,6,4,2,8,7] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,5,8,7,4,2] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,3,7,6,5,8,4,2] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,7,6,8,5,4,2] => ? => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ? => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,4,3,2,7,8,6,5] => ? => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,3,7,6,5,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,4,5,3,6,2,8,7] => ? => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,4,5,3,7,6,2,8] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,4,5,3,7,6,8,2] => ? => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,6,8,7,3,2] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,4,5,7,6,3,2,8] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,4,6,5,3,7,2,8] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,5,3,7,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,4,7,6,5,3,8,2] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,4,7,6,8,5,3,2] => ? => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,5,4,3,7,6,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,5,4,3,7,8,6,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,4,6,3,8,7,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,5,4,7,6,3,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,5,4,7,6,8,3,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,5,6,4,3,7,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,5,6,4,8,7,3,2] => ? => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,5,6,8,7,4,3,2] => ? => ? = 5 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,5,7,6,4,3,8,2] => ? => ? = 4 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,5,7,6,4,8,3,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,5,7,8,6,4,3,2] => ? => ? = 5 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,6,5,4,7,3,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,5,7,4,8,3,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,6,7,5,4,8,3,2] => ? => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,3,5,8,7,6,4] => ? => ? = 4 - 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => ? => ? = 3 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,3,7,6,8,5,4] => ? => ? = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,6,5,4,7,8,3] => ? => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5,8,7,6] => ? => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,1,4,6,8,7,5] => ? => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,4,7,8,6] => ? => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,8,7,6] => ? => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,6,4,8,7] => ? => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,1,5,7,6,4,8] => ? => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,1,5,8,7,6,4] => ? => ? = 4 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001039
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 81%●distinct values known / distinct values provided: 80%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 81%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> {{1}}
=> [1]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> {{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> {{1,2}}
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [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},{2},{3},{4,5}}
=> [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},{2},{3,4},{5}}
=> [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},{2},{3,5},{4}}
=> [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},{2},{3,4,5}}
=> [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},{2,3},{4},{5}}
=> [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},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [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},{2,5},{3},{4}}
=> [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},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [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},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [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},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [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]
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [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]
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [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]
=> {{1,5},{2},{3},{4}}
=> [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]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> [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},{2},{3},{4},{5},{6},{7},{8}}
=> [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]
=> {{1,2,3,4,5,6,7,8},{9}}
=> [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},{2,3,4,5,6,7,8,9}}
=> [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]
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> [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]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> [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},{2,3,4,5,6,7,9},{8}}
=> [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},{2,3,4,5,6,7,8,9,10}}
=> [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]
=> {{1,2,3,4,5,6,7,8,9,10},{11}}
=> [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]
=> {{1,2,3,4,5,6,7,9},{8},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7,8},{6},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8,9},{7}}
=> [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},{2,3,4,5,6,7,8,10},{9}}
=> [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},{2,3,4,5,6,7,8,9,10,11}}
=> [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
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,9},{7},{8}}
=> [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},{2,3,4,5,6,7,10},{8,9}}
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,10},{8},{9}}
=> [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
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7,8},{2},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,9},{6},{7},{8}}
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8,9},{3}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6,7},{8},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,10},{7},{8},{9}}
=> [6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8,9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7,8,9},{2},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6,7,8},{9,10}}
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8,9,10}}
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8,9,10},{3}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> [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
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> [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
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7,8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,7,8,9},{2},{3},{4},{5},{6}}
=> [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
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6,7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6,8},{7}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,6,9},{2},{3},{4},{5},{7},{8}}
=> [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
[1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> {{1,6,8,9},{2},{3},{4},{5},{7}}
=> [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
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> {{1,6,7,9},{2},{3},{4},{5},{8}}
=> [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
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> {{1,5,9},{2},{3},{4},{6},{7},{8}}
=> [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
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> {{1,5,6,9},{2},{3},{4},{7},{8}}
=> [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
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,4,5,9},{2},{3},{6},{7},{8}}
=> [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
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3,4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,3,9},{2},{4},{5},{6},{7},{8}}
=> [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
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,3,4,9},{2},{5},{6},{7},{8}}
=> [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
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,3,4,5,6,7,9},{2},{8}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,3,4,5,6,7,8,9},{2}}
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2,3},{4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2,4},{3},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> {{1,9},{2,8},{3},{4},{5},{6},{7}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,9},{3},{4},{5},{6},{7},{8}}
=> [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
[1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,4,8,9},{3},{5},{6},{7}}
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000013
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> {{1},{2}}
=> [1,1] => [1,0,1,0]
=> 1
[1,1,0,0]
=> {{1,2}}
=> [2] => [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [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},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4,6,7,8},{5}}
=> [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6},{7}}
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,7},{6}}
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6,7}}
=> [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,4},{5},{6,8},{7}}
=> [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> {{1},{2},{3,5},{4},{6,7,8}}
=> [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> {{1},{2},{3,7,8},{4},{5,6}}
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> {{1},{2},{3,5,7,8},{4},{6}}
=> [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> {{1},{2},{3,8},{4},{5,6,7}}
=> [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> {{1},{2},{3,5,8},{4},{6,7}}
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,5},{6},{7}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,6},{5},{7}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> {{1},{2},{3,7,8},{4,6},{5}}
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> {{1},{2},{3,8},{4,7},{5},{6}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> {{1},{2},{3,8},{4,6,7},{5}}
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,5,6},{7}}
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> {{1},{2},{3,4,5,8},{6},{7}}
=> [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2,3},{4},{5},{6,8},{7}}
=> [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> {{1},{2,3},{4,6,7},{5},{8}}
=> [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> {{1},{2,3},{4,6,8},{5},{7}}
=> [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> {{1},{2,3},{4,5,8},{6},{7}}
=> [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3},{5},{6,8},{7}}
=> [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2,4},{3},{5},{6,7,8}}
=> [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> {{1},{2,4},{3},{5,6},{7,8}}
=> [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4},{3},{5,7,8},{6}}
=> [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> {{1},{2,4},{3},{5,8},{6,7}}
=> [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> {{1},{2,5},{3},{4},{6,8},{7}}
=> [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> {{1},{2,8},{3},{4},{5,6,7}}
=> [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> {{1},{2,5,6,8},{3},{4},{7}}
=> [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> {{1},{2,4,5},{3},{6,7},{8}}
=> [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> {{1},{2,4,5},{3},{6,8},{7}}
=> [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> {{1},{2,6},{3},{4,5},{7,8}}
=> [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> {{1},{2,6,8},{3},{4,5},{7}}
=> [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> {{1},{2,4,6},{3},{5},{7,8}}
=> [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> {{1},{2,4,6,7},{3},{5},{8}}
=> [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> {{1},{2,8},{3},{4,6,7},{5}}
=> [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> {{1},{2,4,8},{3},{5},{6,7}}
=> [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,4,6,8},{3},{5},{7}}
=> [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> {{1},{2,4,7},{3},{5,6},{8}}
=> [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> {{1},{2,8},{3},{4,7},{5,6}}
=> [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> {{1},{2,4,8},{3},{5,6},{7}}
=> [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> {{1},{2,4,8},{3},{5,7},{6}}
=> [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,4},{5},{6},{7}}
=> [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> {{1},{2,6,7},{3,4},{5},{8}}
=> [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> {{1},{2,6,8},{3,4},{5},{7}}
=> [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> {{1},{2,6},{3,5},{4},{7,8}}
=> [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> {{1},{2,7},{3,5},{4},{6},{8}}
=> [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,5},{4},{6},{7}}
=> [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> {{1},{2,6,7},{3,5},{4},{8}}
=> [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> {{1},{2,8},{3,5},{4},{6,7}}
=> [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St001291
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> {{1}}
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> {{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0]
=> {{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [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},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [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},{2},{3},{4},{5}}
=> [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},{2},{3},{4,5}}
=> [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},{2},{3,4},{5}}
=> [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},{2},{3,5},{4}}
=> [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},{2},{3,4,5}}
=> [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},{2,3},{4},{5}}
=> [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},{2,3},{4,5}}
=> [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},{2,4},{3},{5}}
=> [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},{2,5},{3},{4}}
=> [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},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [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},{2,5},{3,4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [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},{2,3,4,5}}
=> [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]
=> {{1,2},{3},{4},{5}}
=> [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]
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [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]
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [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]
=> {{1,5},{2},{3},{4}}
=> [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]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6}}
=> [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]
=> {{1,2,3,4,5,6}}
=> [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},{2},{3},{4},{5},{6},{7}}
=> [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},{2},{3},{4},{5},{6,7}}
=> [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},{2},{3},{4},{5,6},{7}}
=> [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},{2},{3},{4},{5,7},{6}}
=> [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},{2},{3},{4,5},{6},{7}}
=> [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},{2},{3},{4,6},{5},{7}}
=> [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},{2},{3},{4,7},{5},{6}}
=> [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},{2},{3,4},{5},{6},{7}}
=> [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},{2},{3,5},{4},{6},{7}}
=> [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},{2},{3,6},{4},{5},{7}}
=> [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},{2},{3,7},{4},{5},{6}}
=> [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},{2,3},{4},{5},{6},{7}}
=> [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},{2,4},{3},{5},{6},{7}}
=> [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},{2,5},{3},{4},{6},{7}}
=> [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},{2,6},{3},{4},{5},{7}}
=> [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},{2,7},{3},{4},{5},{6}}
=> [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},{2,3,4,5,6,7}}
=> [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]
=> {{1,2},{3},{4},{5},{6},{7}}
=> [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]
=> {{1,3},{2},{4},{5},{6},{7}}
=> [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]
=> {{1,4},{2},{3},{5},{6},{7}}
=> [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]
=> {{1,5},{2},{3},{4},{6},{7}}
=> [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]
=> {{1,6},{2},{3},{4},{5},{7}}
=> [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]
=> {{1,7},{2},{3},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,3,4,5,6,7},{2}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,4,5,6,7},{3}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,6,7},{4}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,6,7},{5}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6},{7}}
=> [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]
=> {{1,2,3,4,5,7},{6}}
=> [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]
=> {{1,2,3,4,5,6,7}}
=> [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},{2},{3},{4},{5},{6},{7},{8}}
=> [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},{2},{3},{4},{5},{6},{7,8}}
=> [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},{2},{3},{4},{5},{6,8},{7}}
=> [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},{2},{3},{4},{5},{6,7,8}}
=> [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},{2},{3},{4},{5,6},{7},{8}}
=> [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},{2},{3},{4},{5,6},{7,8}}
=> [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},{2},{3},{4},{5,7},{6},{8}}
=> [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},{2},{3},{4},{5,8},{6},{7}}
=> [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},{2},{3},{4},{5,7,8},{6}}
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> [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},{2},{3},{4},{5,8},{6,7}}
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3},{4},{5,6,8},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5},{6},{7,8}}
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> [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
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6},{7}}
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,7},{6}}
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,4},{5},{6,8},{7}}
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000521
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 70%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [.,.]
=> [[]]
=> 2 = 1 + 1
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [[[]],[]]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [[],[[]],[],[]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [[[]],[],[],[]]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [[],[[]],[],[]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,8,2,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,2,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,8,2,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,2,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,2,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,8,2,3,1] => ?
=> ?
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,8,2,3,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,2,3,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,2,3,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,8,2,3,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,2,4,1] => ?
=> ?
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,2,4,1] => ?
=> ?
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,8,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,2,7,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,2,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,2,3,4,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,2,3,5,1] => ?
=> ?
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,2,3,7,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,2,3,7,1] => ?
=> ?
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,2,3,8,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [7,8,5,3,2,4,6,1] => ?
=> ?
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,2,4,7,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,2,4,8,1] => ?
=> ?
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,2,4,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,2,7,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,2,5,7,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,4,2,5,8,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,2,6,8,1] => ?
=> ?
=> ? = 2 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,2,6,7,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,8,1] => ?
=> ?
=> ? = 5 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [7,3,4,5,2,6,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,2,7,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [6,5,7,2,3,4,8,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [6,7,4,2,3,5,8,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,2,3,7,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [5,6,3,2,4,7,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [7,3,4,2,5,6,8,1] => ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,5,7,8,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,3,5,6,8,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [6,4,2,3,5,7,8,1] => ?
=> ?
=> ? = 3 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,3,2,4,5,7,8,1] => ?
=> ?
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,3,1,2] => ?
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,3,1,2] => ?
=> ?
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,3,8,1,2] => ?
=> ?
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,8,1,2] => ?
=> ?
=> ? = 4 + 1
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,1,2] => ?
=> ?
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,1,2] => ?
=> ?
=> ? = 2 + 1
Description
The number of distinct subtrees of an ordered tree.
A subtree is specified by a node of the tree. Thus, the tree consisting of a single path has as many subtrees as nodes, whereas the tree of height two, having all leaves attached to the root, has only two distinct subtrees. Because we consider ordered trees, the tree $[[[[]], []], [[], [[]]]]$ on nine nodes has five distinct subtrees.
The following 89 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. St000097The order of the largest clique of the graph. St000684The global dimension of the LNakayama algebra associated to 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. St001058The breadth of the ordered tree. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St001644The dimension of a graph. St000686The finitistic dominant dimension of a Dyck path. St001494The Alon-Tarsi number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000171The degree of the graph. St000454The largest eigenvalue of a graph if it is integral. St001826The maximal number of leaves on a vertex of a graph. St000730The maximal arc length of a set partition. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St001118The acyclic chromatic index of a graph. St001120The length of a longest path in a graph. St001963The tree-depth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000662The staircase size of the code of a permutation. St000527The width of the poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. 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. St000308The height of the tree associated to a permutation. St000651The maximal size of a rise in a permutation. St000028The number of stack-sorts needed to sort a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001330The hat guessing number of a graph. St000470The number of runs in a permutation. St000846The maximal number of elements covering an element of a poset. St000542The number of left-to-right-minima of a permutation. St001652The length of a longest interval of consecutive numbers. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000328The maximum number of child nodes in a tree. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St000325The width of the tree associated to a permutation. St000822The Hadwiger number of the graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St000021The number of descents of a permutation. St000094The depth of an ordered tree. St000326The position of the first one in a binary word after appending a 1 at the end. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001117The game chromatic index of a graph. 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. St000982The length of the longest constant subword. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000983The length of the longest alternating subword. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000225Difference between largest and smallest parts in a partition. St000808The number of up steps of the associated bargraph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001624The breadth of a lattice.
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