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Your data matches 72 different statistics following compositions of up to 3 maps.
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Matching statistic: St000381
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Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [2] => 2
[2] => [1,1] => 1
[1,1,1] => [3] => 3
[1,2] => [1,2] => 2
[2,1] => [2,1] => 2
[3] => [1,1,1] => 1
[1,1,1,1] => [4] => 4
[1,1,2] => [1,3] => 3
[1,2,1] => [2,2] => 2
[1,3] => [1,1,2] => 2
[2,1,1] => [3,1] => 3
[2,2] => [1,2,1] => 2
[3,1] => [2,1,1] => 2
[4] => [1,1,1,1] => 1
[1,1,1,1,1] => [5] => 5
[1,1,1,2] => [1,4] => 4
[1,1,2,1] => [2,3] => 3
[1,1,3] => [1,1,3] => 3
[1,2,1,1] => [3,2] => 3
[1,2,2] => [1,2,2] => 2
[1,3,1] => [2,1,2] => 2
[1,4] => [1,1,1,2] => 2
[2,1,1,1] => [4,1] => 4
[2,1,2] => [1,3,1] => 3
[2,2,1] => [2,2,1] => 2
[2,3] => [1,1,2,1] => 2
[3,1,1] => [3,1,1] => 3
[3,2] => [1,2,1,1] => 2
[4,1] => [2,1,1,1] => 2
[5] => [1,1,1,1,1] => 1
[1,1,1,1,1,1] => [6] => 6
[1,1,1,1,2] => [1,5] => 5
[1,1,1,2,1] => [2,4] => 4
[1,1,1,3] => [1,1,4] => 4
[1,1,2,1,1] => [3,3] => 3
[1,1,2,2] => [1,2,3] => 3
[1,1,3,1] => [2,1,3] => 3
[1,1,4] => [1,1,1,3] => 3
[1,2,1,1,1] => [4,2] => 4
[1,2,1,2] => [1,3,2] => 3
[1,2,2,1] => [2,2,2] => 2
[1,2,3] => [1,1,2,2] => 2
[1,3,1,1] => [3,1,2] => 3
[1,3,2] => [1,2,1,2] => 2
[1,4,1] => [2,1,1,2] => 2
[1,5] => [1,1,1,1,2] => 2
[2,1,1,1,1] => [5,1] => 5
[2,1,1,2] => [1,4,1] => 4
[2,1,2,1] => [2,3,1] => 3
Description
The largest part of an integer composition.
Matching statistic: St000147
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,1] => [2] => [2]
=> 2
[2] => [1,1] => [1,1]
=> 1
[1,1,1] => [3] => [3]
=> 3
[1,2] => [2,1] => [2,1]
=> 2
[2,1] => [1,2] => [2,1]
=> 2
[3] => [1,1,1] => [1,1,1]
=> 1
[1,1,1,1] => [4] => [4]
=> 4
[1,1,2] => [3,1] => [3,1]
=> 3
[1,2,1] => [2,2] => [2,2]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> 2
[2,1,1] => [1,3] => [3,1]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> 1
[1,1,1,1,1] => [5] => [5]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> 3
[1,1,2,1,2,2,3] => ? => ?
=> ? = 3
[1,8,3] => ? => ?
=> ? = 2
[1,1,2,2,2,1,3] => ? => ?
=> ? = 3
[9,2,1] => ? => ?
=> ? = 2
[1,10,1] => ? => ?
=> ? = 2
[3,1,1,1,1,1,1,2] => ? => ?
=> ? = 8
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 1
[1,1] => [2] => [2]
=> [1,1]
=> 2
[2] => [1,1] => [1,1]
=> [2]
=> 1
[1,1,1] => [3] => [3]
=> [1,1,1]
=> 3
[1,2] => [2,1] => [2,1]
=> [2,1]
=> 2
[2,1] => [1,2] => [2,1]
=> [2,1]
=> 2
[3] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,1,1,1] => [4] => [4]
=> [1,1,1,1]
=> 4
[1,1,2] => [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,2,1] => [2,2] => [2,2]
=> [2,2]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[2,1,1] => [1,3] => [3,1]
=> [2,1,1]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [1,1,1,1,1]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> [2,1,1,1]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> [2,2,1]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> [2,2,1]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> [3,2]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> [3,2]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> [2,1,1,1]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> [3,2]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,1,1,1,1,1]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> [2,1,1,1,1]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> [2,2,1,1]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [3,1,1,1]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> [2,2,2]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [3,2,1]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [3,2,1]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [4,1,1]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> [2,2,1,1]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [3,2,1]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [3,3]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [4,2]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [3,2,1]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> [2,1,1,1,1]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [3,2,1]
=> 3
[1,1,2,1,2,2,3] => ? => ?
=> ?
=> ? = 3
[1,8,3] => ? => ?
=> ?
=> ? = 2
[1,1,2,2,2,1,3] => ? => ?
=> ?
=> ? = 3
[9,2,1] => ? => ?
=> ?
=> ? = 2
[1,10,1] => ? => ?
=> ?
=> ? = 2
[3,1,1,1,1,1,1,2] => ? => ?
=> ?
=> ? = 8
Description
The length of the partition.
Matching statistic: St000734
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Values
[1] => [1] => [1]
=> [[1]]
=> 1
[1,1] => [2] => [2]
=> [[1,2]]
=> 2
[2] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[1,1,1] => [3] => [3]
=> [[1,2,3]]
=> 3
[1,2] => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
[2,1] => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[3] => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,1,1,1] => [4] => [4]
=> [[1,2,3,4]]
=> 4
[1,1,2] => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,2,1] => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,1] => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [[1,2,3,4,5]]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [[1,2,3,4,5,6]]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,2,2,4,3] => [2,2,2,1,1,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[1,2,4,4,1] => [2,2,1,1,2,1,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[1,2,4,2,3] => [2,2,1,1,2,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[1,3,4,3,1] => [2,1,2,1,1,2,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[1,3,2,3,3] => [2,1,2,2,1,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[1,4,4,2,1] => [2,1,1,2,1,1,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[1,4,2,4,1] => [2,1,1,2,2,1,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[1,4,2,2,3] => [2,1,1,2,2,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
[1,6,5] => [2,1,1,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[3,4,2,2,1] => [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]]
=> ? = 2
[3,6,3] => [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]]
=> ? = 2
[3,3,2,3,1] => [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]]
=> ? = 2
[3,2,4,2,1] => [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]]
=> ? = 2
[3,2,2,4,1] => [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]]
=> ? = 2
[3,2,2,2,3] => [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]]
=> ? = 2
[3,4,5] => [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]]
=> ? = 2
[5,6,1] => [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]]
=> ? = 2
[5,4,3] => [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]]
=> ? = 2
[5,2,5] => [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]]
=> ? = 2
[1,1,1,1,1,1,1,1,1,2] => [10,1] => [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[12] => [1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 1
[3,2,7] => [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]]
=> ? = 2
[7,2,3] => [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]]
=> ? = 2
[1,1,2,1,2,2,3] => ? => ?
=> ?
=> ? = 3
[1,8,3] => ? => ?
=> ?
=> ? = 2
[1,1,2,5,3] => [3,2,1,1,1,2,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
[3,1,2,3,3] => [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]]
=> ? = 3
[1,1,4,3,3] => [3,1,1,2,1,2,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
[1,1,1,2,1,3,3] => [4,3,1,2,1,1] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 4
[3,3,2,1,3] => [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]]
=> ? = 3
[1,5,2,1,3] => [2,1,1,1,2,3,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
[1,1,2,2,2,1,3] => ? => ?
=> ?
=> ? = 3
[1,3,1,2,1,1,3] => [2,1,3,4,1,1] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 4
[1,1,1,4,1,1,3] => [4,1,1,4,1,1] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 4
[9,2,1] => ? => ?
=> ?
=> ? = 2
[5,2,2,2,1] => [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]]
=> ? = 2
[1,1,2,3,2,2,1] => [3,2,1,2,2,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
[1,3,2,1,2,2,1] => [2,1,2,3,2,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
[1,10,1] => ? => ?
=> ?
=> ? = 2
[3,1,2,5,1] => [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]]
=> ? = 3
[1,1,4,5,1] => [3,1,1,2,1,1,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
[1,2,2,1,2,3,1] => [2,2,3,2,1,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
[1,1,2,2,2,3,1] => [3,2,2,2,1,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
[1,1,1,2,3,3,1] => [4,2,1,2,1,2] => [4,2,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 4
[3,1,1,2,1,3,1] => [1,1,4,3,1,2] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 4
[5,3,2,1,1] => [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]]
=> ? = 3
[1,2,2,3,2,1,1] => [2,2,2,1,2,3] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
[3,5,2,1,1] => [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]]
=> ? = 3
[1,3,2,2,2,1,1] => [2,1,2,2,2,3] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
[3,3,4,1,1] => [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]]
=> ? = 3
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000392
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 90%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 90%
Values
[1] => [1] => 1 => 0 => 0 = 1 - 1
[1,1] => [2] => 10 => 01 => 1 = 2 - 1
[2] => [1,1] => 11 => 00 => 0 = 1 - 1
[1,1,1] => [3] => 100 => 011 => 2 = 3 - 1
[1,2] => [1,2] => 110 => 001 => 1 = 2 - 1
[2,1] => [2,1] => 101 => 010 => 1 = 2 - 1
[3] => [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,1,1,1] => [4] => 1000 => 0111 => 3 = 4 - 1
[1,1,2] => [1,3] => 1100 => 0011 => 2 = 3 - 1
[1,2,1] => [2,2] => 1010 => 0101 => 1 = 2 - 1
[1,3] => [1,1,2] => 1110 => 0001 => 1 = 2 - 1
[2,1,1] => [3,1] => 1001 => 0110 => 2 = 3 - 1
[2,2] => [1,2,1] => 1101 => 0010 => 1 = 2 - 1
[3,1] => [2,1,1] => 1011 => 0100 => 1 = 2 - 1
[4] => [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
[1,1,1,1,1] => [5] => 10000 => 01111 => 4 = 5 - 1
[1,1,1,2] => [1,4] => 11000 => 00111 => 3 = 4 - 1
[1,1,2,1] => [2,3] => 10100 => 01011 => 2 = 3 - 1
[1,1,3] => [1,1,3] => 11100 => 00011 => 2 = 3 - 1
[1,2,1,1] => [3,2] => 10010 => 01101 => 2 = 3 - 1
[1,2,2] => [1,2,2] => 11010 => 00101 => 1 = 2 - 1
[1,3,1] => [2,1,2] => 10110 => 01001 => 1 = 2 - 1
[1,4] => [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
[2,1,1,1] => [4,1] => 10001 => 01110 => 3 = 4 - 1
[2,1,2] => [1,3,1] => 11001 => 00110 => 2 = 3 - 1
[2,2,1] => [2,2,1] => 10101 => 01010 => 1 = 2 - 1
[2,3] => [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
[3,1,1] => [3,1,1] => 10011 => 01100 => 2 = 3 - 1
[3,2] => [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
[4,1] => [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
[5] => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
[1,1,1,1,1,1] => [6] => 100000 => 011111 => 5 = 6 - 1
[1,1,1,1,2] => [1,5] => 110000 => 001111 => 4 = 5 - 1
[1,1,1,2,1] => [2,4] => 101000 => 010111 => 3 = 4 - 1
[1,1,1,3] => [1,1,4] => 111000 => 000111 => 3 = 4 - 1
[1,1,2,1,1] => [3,3] => 100100 => 011011 => 2 = 3 - 1
[1,1,2,2] => [1,2,3] => 110100 => 001011 => 2 = 3 - 1
[1,1,3,1] => [2,1,3] => 101100 => 010011 => 2 = 3 - 1
[1,1,4] => [1,1,1,3] => 111100 => 000011 => 2 = 3 - 1
[1,2,1,1,1] => [4,2] => 100010 => 011101 => 3 = 4 - 1
[1,2,1,2] => [1,3,2] => 110010 => 001101 => 2 = 3 - 1
[1,2,2,1] => [2,2,2] => 101010 => 010101 => 1 = 2 - 1
[1,2,3] => [1,1,2,2] => 111010 => 000101 => 1 = 2 - 1
[1,3,1,1] => [3,1,2] => 100110 => 011001 => 2 = 3 - 1
[1,3,2] => [1,2,1,2] => 110110 => 001001 => 1 = 2 - 1
[1,4,1] => [2,1,1,2] => 101110 => 010001 => 1 = 2 - 1
[1,5] => [1,1,1,1,2] => 111110 => 000001 => 1 = 2 - 1
[2,1,1,1,1] => [5,1] => 100001 => 011110 => 4 = 5 - 1
[2,1,1,2] => [1,4,1] => 110001 => 001110 => 3 = 4 - 1
[2,1,2,1] => [2,3,1] => 101001 => 010110 => 2 = 3 - 1
[1,1,1,1,1,1,1,1,2] => [1,9] => 1100000000 => 0011111111 => ? = 9 - 1
[1,1,1,1,2,1,1,1,1] => [5,5] => 1000010000 => 0111101111 => ? = 5 - 1
[1,1,1,2,1,3,1] => [2,1,3,4] => 1011001000 => 0100110111 => ? = 4 - 1
[1,1,1,2,3,1,1] => [3,1,2,4] => 1001101000 => 0110010111 => ? = 4 - 1
[1,1,1,4,1,1,1] => [4,1,1,4] => 1000111000 => 0111000111 => ? = 4 - 1
[1,1,2,1,2,2,1] => [2,2,3,3] => 1010100100 => 0101011011 => ? = 3 - 1
[1,1,2,2,2,1,1] => [3,2,2,3] => 1001010100 => 0110101011 => ? = 3 - 1
[1,1,3,2,1,1,1] => [4,2,1,3] => 1000101100 => 0111010011 => ? = 4 - 1
[1,1,4,3,1] => [2,1,2,1,1,3] => 1011011100 => 0100100011 => ? = 3 - 1
[1,1,6,1,1] => [3,1,1,1,1,3] => 1001111100 => 0110000011 => ? = 3 - 1
[1,2,2,1,2,1,1] => [3,3,2,2] => 1001001010 => 0110110101 => ? = 3 - 1
[1,2,2,4,1] => [2,1,1,2,2,2] => 1011101010 => 0100010101 => ? = 2 - 1
[1,2,4,2,1] => [2,2,1,1,2,2] => 1010111010 => 0101000101 => ? = 2 - 1
[1,3,1,1,1,1,2] => [1,6,1,2] => 1100000110 => 0011111001 => ? = 6 - 1
[1,3,1,2,1,1,1] => [4,3,1,2] => 1000100110 => 0111011001 => ? = 4 - 1
[1,3,2,3,1] => [2,1,2,2,1,2] => 1011010110 => 0100101001 => ? = 2 - 1
[1,3,4,1,1] => [3,1,1,2,1,2] => 1001110110 => 0110001001 => ? = 3 - 1
[1,4,2,2,1] => [2,2,2,1,1,2] => 1010101110 => 0101010001 => ? = 2 - 1
[2,1,1,1,1,1,1,1,1] => [9,1] => 1000000001 => 0111111110 => ? = 9 - 1
[2,1,1,1,1,1,1,2] => [1,8,1] => 1100000001 => 0011111110 => ? = 8 - 1
[2,1,1,1,1,2,2] => [1,2,6,1] => 1101000001 => 0010111110 => ? = 6 - 1
[2,1,1,1,1,3,1] => [2,1,6,1] => 1011000001 => 0100111110 => ? = 6 - 1
[2,1,1,1,2,1,2] => [1,3,5,1] => 1100100001 => 0011011110 => ? = 5 - 1
[2,1,1,2,1,1,2] => [1,4,4,1] => 1100010001 => 0011101110 => ? = 4 - 1
[2,1,2,1,1,1,2] => [1,5,3,1] => 1100001001 => 0011110110 => ? = 5 - 1
[2,1,4,1,2] => [1,3,1,1,3,1] => 1100111001 => 0011000110 => ? = 3 - 1
[2,2,1,1,1,1,2] => [1,6,2,1] => 1100000101 => 0011111010 => ? = 6 - 1
[2,4,1,1,2] => [1,4,1,1,2,1] => 1100011101 => 0011100010 => ? = 4 - 1
[3,1,1,1,1,1,1,1] => [8,1,1] => 1000000011 => 0111111100 => ? = 8 - 1
[3,1,1,1,1,1,2] => [1,7,1,1] => 1100000011 => 0011111100 => ? = 7 - 1
[3,1,1,2,1,1,1] => [4,4,1,1] => 1000100011 => 0111011100 => ? = 4 - 1
[3,1,2,3,1] => [2,1,2,3,1,1] => 1011010011 => 0100101100 => ? = 3 - 1
[3,1,4,1,1] => [3,1,1,3,1,1] => 1001110011 => 0110001100 => ? = 3 - 1
[3,2,2,2,1] => [2,2,2,2,1,1] => 1010101011 => 0101010100 => ? = 2 - 1
[3,3,2,1,1] => [3,2,1,2,1,1] => 1001011011 => 0110100100 => ? = 3 - 1
[5,1,1,1,2] => [1,5,1,1,1,1] => 1100001111 => 0011110000 => ? = 5 - 1
[5,1,2,1,1] => [3,3,1,1,1,1] => 1001001111 => 0110110000 => ? = 3 - 1
[5,4,1] => [2,1,1,2,1,1,1,1] => 1011101111 => 0100010000 => ? = 2 - 1
[7,2,1] => [2,2,1,1,1,1,1,1] => 1010111111 => 0101000000 => ? = 2 - 1
[1,2,2,4,3] => [1,1,2,1,1,2,2,2] => 111011101010 => 000100010101 => ? = 2 - 1
[1,2,4,4,1] => [2,1,1,2,1,1,2,2] => 101110111010 => 010001000101 => ? = 2 - 1
[1,2,4,2,3] => [1,1,2,2,1,1,2,2] => 111010111010 => 000101000101 => ? = 2 - 1
[1,3,4,3,1] => [2,1,2,1,1,2,1,2] => 101101110110 => 010010001001 => ? = 2 - 1
[1,3,2,3,3] => [1,1,2,1,2,2,1,2] => 111011010110 => 000100101001 => ? = 2 - 1
[1,4,4,2,1] => [2,2,1,1,2,1,1,2] => 101011101110 => 010100010001 => ? = 2 - 1
[1,4,2,4,1] => [2,1,1,2,2,1,1,2] => 101110101110 => 010001010001 => ? = 2 - 1
[1,4,2,2,3] => [1,1,2,2,2,1,1,2] => 111010101110 => 000101010001 => ? = 2 - 1
[1,6,5] => [1,1,1,1,2,1,1,1,1,2] => 111110111110 => 000001000001 => ? = 2 - 1
[3,4,2,2,1] => [2,2,2,1,1,2,1,1] => 101010111011 => 010101000100 => ? = 2 - 1
[3,6,3] => [1,1,2,1,1,1,1,2,1,1] => 111011111011 => 000100000100 => ? = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000982
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1 => 1
[1,1] => 11 => 11 => 2
[2] => 10 => 01 => 1
[1,1,1] => 111 => 111 => 3
[1,2] => 110 => 011 => 2
[2,1] => 101 => 001 => 2
[3] => 100 => 101 => 1
[1,1,1,1] => 1111 => 1111 => 4
[1,1,2] => 1110 => 0111 => 3
[1,2,1] => 1101 => 0011 => 2
[1,3] => 1100 => 1011 => 2
[2,1,1] => 1011 => 0001 => 3
[2,2] => 1010 => 1001 => 2
[3,1] => 1001 => 1101 => 2
[4] => 1000 => 0101 => 1
[1,1,1,1,1] => 11111 => 11111 => 5
[1,1,1,2] => 11110 => 01111 => 4
[1,1,2,1] => 11101 => 00111 => 3
[1,1,3] => 11100 => 10111 => 3
[1,2,1,1] => 11011 => 00011 => 3
[1,2,2] => 11010 => 10011 => 2
[1,3,1] => 11001 => 11011 => 2
[1,4] => 11000 => 01011 => 2
[2,1,1,1] => 10111 => 00001 => 4
[2,1,2] => 10110 => 10001 => 3
[2,2,1] => 10101 => 11001 => 2
[2,3] => 10100 => 01001 => 2
[3,1,1] => 10011 => 11101 => 3
[3,2] => 10010 => 01101 => 2
[4,1] => 10001 => 00101 => 2
[5] => 10000 => 10101 => 1
[1,1,1,1,1,1] => 111111 => 111111 => 6
[1,1,1,1,2] => 111110 => 011111 => 5
[1,1,1,2,1] => 111101 => 001111 => 4
[1,1,1,3] => 111100 => 101111 => 4
[1,1,2,1,1] => 111011 => 000111 => 3
[1,1,2,2] => 111010 => 100111 => 3
[1,1,3,1] => 111001 => 110111 => 3
[1,1,4] => 111000 => 010111 => 3
[1,2,1,1,1] => 110111 => 000011 => 4
[1,2,1,2] => 110110 => 100011 => 3
[1,2,2,1] => 110101 => 110011 => 2
[1,2,3] => 110100 => 010011 => 2
[1,3,1,1] => 110011 => 111011 => 3
[1,3,2] => 110010 => 011011 => 2
[1,4,1] => 110001 => 001011 => 2
[1,5] => 110000 => 101011 => 2
[2,1,1,1,1] => 101111 => 000001 => 5
[2,1,1,2] => 101110 => 100001 => 4
[2,1,2,1] => 101101 => 110001 => 3
[1,1,1,1,2,1,1,1,1] => 1111101111 => ? => ? = 5
[1,1,1,2,1,1,3] => 1111011100 => ? => ? = 4
[1,1,1,2,1,3,1] => 1111011001 => ? => ? = 4
[1,1,1,2,3,1,1] => 1111010011 => ? => ? = 4
[1,1,1,4,1,1,1] => 1111000111 => ? => ? = 4
[1,1,2,1,2,2,1] => 1110110101 => ? => ? = 3
[1,1,3,2,1,1,1] => 1110010111 => ? => ? = 4
[1,1,6,1,1] => 1110000011 => ? => ? = 3
[1,2,7] => 1101000000 => ? => ? = 2
[1,3,1,1,1,1,2] => 1100111110 => ? => ? = 6
[1,3,1,2,1,1,1] => 1100110111 => ? => ? = 4
[1,3,4,1,1] => 1100100011 => ? => ? = 3
[1,4,5] => 1100010000 => ? => ? = 2
[1,6,3] => 1100000100 => ? => ? = 2
[1,8,1] => 1100000001 => ? => ? = 2
[2,1,1,1,1,1,1,1,1] => 1011111111 => ? => ? = 9
[2,1,1,1,1,1,1,2] => 1011111110 => 1000000001 => ? = 8
[2,1,1,1,1,1,3] => 1011111100 => ? => ? = 7
[2,1,1,1,1,2,2] => 1011111010 => ? => ? = 6
[2,1,1,1,1,3,1] => 1011111001 => ? => ? = 6
[2,1,1,1,2,1,2] => 1011110110 => ? => ? = 5
[2,1,1,2,1,1,2] => 1011101110 => ? => ? = 4
[2,1,2,1,1,1,2] => 1011011110 => ? => ? = 5
[2,1,4,1,2] => 1011000110 => ? => ? = 3
[3,1,1,2,1,1,1] => 1001110111 => ? => ? = 4
[3,1,4,1,1] => 1001100011 => ? => ? = 3
[3,2,5] => 1001010000 => ? => ? = 2
[3,3,2,1,1] => 1001001011 => ? => ? = 3
[3,4,3] => 1001000100 => ? => ? = 2
[3,6,1] => 1001000001 => ? => ? = 2
[5,1,2,1,1] => 1000011011 => ? => ? = 3
[5,2,3] => 1000010100 => ? => ? = 2
[5,4,1] => 1000010001 => ? => ? = 2
[7,2,1] => 1000000101 => ? => ? = 2
[1,2,2,2,2,2,1] => 110101010101 => ? => ? = 2
[1,2,2,4,3] => 110101000100 => ? => ? = 2
[1,2,4,4,1] => 110100010001 => ? => ? = 2
[1,2,4,2,3] => 110100010100 => ? => ? = 2
[1,3,4,3,1] => 110010001001 => ? => ? = 2
[1,3,2,3,3] => 110010100100 => ? => ? = 2
[1,4,4,2,1] => 110001000101 => ? => ? = 2
[1,4,2,4,1] => 110001010001 => ? => ? = 2
[1,4,2,2,3] => 110001010100 => ? => ? = 2
[1,6,5] => 110000010000 => ? => ? = 2
[3,4,2,2,1] => 100100010101 => ? => ? = 2
[3,6,3] => 100100000100 => ? => ? = 2
[3,3,2,3,1] => 100100101001 => ? => ? = 2
[3,2,4,2,1] => 100101000101 => ? => ? = 2
[3,2,2,4,1] => 100101010001 => ? => ? = 2
[3,2,2,2,3] => 100101010100 => ? => ? = 2
Description
The length of the longest constant subword.
Matching statistic: St000983
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 90%
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 90%
Values
[1] => 1 => 1 => 1 => 1
[1,1] => 11 => 11 => 10 => 2
[2] => 10 => 01 => 00 => 1
[1,1,1] => 111 => 111 => 101 => 3
[1,2] => 110 => 011 => 001 => 2
[2,1] => 101 => 001 => 011 => 2
[3] => 100 => 101 => 111 => 1
[1,1,1,1] => 1111 => 1111 => 1010 => 4
[1,1,2] => 1110 => 0111 => 0010 => 3
[1,2,1] => 1101 => 0011 => 0110 => 2
[1,3] => 1100 => 1011 => 1110 => 2
[2,1,1] => 1011 => 0001 => 0100 => 3
[2,2] => 1010 => 1001 => 1100 => 2
[3,1] => 1001 => 1101 => 1000 => 2
[4] => 1000 => 0101 => 0000 => 1
[1,1,1,1,1] => 11111 => 11111 => 10101 => 5
[1,1,1,2] => 11110 => 01111 => 00101 => 4
[1,1,2,1] => 11101 => 00111 => 01101 => 3
[1,1,3] => 11100 => 10111 => 11101 => 3
[1,2,1,1] => 11011 => 00011 => 01001 => 3
[1,2,2] => 11010 => 10011 => 11001 => 2
[1,3,1] => 11001 => 11011 => 10001 => 2
[1,4] => 11000 => 01011 => 00001 => 2
[2,1,1,1] => 10111 => 00001 => 01011 => 4
[2,1,2] => 10110 => 10001 => 11011 => 3
[2,2,1] => 10101 => 11001 => 10011 => 2
[2,3] => 10100 => 01001 => 00011 => 2
[3,1,1] => 10011 => 11101 => 10111 => 3
[3,2] => 10010 => 01101 => 00111 => 2
[4,1] => 10001 => 00101 => 01111 => 2
[5] => 10000 => 10101 => 11111 => 1
[1,1,1,1,1,1] => 111111 => 111111 => 101010 => 6
[1,1,1,1,2] => 111110 => 011111 => 001010 => 5
[1,1,1,2,1] => 111101 => 001111 => 011010 => 4
[1,1,1,3] => 111100 => 101111 => 111010 => 4
[1,1,2,1,1] => 111011 => 000111 => 010010 => 3
[1,1,2,2] => 111010 => 100111 => 110010 => 3
[1,1,3,1] => 111001 => 110111 => 100010 => 3
[1,1,4] => 111000 => 010111 => 000010 => 3
[1,2,1,1,1] => 110111 => 000011 => 010110 => 4
[1,2,1,2] => 110110 => 100011 => 110110 => 3
[1,2,2,1] => 110101 => 110011 => 100110 => 2
[1,2,3] => 110100 => 010011 => 000110 => 2
[1,3,1,1] => 110011 => 111011 => 101110 => 3
[1,3,2] => 110010 => 011011 => 001110 => 2
[1,4,1] => 110001 => 001011 => 011110 => 2
[1,5] => 110000 => 101011 => 111110 => 2
[2,1,1,1,1] => 101111 => 000001 => 010100 => 5
[2,1,1,2] => 101110 => 100001 => 110100 => 4
[2,1,2,1] => 101101 => 110001 => 100100 => 3
[1,1,1,1,2,1,1,1,1] => 1111101111 => ? => ? => ? = 5
[1,1,1,2,1,1,3] => 1111011100 => ? => ? => ? = 4
[1,1,1,2,1,3,1] => 1111011001 => ? => ? => ? = 4
[1,1,1,2,3,1,1] => 1111010011 => ? => ? => ? = 4
[1,1,1,4,1,1,1] => 1111000111 => ? => ? => ? = 4
[1,1,2,1,2,2,1] => 1110110101 => ? => ? => ? = 3
[1,1,2,2,2,1,1] => 1110101011 => 0001100111 => 0100110010 => ? = 3
[1,1,2,5,1] => 1110100001 => 0010100111 => 0111110010 => ? = 3
[1,1,3,2,1,1,1] => 1110010111 => ? => ? => ? = 4
[1,1,4,3,1] => 1110001001 => 0010010111 => 0111000010 => ? = 3
[1,1,6,1,1] => 1110000011 => ? => ? => ? = 3
[1,2,2,1,2,1,1] => 1101011011 => 0001110011 => 0100100110 => ? = 3
[1,2,2,4,1] => 1101010001 => 0010110011 => 0111100110 => ? = 2
[1,2,4,2,1] => 1101000101 => 0011010011 => 0110000110 => ? = 2
[1,2,7] => 1101000000 => ? => ? => ? = 2
[1,3,1,1,1,1,2] => 1100111110 => ? => ? => ? = 6
[1,3,1,2,1,1,1] => 1100110111 => ? => ? => ? = 4
[1,3,2,3,1] => 1100101001 => 0010011011 => 0111001110 => ? = 2
[1,3,4,1,1] => 1100100011 => ? => ? => ? = 3
[1,4,2,2,1] => 1100010101 => 0011001011 => 0110011110 => ? = 2
[1,4,5] => 1100010000 => ? => ? => ? = 2
[1,5,2,1,1] => 1100001011 => 0001101011 => 0100111110 => ? = 3
[1,6,3] => 1100000100 => ? => ? => ? = 2
[1,8,1] => 1100000001 => ? => ? => ? = 2
[2,1,1,1,1,1,1,1,1] => 1011111111 => ? => ? => ? = 9
[2,1,1,1,1,1,1,2] => 1011111110 => 1000000001 => 1101010100 => ? = 8
[2,1,1,1,1,1,3] => 1011111100 => ? => ? => ? = 7
[2,1,1,1,1,2,2] => 1011111010 => ? => ? => ? = 6
[2,1,1,1,1,3,1] => 1011111001 => ? => ? => ? = 6
[2,1,1,1,2,1,2] => 1011110110 => ? => ? => ? = 5
[2,1,1,2,1,1,2] => 1011101110 => ? => ? => ? = 4
[2,1,1,4,2] => 1011100010 => 0110100001 => 0011110100 => ? = 4
[2,1,2,1,1,1,2] => 1011011110 => ? => ? => ? = 5
[2,1,4,1,2] => 1011000110 => ? => ? => ? = 3
[2,4,1,1,2] => 1010001110 => 0111101001 => 0010111100 => ? = 4
[3,1,1,1,1,1,1,1] => 1001111111 => 1111111101 => 1010101000 => ? = 8
[3,1,1,2,1,1,1] => 1001110111 => ? => ? => ? = 4
[3,1,2,3,1] => 1001101001 => 0010011101 => 0111001000 => ? = 3
[3,1,4,1,1] => 1001100011 => ? => ? => ? = 3
[3,2,2,2,1] => 1001010101 => 0011001101 => 0110011000 => ? = 2
[3,2,5] => 1001010000 => ? => ? => ? = 2
[3,3,2,1,1] => 1001001011 => ? => ? => ? = 3
[3,4,3] => 1001000100 => ? => ? => ? = 2
[3,6,1] => 1001000001 => ? => ? => ? = 2
[5,1,2,1,1] => 1000011011 => ? => ? => ? = 3
[5,2,3] => 1000010100 => ? => ? => ? = 2
[5,4,1] => 1000010001 => ? => ? => ? = 2
[7,2,1] => 1000000101 => ? => ? => ? = 2
[1,2,2,2,2,2,1] => 110101010101 => ? => ? => ? = 2
[1,2,2,4,3] => 110101000100 => ? => ? => ? = 2
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000676
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 80%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1]
=> [1,0]
=> 1
[1,1] => [2] => [2]
=> [1,0,1,0]
=> 2
[2] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[1,1,1] => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,1] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[8] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,2] => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,1,1,1,3] => [7,1,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,4] => [6,1,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[1,1,1,1,5] => [5,1,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,1,6] => [4,1,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,7] => [3,1,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,2,6] => [2,2,1,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,3,5] => [2,1,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,4,4] => [2,1,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,5,3] => [2,1,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,6,2] => [2,1,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,7,1] => [2,1,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,8] => [2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1] => [1,8] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[2,1,1,1,1,1,2] => [1,7,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[2,1,1,1,1,3] => [1,6,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[2,1,1,1,4] => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[2,1,1,5] => [1,4,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[2,1,6] => [1,3,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[2,2,5] => [1,2,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,3,4] => [1,2,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,4,3] => [1,2,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,5,2] => [1,2,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,6,1] => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,7] => [1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,1,1,1,1,1,1] => [1,1,7] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[3,1,1,1,1,2] => [1,1,6,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[3,1,1,1,3] => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[3,1,1,4] => [1,1,4,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[3,1,5] => [1,1,3,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[3,2,4] => [1,1,2,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,3,3] => [1,1,2,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,4,2] => [1,1,2,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,5,1] => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,6] => [1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,1,1,1,1,1] => [1,1,1,6] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[4,1,1,1,2] => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[4,1,1,3] => [1,1,1,4,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[4,1,4] => [1,1,1,3,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[4,2,3] => [1,1,1,2,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,3,2] => [1,1,1,2,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,4,1] => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,5] => [1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[5,1,1,1,1] => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[5,1,1,2] => [1,1,1,1,4,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[5,1,3] => [1,1,1,1,3,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[5,2,2] => [1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[5,3,1] => [1,1,1,1,2,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001039
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 80%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1]
=> [1,0]
=> ? = 1
[1,1] => [2] => [2]
=> [1,0,1,0]
=> 2
[2] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[1,1,1] => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,1] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4] => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,2,1] => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,3] => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,1,1] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,2,2] => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,1] => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,1,1,1] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[2,2,1] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[3,1,1] => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,2,1] => [4,2] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,2,1,1] => [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[1,2,1,1,1] => [2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,1,1] => [1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[2,1,3] => [1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[8] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,2] => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,1,1,1,3] => [7,1,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,4] => [6,1,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[1,1,1,1,5] => [5,1,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,1,6] => [4,1,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,7] => [3,1,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,2,6] => [2,2,1,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,3,5] => [2,1,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,4,4] => [2,1,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,5,3] => [2,1,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,6,2] => [2,1,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,7,1] => [2,1,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,8] => [2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1] => [1,8] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[2,1,1,1,1,1,2] => [1,7,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[2,1,1,1,1,3] => [1,6,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[2,1,1,1,4] => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[2,1,1,5] => [1,4,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[2,1,6] => [1,3,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[2,2,5] => [1,2,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,3,4] => [1,2,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,4,3] => [1,2,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,5,2] => [1,2,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,6,1] => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,7] => [1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,1,1,1,1,1,1] => [1,1,7] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[3,1,1,1,1,2] => [1,1,6,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[3,1,1,1,3] => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[3,1,1,4] => [1,1,4,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[3,1,5] => [1,1,3,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[3,2,4] => [1,1,2,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,3,3] => [1,1,2,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,4,2] => [1,1,2,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,5,1] => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,6] => [1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,1,1,1,1,1] => [1,1,1,6] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[4,1,1,1,2] => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[4,1,1,3] => [1,1,1,4,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[4,1,4] => [1,1,1,3,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[4,2,3] => [1,1,1,2,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,3,2] => [1,1,1,2,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,4,1] => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,5] => [1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[5,1,1,1,1] => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[5,1,1,2] => [1,1,1,1,4,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[5,1,3] => [1,1,1,1,3,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[5,2,2] => [1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001291
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1] => [2] => [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2] => [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,2] => [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1] => [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3] => [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,2] => [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,2,1] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,1] => [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[2,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4] => [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,2] => [4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,2,1] => [3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,3] => [3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,2,1,1] => [2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,2] => [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1] => [2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[2,1,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[2,2,1] => [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,2] => [5,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,2,1] => [4,2] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,2,1,1] => [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,2,1,1,1] => [2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[2,1,1,1,1] => [1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 6 = 5 + 1
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,3] => [1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[6] => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1] => [7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,1,1,1,1,2] => [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,6] => [2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[2,1,1,1,1,1] => [1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[2,5] => [1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[3,4] => [1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[4,3] => [1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[5,2] => [1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[6,1] => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1] => [8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,1,1,1,1,1,2] => [7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,1,1,1,1,2,1] => [6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,3] => [6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,6] => [3,1,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[1,2,1,1,1,1,1] => [2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 6 + 1
[1,2,5] => [2,2,1,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,3,4] => [2,1,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,4,3] => [2,1,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,5,2] => [2,1,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,1] => [2,1,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,7] => [2,1,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[2,1,1,1,1,1,1] => [1,7] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[2,1,1,1,1,2] => [1,6,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[2,1,5] => [1,3,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[2,2,4] => [1,2,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,3,3] => [1,2,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,4,2] => [1,2,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,5,1] => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,6] => [1,2,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[3,1,1,1,1,1] => [1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[3,1,4] => [1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[3,2,3] => [1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[3,3,2] => [1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[3,4,1] => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[3,5] => [1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[4,1,3] => [1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[4,2,2] => [1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[4,3,1] => [1,1,1,2,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[4,4] => [1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[5,1,2] => [1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[5,2,1] => [1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[5,3] => [1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[6,1,1] => [1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[6,2] => [1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[7,1] => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St001933The largest multiplicity of a part in an integer partition. St000439The position of the first down step of a Dyck path. St000306The bounce count of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000662The staircase size of the code of a permutation. St000094The depth of an ordered tree. St000141The maximum drop size of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St001058The breadth of the ordered tree. St000686The finitistic dominant dimension of a Dyck path. St000025The number of initial rises of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001674The number of vertices of the largest induced star graph in the graph. St000171The degree of the graph. St001826The maximal number of leaves on a vertex of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000730The maximal arc length of a set partition. St001118The acyclic chromatic index of a graph. St000028The number of stack-sorts needed to sort a permutation. St000308The height of the tree associated to a permutation. St000651The maximal size of a rise in a permutation. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St001268The size of the largest ordinal summand in the poset. St001372The length of a longest cyclic run of ones of a binary word. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001652The length of a longest interval of consecutive numbers. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000299The number of nonisomorphic vertex-induced subtrees. St000328The maximum number of child nodes in a tree. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001117The game chromatic index of a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000652The maximal difference between successive positions of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001651The Frankl number of a lattice.
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