searching the database
Your data matches 58 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 1
[1,2] => [1,1]
=> [[1],[2]]
=> 1
[2,1] => [2]
=> [[1,2]]
=> 2
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> 2
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> 2
[2,3,1] => [3]
=> [[1,2,3]]
=> 3
[3,1,2] => [3]
=> [[1,2,3]]
=> 3
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> 2
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> 3
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> 3
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 3
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 3
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> 4
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> 4
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 3
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 3
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> 4
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 3
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 3
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> 4
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> 4
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 3
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> 3
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> 4
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> 3
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 4
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> 4
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St001176
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Values
[1] => [1]
=> [1]
=> 0 = 1 - 1
[1,2] => [1,1]
=> [2]
=> 0 = 1 - 1
[2,1] => [2]
=> [1,1]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,3,1] => [3]
=> [1,1,1]
=> 2 = 3 - 1
[3,1,2] => [3]
=> [1,1,1]
=> 2 = 3 - 1
[3,2,1] => [2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 2 = 3 - 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 3 = 4 - 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 3 = 4 - 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[3,4,1,2] => [2,2]
=> [2,2]
=> 2 = 3 - 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 3 = 4 - 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 3 = 4 - 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 3 = 4 - 1
[4,3,2,1] => [2,2]
=> [2,2]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3 = 4 - 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 3 = 4 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> 2 = 3 - 1
[] => []
=> []
=> ? = 0 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000738
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Values
[1] => [1]
=> [1]
=> [[1]]
=> 1
[1,2] => [1,1]
=> [2]
=> [[1,2]]
=> 1
[2,1] => [2]
=> [1,1]
=> [[1],[2]]
=> 2
[1,2,3] => [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[1,3,2] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2
[2,1,3] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2
[2,3,1] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[3,1,2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[3,2,1] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2
[1,2,3,4] => [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[2,1,4,3] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[3,4,1,2] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[4,3,2,1] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3
[] => []
=> []
=> []
=> ? = 0
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,5,9,10,11,12,1,2,4,6,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,6,8,10,11,12,1,2,4,5,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,7,9,10,11,12,1,2,4,5,6,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,5,6,8,11,12,1,2,3,7,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,5,6,9,10,12,1,2,3,7,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,5,7,8,10,12,1,2,3,6,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,5,8,9,10,12,1,2,3,6,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,5,8,10,11,12,1,2,3,6,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,6,7,8,11,12,1,2,3,5,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,6,7,10,11,12,1,2,3,5,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,6,8,9,11,12,1,2,3,5,7,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,7,8,10,11,12,1,2,3,5,6,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[5,6,7,8,10,12,1,2,3,4,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[5,6,7,9,11,12,1,2,3,4,8,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[5,6,8,9,10,12,1,2,3,4,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[5,7,8,9,11,12,1,2,3,4,6,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[6,7,8,9,10,12,1,2,3,4,5,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,3,4,5,6,7,8,9,10,12,1,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,1,4,5,6,7,8,9,10,11,12,2] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,5,2,7,4,9,6,11,8,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,5,2,8,4,10,11,7,12,9,6,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,6,2,8,9,5,11,7,4,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,6,2,9,10,5,11,12,8,7,4,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,7,2,8,10,11,6,5,12,9,4,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,6,7,3,9,5,2,11,8,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,6,8,3,10,5,11,7,12,9,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[4,7,8,3,9,11,6,5,2,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[5,6,8,9,4,3,11,7,2,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[5,6,9,10,4,3,11,12,8,7,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[5,8,9,10,4,11,12,7,6,3,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[6,7,9,10,11,5,4,12,8,3,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[3,10,12,6,7,11,9,4,8,5,1,2] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000228
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Values
[1] => [1]
=> [1]
=> []
=> 0 = 1 - 1
[1,2] => [1,1]
=> [2]
=> []
=> 0 = 1 - 1
[2,1] => [2]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> [3]
=> []
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[] => []
=> []
=> ?
=> ? = 0 - 1
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,5,9,10,11,12,1,2,4,6,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,6,8,10,11,12,1,2,4,5,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,7,9,10,11,12,1,2,4,5,6,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,5,6,8,11,12,1,2,3,7,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,5,6,9,10,12,1,2,3,7,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,5,7,8,10,12,1,2,3,6,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,5,8,9,10,12,1,2,3,6,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,5,8,10,11,12,1,2,3,6,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,6,7,8,11,12,1,2,3,5,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,6,7,10,11,12,1,2,3,5,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,6,8,9,11,12,1,2,3,5,7,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,7,8,10,11,12,1,2,3,5,6,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[5,6,7,8,10,12,1,2,3,4,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[5,6,7,9,11,12,1,2,3,4,8,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[5,6,8,9,10,12,1,2,3,4,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[5,7,8,9,11,12,1,2,3,4,6,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[6,7,8,9,10,12,1,2,3,4,5,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[2,3,4,5,6,7,8,9,10,12,1,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,1,4,5,6,7,8,9,10,11,12,2] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,5,2,7,4,9,6,11,8,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,5,2,8,4,10,11,7,12,9,6,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,6,2,8,9,5,11,7,4,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,6,2,9,10,5,11,12,8,7,4,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,7,2,8,10,11,6,5,12,9,4,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,6,7,3,9,5,2,11,8,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,6,8,3,10,5,11,7,12,9,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[4,7,8,3,9,11,6,5,2,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[5,6,8,9,4,3,11,7,2,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[5,6,9,10,4,3,11,12,8,7,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[5,8,9,10,4,11,12,7,6,3,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[6,7,9,10,11,5,4,12,8,3,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
[3,10,12,6,7,11,9,4,8,5,1,2] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000293
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Values
[1] => [1]
=> 10 => 01 => 0 = 1 - 1
[1,2] => [1,1]
=> 110 => 011 => 0 = 1 - 1
[2,1] => [2]
=> 100 => 010 => 1 = 2 - 1
[1,2,3] => [1,1,1]
=> 1110 => 0111 => 0 = 1 - 1
[1,3,2] => [2,1]
=> 1010 => 0101 => 1 = 2 - 1
[2,1,3] => [2,1]
=> 1010 => 0101 => 1 = 2 - 1
[2,3,1] => [3]
=> 1000 => 0100 => 2 = 3 - 1
[3,1,2] => [3]
=> 1000 => 0100 => 2 = 3 - 1
[3,2,1] => [2,1]
=> 1010 => 0101 => 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> 11110 => 01111 => 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> 10110 => 01011 => 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> 10110 => 01011 => 1 = 2 - 1
[1,3,4,2] => [3,1]
=> 10010 => 01001 => 2 = 3 - 1
[1,4,2,3] => [3,1]
=> 10010 => 01001 => 2 = 3 - 1
[1,4,3,2] => [2,1,1]
=> 10110 => 01011 => 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> 10110 => 01011 => 1 = 2 - 1
[2,1,4,3] => [2,2]
=> 1100 => 0110 => 2 = 3 - 1
[2,3,1,4] => [3,1]
=> 10010 => 01001 => 2 = 3 - 1
[2,3,4,1] => [4]
=> 10000 => 01000 => 3 = 4 - 1
[2,4,1,3] => [4]
=> 10000 => 01000 => 3 = 4 - 1
[2,4,3,1] => [3,1]
=> 10010 => 01001 => 2 = 3 - 1
[3,1,2,4] => [3,1]
=> 10010 => 01001 => 2 = 3 - 1
[3,1,4,2] => [4]
=> 10000 => 01000 => 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> 10110 => 01011 => 1 = 2 - 1
[3,2,4,1] => [3,1]
=> 10010 => 01001 => 2 = 3 - 1
[3,4,1,2] => [2,2]
=> 1100 => 0110 => 2 = 3 - 1
[3,4,2,1] => [4]
=> 10000 => 01000 => 3 = 4 - 1
[4,1,2,3] => [4]
=> 10000 => 01000 => 3 = 4 - 1
[4,1,3,2] => [3,1]
=> 10010 => 01001 => 2 = 3 - 1
[4,2,1,3] => [3,1]
=> 10010 => 01001 => 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> 10110 => 01011 => 1 = 2 - 1
[4,3,1,2] => [4]
=> 10000 => 01000 => 3 = 4 - 1
[4,3,2,1] => [2,2]
=> 1100 => 0110 => 2 = 3 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> 111110 => 011111 => 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> 101110 => 010111 => 1 = 2 - 1
[1,2,4,3,5] => [2,1,1,1]
=> 101110 => 010111 => 1 = 2 - 1
[1,2,4,5,3] => [3,1,1]
=> 100110 => 010011 => 2 = 3 - 1
[1,2,5,3,4] => [3,1,1]
=> 100110 => 010011 => 2 = 3 - 1
[1,2,5,4,3] => [2,1,1,1]
=> 101110 => 010111 => 1 = 2 - 1
[1,3,2,4,5] => [2,1,1,1]
=> 101110 => 010111 => 1 = 2 - 1
[1,3,2,5,4] => [2,2,1]
=> 11010 => 01101 => 2 = 3 - 1
[1,3,4,2,5] => [3,1,1]
=> 100110 => 010011 => 2 = 3 - 1
[1,3,4,5,2] => [4,1]
=> 100010 => 010001 => 3 = 4 - 1
[1,3,5,2,4] => [4,1]
=> 100010 => 010001 => 3 = 4 - 1
[1,3,5,4,2] => [3,1,1]
=> 100110 => 010011 => 2 = 3 - 1
[1,4,2,3,5] => [3,1,1]
=> 100110 => 010011 => 2 = 3 - 1
[1,4,2,5,3] => [4,1]
=> 100010 => 010001 => 3 = 4 - 1
[1,4,3,2,5] => [2,1,1,1]
=> 101110 => 010111 => 1 = 2 - 1
[1,4,3,5,2] => [3,1,1]
=> 100110 => 010011 => 2 = 3 - 1
[1,4,5,2,3] => [2,2,1]
=> 11010 => 01101 => 2 = 3 - 1
[] => []
=> => ? => ? = 0 - 1
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,5,9,10,11,12,1,2,4,6,7,8] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,6,8,10,11,12,1,2,4,5,7,9] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,7,9,10,11,12,1,2,4,5,6,8] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,5,6,8,11,12,1,2,3,7,9,10] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,5,6,9,10,12,1,2,3,7,8,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,5,7,8,10,12,1,2,3,6,9,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,5,8,9,10,12,1,2,3,6,7,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,5,8,10,11,12,1,2,3,6,7,9] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,6,7,8,11,12,1,2,3,5,9,10] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,6,7,10,11,12,1,2,3,5,8,9] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,6,8,9,11,12,1,2,3,5,7,10] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,7,8,10,11,12,1,2,3,5,6,9] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[5,6,7,8,10,12,1,2,3,4,9,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[5,6,7,9,11,12,1,2,3,4,8,10] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[5,6,8,9,10,12,1,2,3,4,7,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[5,7,8,9,11,12,1,2,3,4,6,10] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[6,7,8,9,10,12,1,2,3,4,5,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[2,3,4,5,6,7,8,9,10,12,1,11] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,1,4,5,6,7,8,9,10,11,12,2] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,5,2,7,4,9,6,11,8,12,10,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,5,2,8,4,10,11,7,12,9,6,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,6,2,8,9,5,11,7,4,12,10,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,6,2,9,10,5,11,12,8,7,4,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,7,2,8,10,11,6,5,12,9,4,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,6,7,3,9,5,2,11,8,12,10,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,6,8,3,10,5,11,7,12,9,2,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[4,7,8,3,9,11,6,5,2,12,10,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[5,6,8,9,4,3,11,7,2,12,10,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[5,6,9,10,4,3,11,12,8,7,2,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[5,8,9,10,4,11,12,7,6,3,2,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[6,7,9,10,11,5,4,12,8,3,2,1] => [12]
=> 1000000000000 => ? => ? = 12 - 1
[3,10,12,6,7,11,9,4,8,5,1,2] => [12]
=> 1000000000000 => ? => ? = 12 - 1
Description
The number of inversions of a binary word.
Matching statistic: St000394
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 98%●distinct values known / distinct values provided: 85%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 98%●distinct values known / distinct values provided: 85%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[4,1,2,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,3,1,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[] => []
=> []
=> []
=> ? = 0 - 1
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 9 - 1
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 9 - 1
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,5,9,10,11,12,1,2,4,6,7,8] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,6,7,9,10,12,1,2,4,5,8,11] => [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 9 - 1
[3,6,8,10,11,12,1,2,4,5,7,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,7,8,9,10,12,1,2,4,5,6,11] => [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 9 - 1
[3,7,9,10,11,12,1,2,4,5,6,8] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,5,6,8,11,12,1,2,3,7,9,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,5,6,9,10,12,1,2,3,7,8,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,5,7,8,10,12,1,2,3,6,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,5,8,9,10,12,1,2,3,6,7,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,5,8,10,11,12,1,2,3,6,7,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,6,7,8,11,12,1,2,3,5,9,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,6,7,10,11,12,1,2,3,5,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,6,8,9,11,12,1,2,3,5,7,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,7,8,10,11,12,1,2,3,5,6,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[5,6,7,8,10,12,1,2,3,4,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[5,6,7,9,11,12,1,2,3,4,8,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[5,6,8,9,10,12,1,2,3,4,7,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[5,7,8,9,11,12,1,2,3,4,6,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[6,7,8,9,10,12,1,2,3,4,5,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[2,3,4,5,6,7,8,9,10,12,1,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,1,4,5,6,7,8,9,10,11,12,2] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,5,2,7,4,9,6,11,8,12,10,1] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,5,2,8,4,10,11,7,12,9,6,1] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,6,2,8,9,5,11,7,4,12,10,1] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,6,2,9,10,5,11,12,8,7,4,1] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[3,7,2,8,10,11,6,5,12,9,4,1] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,6,7,3,9,5,2,11,8,12,10,1] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,6,8,3,10,5,11,7,12,9,2,1] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[4,7,8,3,9,11,6,5,2,12,10,1] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[5,6,8,9,4,3,11,7,2,12,10,1] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000074
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 85% ●values known / values provided: 97%●distinct values known / distinct values provided: 85%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 85% ●values known / values provided: 97%●distinct values known / distinct values provided: 85%
Values
[1] => [1]
=> [[1]]
=> [[1]]
=> 0 = 1 - 1
[1,2] => [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0 = 1 - 1
[2,1] => [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[3,1,2] => [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2 = 3 - 1
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3 = 4 - 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3 = 4 - 1
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2 = 3 - 1
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3 = 4 - 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3 = 4 - 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3 = 4 - 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3 = 4 - 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3 = 4 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1 = 2 - 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2 = 3 - 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2 = 3 - 1
[] => []
=> []
=> ?
=> ? = 0 - 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[2,2,2,2,2,2,0,0,0,0,0,0],[2,2,2,2,2,1,0,0,0,0,0],[2,2,2,2,2,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 7 - 1
[6,5,4,3,2,1,12,11,10,9,8,7] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[2,2,2,2,2,2,0,0,0,0,0,0],[2,2,2,2,2,1,0,0,0,0,0],[2,2,2,2,2,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 7 - 1
[12,11,10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[2,2,2,2,2,2,0,0,0,0,0,0],[2,2,2,2,2,1,0,0,0,0,0],[2,2,2,2,2,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 7 - 1
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9 - 1
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9 - 1
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[3,5,6,9,11,12,1,2,4,7,8,10] => [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10 - 1
[3,5,7,9,11,12,1,2,4,6,8,10] => [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [[4,3,3,2,0,0,0,0,0,0,0,0],[4,3,3,1,0,0,0,0,0,0,0],[4,3,3,0,0,0,0,0,0,0],[4,3,2,0,0,0,0,0,0],[4,3,1,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9 - 1
[3,5,7,10,11,12,1,2,4,6,8,9] => [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> [[5,4,3,0,0,0,0,0,0,0,0,0],[5,4,2,0,0,0,0,0,0,0,0],[5,4,1,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10 - 1
[3,5,9,10,11,12,1,2,4,6,7,8] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[3,6,7,9,10,12,1,2,4,5,8,11] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9 - 1
[3,6,8,10,11,12,1,2,4,5,7,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[3,7,8,9,10,12,1,2,4,5,6,11] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9 - 1
[3,7,9,10,11,12,1,2,4,5,6,8] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[4,5,6,8,11,12,1,2,3,7,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[4,5,6,9,10,12,1,2,3,7,8,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[4,5,7,8,10,12,1,2,3,6,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[4,5,7,9,11,12,1,2,3,6,8,10] => [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> [[5,4,3,0,0,0,0,0,0,0,0,0],[5,4,2,0,0,0,0,0,0,0,0],[5,4,1,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10 - 1
[4,5,8,9,10,12,1,2,3,6,7,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[4,5,8,10,11,12,1,2,3,6,7,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[4,6,7,8,11,12,1,2,3,5,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[4,6,7,10,11,12,1,2,3,5,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[4,6,8,9,11,12,1,2,3,5,7,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[4,6,9,10,11,12,1,2,3,5,7,8] => [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10 - 1
[4,7,8,10,11,12,1,2,3,5,6,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[5,6,7,8,10,12,1,2,3,4,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[5,6,7,9,11,12,1,2,3,4,8,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[5,6,8,9,10,12,1,2,3,4,7,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[5,6,8,10,11,12,1,2,3,4,7,9] => [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10 - 1
[5,6,9,10,11,12,1,2,3,4,7,8] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [[4,4,2,2,0,0,0,0,0,0,0,0],[4,4,2,1,0,0,0,0,0,0,0],[4,4,2,0,0,0,0,0,0,0],[4,4,1,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9 - 1
[5,7,8,9,11,12,1,2,3,4,6,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[6,7,8,9,10,12,1,2,3,4,5,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[7,8,9,10,11,12,1,2,3,4,5,6] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[2,2,2,2,2,2,0,0,0,0,0,0],[2,2,2,2,2,1,0,0,0,0,0],[2,2,2,2,2,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 7 - 1
Description
The number of special entries.
An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000369
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 77%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 77%
Values
[1] => [1]
=> [1]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2] => [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1] => [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[3,2,1] => [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,4,1,2] => [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[4,3,2,1] => [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1,5,6,7,4] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,1,5,7,4,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,1,6,4,7,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,1,6,7,5,4] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,1,7,4,5,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,1,7,6,4,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,4,1,6,7,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,4,1,7,5,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,5,6,1,7,4] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,5,7,1,4,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,6,5,7,1,4] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,6,7,4,1,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,7,5,6,4,1] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,3,7,6,4,5,1] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,1,3,6,7,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,1,3,7,5,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,5,1,6,7,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,5,1,7,3,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,5,6,7,1,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,5,7,6,3,1] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,6,1,3,7,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,6,1,7,5,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,6,5,1,7,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,6,7,3,5,1] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,7,1,3,5,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,7,1,6,3,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,7,5,1,3,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,4,7,6,3,1,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,1,6,3,7,4] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,1,7,3,4,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,4,6,1,7,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,4,6,7,3,1] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,4,7,1,3,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,4,7,6,1,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,6,1,4,7,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,6,3,1,7,4] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,6,3,7,4,1] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,6,7,1,4,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,7,1,4,3,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,7,3,1,4,6] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,7,3,6,1,4] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,5,7,6,1,3,4] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,6,1,5,7,3,4] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,6,1,7,4,3,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,6,4,5,3,7,1] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,6,4,5,7,1,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,6,4,7,1,5,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,6,4,7,3,1,5] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,6,5,1,7,4,3] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[2,6,5,3,4,7,1] => [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
Description
The dinv deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$
In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$.
See also [[St000376]] for the bounce deficit.
Matching statistic: St000676
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 69%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 69%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[8,7,6,4,5,3,2,1] => [4,5,6,3,7,2,8,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
[7,8,6,4,5,3,2,1] => [4,5,6,3,8,1,7,2] => [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[8,6,7,4,5,3,2,1] => [4,5,7,2,6,3,8,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 5
[7,6,8,4,5,3,2,1] => [4,5,8,1,7,2,6,3] => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 6
[8,7,5,4,6,3,2,1] => [4,6,3,5,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 5
[8,6,5,4,7,3,2,1] => [4,7,2,6,3,5,8,1] => ?
=> ?
=> ? = 6
[8,5,6,4,7,3,2,1] => [4,6,3,7,2,5,8,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 5
[8,4,5,6,7,3,2,1] => [7,2,4,6,3,5,8,1] => ?
=> ?
=> ? = 7
[7,6,5,4,8,3,2,1] => [4,8,1,7,2,6,3,5] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7
[6,7,5,4,8,3,2,1] => [4,7,2,8,1,6,3,5] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 6
[7,5,6,4,8,3,2,1] => [4,6,3,8,1,7,2,5] => [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 6
[6,5,7,4,8,3,2,1] => [4,8,1,6,3,7,2,5] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7
[5,6,7,4,8,3,2,1] => [4,7,2,6,3,8,1,5] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 6
[5,6,4,7,8,3,2,1] => [7,2,6,3,4,8,1,5] => ?
=> ?
=> ? = 7
[6,4,5,7,8,3,2,1] => [7,2,4,8,1,6,3,5] => ?
=> ?
=> ? = 7
[7,8,5,6,3,4,2,1] => [5,3,6,4,8,1,7,2] => ?
=> ?
=> ? = 6
[8,5,6,7,3,4,2,1] => [7,2,5,3,6,4,8,1] => ?
=> ?
=> ? = 7
[6,5,7,8,3,4,2,1] => [7,2,5,3,8,1,6,4] => ?
=> ?
=> ? = 7
[8,7,6,4,3,5,2,1] => [4,6,5,3,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 5
[7,8,6,4,3,5,2,1] => [4,6,5,3,8,1,7,2] => [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 6
[8,6,7,4,3,5,2,1] => [4,7,2,6,5,3,8,1] => ?
=> ?
=> ? = 6
[7,6,8,4,3,5,2,1] => [4,8,1,7,2,6,5,3] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7
[6,7,8,4,3,5,2,1] => [4,7,2,8,1,6,5,3] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 6
[8,7,5,4,3,6,2,1] => [4,5,3,6,7,2,8,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
[7,8,5,4,3,6,2,1] => [4,5,3,6,8,1,7,2] => [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[8,7,4,5,3,6,2,1] => [5,3,4,6,7,2,8,1] => ?
=> ?
=> ? = 5
[8,7,4,3,5,6,2,1] => [4,3,5,6,7,2,8,1] => ?
=> ?
=> ? = 4
[8,7,3,4,5,6,2,1] => [3,4,5,6,7,2,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 3
[7,8,3,4,5,6,2,1] => [3,4,5,6,8,1,7,2] => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 4
[8,6,5,4,3,7,2,1] => [4,5,3,7,2,6,8,1] => ?
=> ?
=> ? = 5
[8,5,3,4,6,7,2,1] => [3,4,7,2,5,6,8,1] => [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[8,4,3,5,6,7,2,1] => [3,7,2,4,5,6,8,1] => ?
=> ?
=> ? = 6
[7,6,5,4,3,8,2,1] => [4,5,3,8,1,7,2,6] => [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 6
[6,7,5,4,3,8,2,1] => [4,5,3,7,2,8,1,6] => [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 5
[7,5,6,4,3,8,2,1] => [4,8,1,7,2,5,3,6] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7
[6,5,7,4,3,8,2,1] => [4,7,2,5,3,8,1,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 6
[5,6,7,4,3,8,2,1] => [4,8,1,5,3,7,2,6] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7
[7,6,3,4,5,8,2,1] => [3,4,5,8,1,7,2,6] => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5
[7,5,3,4,6,8,2,1] => [3,4,8,1,7,2,5,6] => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 6
[5,4,3,6,7,8,2,1] => [3,8,1,5,7,2,4,6] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[5,3,4,6,7,8,2,1] => [8,1,5,7,2,3,4,6] => ?
=> ?
=> ? = 8
[8,6,5,7,4,2,3,1] => [6,2,7,3,5,4,8,1] => ?
=> ?
=> ? = 6
[7,5,6,8,4,2,3,1] => ? => ?
=> ?
=> ? = 8
[8,7,6,4,5,2,3,1] => [4,5,7,3,6,2,8,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 5
[8,6,7,4,5,2,3,1] => [4,5,6,2,7,3,8,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
[7,6,8,4,5,2,3,1] => [4,5,6,2,8,1,7,3] => ?
=> ?
=> ? = 5
[8,7,5,4,6,2,3,1] => [4,7,3,5,6,2,8,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[8,6,5,4,7,2,3,1] => [4,6,2,7,3,5,8,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 5
[8,5,6,4,7,2,3,1] => [4,7,3,6,2,5,8,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[8,4,5,6,7,2,3,1] => [6,2,4,7,3,5,8,1] => ?
=> ?
=> ? = 6
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: St000093
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 54%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 54%
Values
[1] => {{1}}
=> [1] => ([],1)
=> 1
[1,2] => {{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 1
[2,1] => {{1,2}}
=> [2] => ([],2)
=> 2
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => {{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => {{1,2,3}}
=> [3] => ([],3)
=> 3
[3,1,2] => {{1,2,3}}
=> [3] => ([],3)
=> 3
[3,2,1] => {{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 3
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 3
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => {{1,2,3,4}}
=> [4] => ([],4)
=> 4
[2,4,1,3] => {{1,2,3,4}}
=> [4] => ([],4)
=> 4
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => {{1,2,3,4}}
=> [4] => ([],4)
=> 4
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,4,2,1] => {{1,2,3,4}}
=> [4] => ([],4)
=> 4
[4,1,2,3] => {{1,2,3,4}}
=> [4] => ([],4)
=> 4
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => {{1,2,3,4}}
=> [4] => ([],4)
=> 4
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 4
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 4
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 4
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[7,8,6,5,4,3,2,1] => {{1,2,7,8},{3,6},{4,5}}
=> ? => ?
=> ? = 6
[8,6,7,5,4,3,2,1] => {{1,8},{2,3,6,7},{4,5}}
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[7,6,8,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[6,7,8,5,4,3,2,1] => {{1,3,6,8},{2,7},{4,5}}
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 6
[7,8,5,6,4,3,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[8,6,5,7,4,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 7
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ?
=> ? = 6
[7,6,5,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => ([],8)
=> ? = 8
[6,7,5,8,4,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[7,5,6,8,4,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[6,5,7,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => ([],8)
=> ? = 8
[5,6,7,8,4,3,2,1] => {{1,4,5,8},{2,3,6,7}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? => ?
=> ? = 4
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 6
[8,6,4,5,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 7
[8,4,5,6,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 7
[5,6,7,4,8,3,2,1] => {{1,5,8},{2,3,6,7},{4}}
=> ? => ?
=> ? = 6
[7,6,4,5,8,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => ([],8)
=> ? = 8
[6,7,4,5,8,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[7,5,4,6,8,3,2,1] => {{1,2,5,7,8},{3,4,6}}
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[7,4,5,6,8,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => ([],8)
=> ? = 8
[6,5,4,7,8,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => ([],8)
=> ? = 8
[5,6,4,7,8,3,2,1] => {{1,5,8},{2,3,4,6,7}}
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 7
[6,4,5,7,8,3,2,1] => {{1,3,5,6,8},{2,4,7}}
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[5,4,6,7,8,3,2,1] => {{1,5,8},{2,4,7},{3,6}}
=> ? => ?
=> ? = 6
[4,5,6,7,8,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 6
[7,8,6,5,3,4,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[8,6,7,5,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 7
[7,6,8,5,3,4,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => ([],8)
=> ? = 8
[8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[7,8,5,6,3,4,2,1] => {{1,2,7,8},{3,5},{4,6}}
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[8,5,6,7,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 7
[7,6,5,8,3,4,2,1] => {{1,2,4,6,7,8},{3,5}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[6,5,7,8,3,4,2,1] => {{1,4,6,8},{2,3,5,7}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[5,6,7,8,3,4,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => ([],8)
=> ? = 8
[7,8,6,4,3,5,2,1] => {{1,2,7,8},{3,5,6},{4}}
=> ? => ?
=> ? = 6
[8,7,6,3,4,5,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 6
[7,8,6,3,4,5,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[8,6,7,3,4,5,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 7
[7,6,8,3,4,5,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => ([],8)
=> ? = 8
[6,7,8,3,4,5,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[8,7,5,4,3,6,2,1] => {{1,8},{2,7},{3,5},{4},{6}}
=> ? => ?
=> ? = 4
[7,8,5,3,4,6,2,1] => {{1,2,7,8},{3,4,5},{6}}
=> ? => ?
=> ? = 6
[8,5,4,3,6,7,2,1] => {{1,8},{2,5,6,7},{3,4}}
=> ? => ?
=> ? = 6
[8,3,4,5,6,7,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 7
[6,5,7,4,3,8,2,1] => {{1,6,8},{2,3,5,7},{4}}
=> ? => ?
=> ? = 6
[6,7,4,5,3,8,2,1] => {{1,6,8},{2,7},{3,4,5}}
=> ? => ?
=> ? = 6
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000502The number of successions of a set partitions. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000211The rank of the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000702The number of weak deficiencies of a permutation. St000332The positive inversions of an alternating sign matrix. St000740The last entry of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000316The number of non-left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000216The absolute length of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001480The number of simple summands of the module J^2/J^3. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!