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Your data matches 186 different statistics following compositions of up to 3 maps.
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Matching statistic: St000667
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,4] => [4,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,3] => [3,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2] => [3,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,1] => [4,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3] => [3,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,2] => [2,2,2]
=> [2,2]
=> [2]
=> 2
[2,3,1] => [3,2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,2] => [3,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1] => [3,2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1] => [4,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,3] => [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,2,2] => [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,3,1] => [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,4] => [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000733
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 75%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 75%
Values
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,1,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[1,2,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[2,1,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[2,2,1,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[2,2,2] => [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
[2,3,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[3,1,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[3,1,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[3,2,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[4,1,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> 1
[1,1,1,1,3] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> 1
[1,1,1,2,2] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> 1
[1,1,1,3,1] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> 1
[1,1,1,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> 1
[1,8,1,1] => [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1
[1,7,2,1] => [7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> [[1,8,10,11],[2,9],[3],[4],[5],[6],[7]]
=> ? = 1
[1,6,3,1] => [6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> [[1,7,10,11],[2,8],[3,9],[4],[5],[6]]
=> ? = 1
[1,5,4,1] => [5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [[1,6,10,11],[2,7],[3,8],[4,9],[5]]
=> ? = 1
[1,4,5,1] => [5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [[1,6,10,11],[2,7],[3,8],[4,9],[5]]
=> ? = 1
[1,3,6,1] => [6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> [[1,7,10,11],[2,8],[3,9],[4],[5],[6]]
=> ? = 1
[1,2,7,1] => [7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> [[1,8,10,11],[2,9],[3],[4],[5],[6],[7]]
=> ? = 1
[1,1,8,1] => [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1
[1,1,1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,1,1,1,1,3,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,1,1,2,2,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> [[1,3,5,7,9,10,11,12],[2,4,6,8]]
=> ? = 1
[1,1,1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,1,1,2,1,2,3] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,1,1,3,3,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,1,1,3,2,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,1,1,3,1,1,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,1,1,4,4] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,2,2,1,1,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> [[1,3,5,7,9,10,11,12],[2,4,6,8]]
=> ? = 1
[1,1,2,2,2,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> [[1,3,5,7,9,10,11,12],[2,4,6,8]]
=> ? = 1
[1,1,2,2,2,1,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> [[1,3,5,7,9,10,11,12],[2,4,6,8]]
=> ? = 1
[1,1,2,2,3,3] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [[1,4,7,9,11,12],[2,5,8,10],[3,6]]
=> ? = 1
[1,1,2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,2,1,1,2,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> [[1,3,5,7,9,10,11,12],[2,4,6,8]]
=> ? = 1
[1,1,2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
[1,1,2,1,1,1,2,3] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,2,1,2,3,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,2,1,2,2,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> [[1,3,5,7,9,10,11,12],[2,4,6,8]]
=> ? = 1
[1,1,2,1,2,1,1,3] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,2,1,3,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,3,3,1,1,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,3,3,2,2] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [[1,4,7,9,11,12],[2,5,8,10],[3,6]]
=> ? = 1
[1,1,3,2,1,2,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,3,2,1,1,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,3,2,2,3] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [[1,4,7,9,11,12],[2,5,8,10],[3,6]]
=> ? = 1
[1,1,3,1,1,3,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,3,1,1,2,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,3,1,1,1,1,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,3,1,2,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,4,4,1,1] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,4,3,1,2] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,4,2,1,3] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,4,1,1,4] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ?
=> ? = 1
[1,1,5,5] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ?
=> ? = 1
[2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1,3,5,6,7,8,9,10,11,12],[2,4]]
=> ? = 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 75%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 75%
Values
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[1,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[1,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,1,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,2,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[1,3,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[2,1,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[1,1,4] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[1,2,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[1,3,2] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[1,4,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[2,1,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[2,2,1,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[2,2,2] => [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
[2,3,1] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[3,1,1,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[3,1,2] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[3,2,1] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[4,1,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> 1
[1,1,1,1,3] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> 1
[1,1,1,2,2] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7]]
=> 1
[1,1,1,3,1] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> 1
[1,1,1,4] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7]]
=> 1
[1,8,1,1] => [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ?
=> ? = 1
[1,7,2,1] => [7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ?
=> ? = 1
[1,6,3,1] => [6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ?
=> ? = 1
[1,5,4,1] => [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
=> ? = 1
[1,4,5,1] => [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
=> ? = 1
[1,3,6,1] => [6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ?
=> ? = 1
[1,2,7,1] => [7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ?
=> ? = 1
[1,1,8,1] => [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ?
=> ? = 1
[1,1,1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,1,1,1,1,3,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 1
[1,1,1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,1,1,2,2,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ?
=> ? = 1
[1,1,1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,1,1,2,1,2,3] => [3,2,2,1,1,1,1,1]
=> [[1,7,12],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ?
=> ? = 1
[1,1,1,1,3,3,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 1
[1,1,1,1,3,2,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,7,12],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ?
=> ? = 1
[1,1,1,1,3,1,1,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 1
[1,1,1,1,4,4] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 1
[1,1,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,2,2,1,1,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ?
=> ? = 1
[1,1,2,2,2,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ?
=> ? = 1
[1,1,2,2,2,1,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ?
=> ? = 1
[1,1,2,2,3,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[1,1,2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,2,1,1,2,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ?
=> ? = 1
[1,1,2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
[1,1,2,1,1,1,2,3] => [3,2,2,1,1,1,1,1]
=> [[1,7,12],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ?
=> ? = 1
[1,1,2,1,2,3,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,7,12],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ?
=> ? = 1
[1,1,2,1,2,2,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ?
=> ? = 1
[1,1,2,1,2,1,1,3] => [3,2,2,1,1,1,1,1]
=> [[1,7,12],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ?
=> ? = 1
[1,1,2,1,3,4] => [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
[1,1,3,3,1,1,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 1
[1,1,3,3,2,2] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[1,1,3,2,1,2,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,7,12],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ?
=> ? = 1
[1,1,3,2,1,1,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,7,12],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ?
=> ? = 1
[1,1,3,2,2,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[1,1,3,1,1,3,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 1
[1,1,3,1,1,2,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,7,12],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ?
=> ? = 1
[1,1,3,1,1,1,1,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 1
[1,1,3,1,2,4] => [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
[1,1,4,4,1,1] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 1
[1,1,4,3,1,2] => [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
[1,1,4,2,1,3] => [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
[1,1,4,1,1,4] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 1
[1,1,5,5] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 1
[2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000996
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 75%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 75%
Values
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[1,1,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[1,1,2,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[1,1,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[1,1,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[1,2,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[1,2,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[1,2,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[2,1,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[2,2,1,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[2,2,2] => [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[2,3,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[3,1,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[3,1,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[3,2,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[4,1,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => 1
[1,1,1,1,3] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => 1
[1,1,1,2,2] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => 1
[1,1,1,3,1] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => 1
[1,1,1,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => 1
[1,8,1,1] => [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 1
[1,7,2,1] => [7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? => ? = 1
[1,6,3,1] => [6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? => ? = 1
[1,5,4,1] => [5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [11,10,6,7,8,9,1,2,3,4,5] => ? = 1
[1,4,5,1] => [5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [11,10,6,7,8,9,1,2,3,4,5] => ? = 1
[1,3,6,1] => [6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? => ? = 1
[1,2,7,1] => [7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? => ? = 1
[1,1,8,1] => [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 1
[1,1,1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,1,1,3,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,2,2,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,2,1,2,3] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,3,3,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,3,2,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,3,1,1,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,1,1,4,4] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,2,1,1,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,2,2,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,2,2,1,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,2,3,3] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [12,11,9,10,7,8,4,5,6,1,2,3] => ? = 1
[1,1,2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,1,1,2,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,1,1,1,2,3] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,1,2,3,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,1,2,2,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,1,2,1,1,3] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,2,1,3,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? => ? = 1
[1,1,3,3,1,1,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,3,3,2,2] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [12,11,9,10,7,8,4,5,6,1,2,3] => ? = 1
[1,1,3,2,1,2,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,3,2,1,1,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,3,2,2,3] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [12,11,9,10,7,8,4,5,6,1,2,3] => ? = 1
[1,1,3,1,1,3,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,3,1,1,2,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,3,1,1,1,1,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,3,1,2,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? => ? = 1
[1,1,4,4,1,1] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,4,3,1,2] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? => ? = 1
[1,1,4,2,1,3] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? => ? = 1
[1,1,4,1,1,4] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 1
[1,1,5,5] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 1
[2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000657
(load all 91 compositions to match this statistic)
(load all 91 compositions to match this statistic)
St000657: Integer compositions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 75%
Values
[1,1,1] => 1
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 1
[2,1,1] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 1
[2,1,1,1] => 1
[2,1,2] => 1
[2,2,1] => 1
[3,1,1] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 1
[1,2,1,2] => 1
[1,2,2,1] => 1
[1,2,3] => 1
[1,3,1,1] => 1
[1,3,2] => 1
[1,4,1] => 1
[2,1,1,1,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 1
[2,1,3] => 1
[2,2,1,1] => 1
[2,2,2] => 2
[2,3,1] => 1
[3,1,1,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[4,1,1] => 1
[1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => 1
[1,1,1,1,2,1] => 1
[1,1,1,1,3] => 1
[1,1,1,2,1,1] => 1
[1,1,1,2,2] => 1
[1,1,1,3,1] => 1
[1,1,1,4] => 1
[1,1,1,1,1,1,1,1,1,1] => ? = 1
[1,1,1,1,1,1,1,1,2] => ? = 1
[1,1,1,1,1,1,1,2,1] => ? = 1
[1,1,1,1,1,1,1,3] => ? = 1
[1,1,1,1,1,1,2,1,1] => ? = 1
[1,1,1,1,1,1,2,2] => ? = 1
[1,1,1,1,1,1,3,1] => ? = 1
[1,1,1,1,1,1,4] => ? = 1
[1,1,1,1,1,2,1,1,1] => ? = 1
[1,1,1,1,1,2,1,2] => ? = 1
[1,1,1,1,1,2,2,1] => ? = 1
[1,1,1,1,1,2,3] => ? = 1
[1,1,1,1,1,3,1,1] => ? = 1
[1,1,1,1,1,3,2] => ? = 1
[1,1,1,1,1,4,1] => ? = 1
[1,1,1,1,1,5] => ? = 1
[1,1,1,1,2,1,1,1,1] => ? = 1
[1,1,1,1,2,1,1,2] => ? = 1
[1,1,1,1,2,1,2,1] => ? = 1
[1,1,1,1,2,1,3] => ? = 1
[1,1,1,1,2,2,1,1] => ? = 1
[1,1,1,1,2,2,2] => ? = 1
[1,1,1,1,2,3,1] => ? = 1
[1,1,1,1,2,4] => ? = 1
[1,1,1,1,3,1,1,1] => ? = 1
[1,1,1,1,3,1,2] => ? = 1
[1,1,1,1,3,2,1] => ? = 1
[1,1,1,1,3,3] => ? = 1
[1,1,1,1,4,1,1] => ? = 1
[1,1,1,1,4,2] => ? = 1
[1,1,1,1,5,1] => ? = 1
[1,1,1,1,6] => ? = 1
[1,1,1,2,1,1,1,1,1] => ? = 1
[1,1,1,2,1,1,1,2] => ? = 1
[1,1,1,2,1,1,2,1] => ? = 1
[1,1,1,2,1,1,3] => ? = 1
[1,1,1,2,1,2,1,1] => ? = 1
[1,1,1,2,1,2,2] => ? = 1
[1,1,1,2,1,3,1] => ? = 1
[1,1,1,2,1,4] => ? = 1
[1,1,1,2,2,1,1,1] => ? = 1
[1,1,1,2,2,1,2] => ? = 1
[1,1,1,2,2,2,1] => ? = 1
[1,1,1,2,2,3] => ? = 1
[1,1,1,2,3,1,1] => ? = 1
[1,1,1,2,3,2] => ? = 1
[1,1,1,2,4,1] => ? = 1
[1,1,1,2,5] => ? = 1
[1,1,1,3,1,1,1,1] => ? = 1
[1,1,1,3,1,1,2] => ? = 1
Description
The smallest part of an integer composition.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Values
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[1,1,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[1,1,2,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[1,1,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[1,1,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[1,2,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[1,2,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[1,2,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[2,1,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[2,2,1,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[2,2,2] => [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[2,3,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[3,1,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[3,1,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[3,2,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[4,1,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 1
[1,1,1,1,3] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 1
[1,1,1,2,2] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 1
[1,1,1,3,1] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 1
[1,1,1,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 1
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1
[1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1
[1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1
[1,1,1,1,4] => [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1
[1,1,1,2,3] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1
[1,1,1,3,2] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,1,1,4,1] => [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1
[1,1,2,1,3] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,1,2,3,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,1,2,4] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1
[1,1,3,1,2] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,1,3,2,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,1,4,1,1] => [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[1,1,4,2] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1
[1,2,1,1,3] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,2,1,3,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,2,1,4] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[1,2,3,1,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,2,4,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[1,3,1,1,1,1] => [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1
[1,3,1,1,2] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,3,1,2,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,3,2,1,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[1,4,1,1,1] => [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[1,4,1,2] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[1,4,2,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[2,1,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1
[2,1,1,1,3] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[2,1,1,3,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[2,1,1,4] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[2,1,3,1,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[2,1,4,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[2,2,4] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[2,3,1,1,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[2,4,1,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[2,4,2] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[3,1,1,1,1,1] => [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1
[3,1,1,1,2] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[3,1,1,2,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[3,1,2,1,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[3,2,1,1,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[4,1,1,1,1] => [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[4,1,1,2] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000617
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
Description
The number of global maxima of a Dyck path.
Matching statistic: St000655
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
St000655: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,2,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,2,1,3] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,1,1,2,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,1,1,2,4] => [1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,3,3] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,4,1,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,4,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
Matching statistic: St001803
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,1,1] => [1,1,1]
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
[1,1,1,1] => [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
[1,1,2] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[1,2,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[2,1,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[1,1,3] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[1,2,2] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
[1,3,1] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[2,1,2] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
[2,2,1] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
[3,1,1] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 0 = 1 - 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 0 = 1 - 1
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0 = 1 - 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 0 = 1 - 1
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0 = 1 - 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0 = 1 - 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 0 = 1 - 1
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
[1,2,3] => [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0 = 1 - 1
[1,3,2] => [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0 = 1 - 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 0 = 1 - 1
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
[2,1,3] => [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[2,2,1,1] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
[2,2,2] => [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 2 - 1
[2,3,1] => [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[3,1,1,1] => [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0 = 1 - 1
[3,1,2] => [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[3,2,1] => [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[4,1,1] => [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0 = 1 - 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 0 = 1 - 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 0 = 1 - 1
[1,1,1,1,3] => [3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 0 = 1 - 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 0 = 1 - 1
[1,1,1,2,2] => [2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 0 = 1 - 1
[1,1,1,3,1] => [3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 0 = 1 - 1
[1,1,1,4] => [4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 0 = 1 - 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> [9]
=> [[1,2,3,4,5,6,7,8,9]]
=> ? = 1 - 1
[1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> [8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 1 - 1
[1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1]
=> [8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 1 - 1
[1,1,1,1,1,1,3] => [3,1,1,1,1,1,1]
=> [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 1 - 1
[1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1]
=> [8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 1 - 1
[1,1,1,1,1,2,2] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,1,1,1,3,1] => [3,1,1,1,1,1,1]
=> [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 1 - 1
[1,1,1,1,1,4] => [4,1,1,1,1,1]
=> [6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 1 - 1
[1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1]
=> [8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 1 - 1
[1,1,1,1,2,1,2] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,1,1,2,3] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[1,1,1,1,3,1,1] => [3,1,1,1,1,1,1]
=> [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 1 - 1
[1,1,1,1,3,2] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[1,1,1,1,4,1] => [4,1,1,1,1,1]
=> [6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 1 - 1
[1,1,1,1,5] => [5,1,1,1,1]
=> [5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> ? = 1 - 1
[1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 1 - 1
[1,1,1,2,1,1,2] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,1,2,1,2,1] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,1,2,1,3] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[1,1,1,2,2,1,1] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,1,2,2,2] => [2,2,2,1,1,1]
=> [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> ? = 1 - 1
[1,1,1,2,3,1] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[1,1,1,2,4] => [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
[1,1,1,3,1,1,1] => [3,1,1,1,1,1,1]
=> [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 1 - 1
[1,1,1,3,1,2] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[1,1,1,3,2,1] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[1,1,1,3,3] => [3,3,1,1,1]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 - 1
[1,1,1,4,1,1] => [4,1,1,1,1,1]
=> [6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 1 - 1
[1,1,1,4,2] => [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
[1,1,1,5,1] => [5,1,1,1,1]
=> [5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> ? = 1 - 1
[1,1,1,6] => [6,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> ? = 1 - 1
[1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 1 - 1
[1,1,2,1,1,1,2] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,2,1,1,2,1] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,2,1,1,3] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[1,1,2,1,2,1,1] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,2,1,2,2] => [2,2,2,1,1,1]
=> [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> ? = 1 - 1
[1,1,2,1,3,1] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[1,1,2,1,4] => [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
[1,1,2,2,1,1,1] => [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[1,1,2,2,1,2] => [2,2,2,1,1,1]
=> [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> ? = 1 - 1
[1,1,2,2,2,1] => [2,2,2,1,1,1]
=> [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> ? = 1 - 1
[1,1,2,2,3] => [3,2,2,1,1]
=> [5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> ? = 1 - 1
[1,1,2,3,1,1] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[1,1,2,3,2] => [3,2,2,1,1]
=> [5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> ? = 1 - 1
[1,1,2,4,1] => [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
[1,1,2,5] => [5,2,1,1]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 1 - 1
[1,1,3,1,1,1,1] => [3,1,1,1,1,1,1]
=> [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 1 - 1
[1,1,3,1,1,2] => [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St000700
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Values
[1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[]]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]]]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[[]],[]]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[[[]]]]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[],[],[[]],[],[]]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[],[],[[]],[[]]]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[],[],[[[]]],[]]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[],[],[[[[]]]]]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[],[[]],[],[],[]]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[],[[]],[],[[]]]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[],[[]],[[]],[]]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[],[[]],[[[]]]]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[],[[[]]],[],[]]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[],[[[]]],[[]]]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[],[[[[]]]],[]]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[]],[],[],[],[]]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[]],[],[],[[]]]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[]],[],[[]],[]]
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[]],[],[[[]]]]
=> 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[]],[[]],[],[]]
=> 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[[]],[[]],[[]]]
=> 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[[]],[[[]]],[]]
=> 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[]]],[],[],[]]
=> 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[[]]],[],[[]]]
=> 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[[]]],[[]],[]]
=> 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[[]]]],[],[]]
=> 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[]]
=> 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[],[[]]]
=> 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[],[[]],[]]
=> 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[],[[[]]]]
=> 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[],[],[],[[]],[],[]]
=> 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[],[],[],[[]],[[]]]
=> 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[],[],[],[[[]]],[]]
=> 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[],[],[],[[[[]]]]]
=> 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[[[]]],[],[[[]]],[]]
=> ? = 1
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[[[]]],[],[[[[]]]]]
=> ? = 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[]]],[[]],[],[],[]]
=> ? = 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[]]],[[]],[],[[]]]
=> ? = 1
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[[]]],[[]],[[]],[]]
=> ? = 1
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[[[]]],[[]],[[[]]]]
=> ? = 2
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[[[]]],[],[]]
=> ? = 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[[]]],[[]]]
=> ? = 2
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[[]]],[[[[]]]],[]]
=> ? = 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[]]]],[],[],[],[]]
=> ? = 1
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [[[[[]]]],[],[],[[]]]
=> ? = 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [[[[[]]]],[],[[]],[]]
=> ? = 1
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[[]]]],[],[[[]]]]
=> ? = 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [[[[[]]]],[[]],[],[]]
=> ? = 1
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [[[[[]]]],[[]],[[]]]
=> ? = 2
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [[[[[]]]],[[[]]],[]]
=> ? = 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 1
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [[[[[[]]]]],[],[[]]]
=> ? = 1
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [[[[[[]]]]],[[]],[]]
=> ? = 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[[]]]]]],[],[]]
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1
[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[],[],[],[[]]]
=> ? = 1
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[],[],[],[[]],[]]
=> ? = 1
[1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[],[],[],[[[]]]]
=> ? = 1
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[],[],[],[],[],[[]],[],[]]
=> ? = 1
[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[],[],[],[],[],[[]],[[]]]
=> ? = 1
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[],[],[],[],[],[[[]]],[]]
=> ? = 1
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[],[],[],[],[],[[[[]]]]]
=> ? = 1
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[],[],[],[],[[]],[],[],[]]
=> ? = 1
[1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]],[],[[]]]
=> ? = 1
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[],[],[],[],[[]],[[]],[]]
=> ? = 1
[1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[],[],[],[],[[]],[[[]]]]
=> ? = 1
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[],[],[],[],[[[]]],[],[]]
=> ? = 1
[1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[],[],[],[],[[[]]],[[]]]
=> ? = 1
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[],[],[],[],[[[[]]]],[]]
=> ? = 1
[1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[],[],[],[[[[[]]]]]]
=> ? = 1
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[[]],[],[],[],[]]
=> ? = 1
[1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ?
=> ? = 1
[1,1,1,2,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[[]],[],[[]],[]]
=> ? = 1
[1,1,1,2,1,3] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ?
=> ? = 1
[1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[],[],[],[[]],[[]],[],[]]
=> ? = 1
[1,1,1,2,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[],[],[],[[]],[[]],[[]]]
=> ? = 1
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ?
=> ? = 1
[1,1,1,2,4] => [1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[],[],[],[[]],[[[[]]]]]
=> ? = 1
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[],[],[],[[[]]],[],[],[]]
=> ? = 1
[1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ?
=> ? = 1
[1,1,1,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [[],[],[],[[[]]],[[]],[]]
=> ? = 1
[1,1,1,3,3] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[],[],[],[[[]]],[[[]]]]
=> ? = 1
[1,1,1,4,1,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ?
=> ? = 1
[1,1,1,4,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ?
=> ? = 1
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
The following 176 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000654The first descent of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001075The minimal size of a block of a set partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000260The radius of a connected graph. St000273The domination number of a graph. St000544The cop number of a graph. St001316The domatic number of a graph. St001672The restrained domination number of a graph. St001829The common independence number of a graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St000908The length of the shortest maximal antichain in a poset. St001568The smallest positive integer that does not appear twice in the partition. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000487The length of the shortest cycle of a permutation. St000210Minimum over maximum difference of elements in cycles. St000906The length of the shortest maximal chain in a poset. St001890The maximum magnitude of the Möbius function of a poset. St000914The sum of the values of the Möbius function of a poset. St000666The number of right tethers of a permutation. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000699The toughness times the least common multiple of 1,. St000455The second largest eigenvalue of a graph if it is integral. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000463The number of admissible inversions of a permutation. St000731The number of double exceedences of a permutation. St000842The breadth of a permutation. St000884The number of isolated descents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000750The number of occurrences of the pattern 4213 in a permutation. St000649The number of 3-excedences of a permutation. St000929The constant term of the character polynomial of an integer partition. St001513The number of nested exceedences of a permutation. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001793The difference between the clique number and the chromatic number of a graph. St000090The variation of a composition. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000314The number of left-to-right-maxima of a permutation. St000536The pathwidth of a graph. St000570The Edelman-Greene number of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St000664The number of right ropes of a permutation. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001307The number of induced stars on four vertices in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St000264The girth of a graph, which is not a tree. St000054The first entry of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000218The number of occurrences of the pattern 213 in a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000546The number of global descents of a permutation. St000862The number of parts of the shifted shape of a permutation. St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000526The number of posets with combinatorially isomorphic order polytopes. St000648The number of 2-excedences of a permutation. St000717The number of ordinal summands of a poset. St001301The first Betti number of the order complex associated with the poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000237The number of small exceedances. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000534The number of 2-rises of a permutation. St000234The number of global ascents of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000754The Grundy value for the game of removing nestings in a perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000456The monochromatic index of a connected graph. St000374The number of exclusive right-to-left minima of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St001665The number of pure excedances of a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000441The number of successions of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000665The number of rafts of a permutation. St001381The fertility of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001728The number of invisible descents of a permutation. St001964The interval resolution global dimension of a poset. St001344The neighbouring number of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001552The number of inversions between excedances and fixed points of a permutation. St001715The number of non-records in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001846The number of elements which do not have a complement in the lattice. St001847The number of occurrences of the pattern 1432 in a permutation.
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