Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00108: Permutations cycle typeInteger partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [1,1]
=> 0
[2,1] => [2]
=> 1
[1,2,3] => [1,1,1]
=> 0
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [3]
=> 1
[3,1,2] => [3]
=> 1
[3,2,1] => [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> 1
[1,3,4,2] => [3,1]
=> 1
[1,4,2,3] => [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> 1
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1]
=> 1
[2,3,4,1] => [4]
=> 1
[2,4,1,3] => [4]
=> 1
[2,4,3,1] => [3,1]
=> 1
[3,1,2,4] => [3,1]
=> 1
[3,1,4,2] => [4]
=> 1
[3,2,1,4] => [2,1,1]
=> 1
[3,2,4,1] => [3,1]
=> 1
[3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [4]
=> 1
[4,1,2,3] => [4]
=> 1
[4,1,3,2] => [3,1]
=> 1
[4,2,1,3] => [3,1]
=> 1
[4,2,3,1] => [2,1,1]
=> 1
[4,3,1,2] => [4]
=> 1
[4,3,2,1] => [2,2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> 1
[1,3,5,2,4] => [4,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000010
Mp00108: Permutations cycle typeInteger partitions
Mp00308: Integer partitions Bulgarian solitaireInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[2,1] => [2]
=> [1,1]
=> 2 = 1 + 1
[1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,3,1] => [3]
=> [2,1]
=> 2 = 1 + 1
[3,1,2] => [3]
=> [2,1]
=> 2 = 1 + 1
[3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[1,4,2,3] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [2,1,1]
=> 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[2,3,4,1] => [4]
=> [3,1]
=> 2 = 1 + 1
[2,4,1,3] => [4]
=> [3,1]
=> 2 = 1 + 1
[2,4,3,1] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[3,1,2,4] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[3,1,4,2] => [4]
=> [3,1]
=> 2 = 1 + 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[3,4,1,2] => [2,2]
=> [2,1,1]
=> 3 = 2 + 1
[3,4,2,1] => [4]
=> [3,1]
=> 2 = 1 + 1
[4,1,2,3] => [4]
=> [3,1]
=> 2 = 1 + 1
[4,1,3,2] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[4,2,1,3] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[4,3,1,2] => [4]
=> [3,1]
=> 2 = 1 + 1
[4,3,2,1] => [2,2]
=> [2,1,1]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 1 = 0 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,4,5,3] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,5,2,4] => [4,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,4,2,5,3] => [4,1]
=> [3,2]
=> 2 = 1 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,4,3,5,2] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
Description
The length of the partition.
Mp00108: Permutations cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> []
=> 0
[1,2] => [1,1]
=> [2]
=> []
=> 0
[2,1] => [2]
=> [1,1]
=> [1]
=> 1
[1,2,3] => [1,1,1]
=> [3]
=> []
=> 0
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000665
Mp00108: Permutations cycle typeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000665: Permutations ⟶ ℤResult quality: 86% values known / values provided: 91%distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[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]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[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]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[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]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? = 2
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 2
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 2
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? = 2
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 5
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 5
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 2
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 3
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 3
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 2
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 3
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 3
[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]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 4
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 3
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 3
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
Description
The number of rafts of a permutation. Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Mp00108: Permutations cycle typeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000834: Permutations ⟶ ℤResult quality: 86% values known / values provided: 91%distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[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]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[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]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[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]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? = 2
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 2
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 2
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? = 2
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 5
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 5
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 2
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 3
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 3
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 2
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 3
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 3
[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]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 4
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 3
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 2
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 3
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[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]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1
Description
The number of right outer peaks of a permutation. A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$. In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000291
Mp00151: Permutations to cycle typeSet partitions
Mp00128: Set partitions to compositionInteger compositions
Mp00094: Integer compositions to binary wordBinary words
St000291: Binary words ⟶ ℤResult quality: 72% values known / values provided: 72%distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1] => 1 => 0
[1,2] => {{1},{2}}
=> [1,1] => 11 => 0
[2,1] => {{1,2}}
=> [2] => 10 => 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => 111 => 0
[1,3,2] => {{1},{2,3}}
=> [1,2] => 110 => 1
[2,1,3] => {{1,2},{3}}
=> [2,1] => 101 => 1
[2,3,1] => {{1,2,3}}
=> [3] => 100 => 1
[3,1,2] => {{1,2,3}}
=> [3] => 100 => 1
[3,2,1] => {{1,3},{2}}
=> [2,1] => 101 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => 1110 => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => 1101 => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => 1100 => 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => 1100 => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => 1101 => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => 1011 => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => 1010 => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => 1001 => 1
[2,3,4,1] => {{1,2,3,4}}
=> [4] => 1000 => 1
[2,4,1,3] => {{1,2,3,4}}
=> [4] => 1000 => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => 1001 => 1
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => 1001 => 1
[3,1,4,2] => {{1,2,3,4}}
=> [4] => 1000 => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => 1011 => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => 1001 => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => 1010 => 2
[3,4,2,1] => {{1,2,3,4}}
=> [4] => 1000 => 1
[4,1,2,3] => {{1,2,3,4}}
=> [4] => 1000 => 1
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => 1001 => 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => 1001 => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
[4,3,1,2] => {{1,2,3,4}}
=> [4] => 1000 => 1
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => 1010 => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 11111 => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => 11110 => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 11101 => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => 11100 => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => 11100 => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 11101 => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 11011 => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => 11010 => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 11001 => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => 11001 => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 11001 => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 11011 => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => 11001 => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => 11010 => 2
[7,8,6,5,4,3,2,1] => {{1,2,7,8},{3,6},{4,5}}
=> ? => ? => ? = 3
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ? => ? = 3
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ? => ? = 3
[8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? => ? => ? = 3
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ? => ? = 3
[5,6,7,4,8,3,2,1] => {{1,5,8},{2,3,6,7},{4}}
=> ? => ? => ? = 2
[5,4,6,7,8,3,2,1] => {{1,5,8},{2,4,7},{3,6}}
=> ? => ? => ? = 3
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ? => ? = 3
[7,8,6,4,3,5,2,1] => {{1,2,7,8},{3,5,6},{4}}
=> ? => ? => ? = 2
[8,7,6,3,4,5,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ? => ? = 3
[8,7,5,4,3,6,2,1] => {{1,8},{2,7},{3,5},{4},{6}}
=> ? => ? => ? = 3
[7,8,5,3,4,6,2,1] => {{1,2,7,8},{3,4,5},{6}}
=> ? => ? => ? = 2
[8,5,4,3,6,7,2,1] => {{1,8},{2,5,6,7},{3,4}}
=> ? => ? => ? = 3
[6,5,7,4,3,8,2,1] => {{1,6,8},{2,3,5,7},{4}}
=> ? => ? => ? = 2
[6,7,4,5,3,8,2,1] => {{1,6,8},{2,7},{3,4,5}}
=> ? => ? => ? = 3
[7,6,4,3,5,8,2,1] => {{1,2,6,7,8},{3,4},{5}}
=> ? => ? => ? = 2
[7,6,8,5,4,2,3,1] => {{1,3,7,8},{2,6},{4,5}}
=> ? => ? => ? = 3
[8,6,5,7,4,2,3,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ? => ? = 3
[8,6,7,4,5,2,3,1] => {{1,8},{2,6},{3,7},{4},{5}}
=> ? => ? => ? = 3
[8,6,4,5,7,2,3,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ? => ? = 3
[8,4,5,6,7,2,3,1] => {{1,8},{2,4,6},{3,5,7}}
=> ? => ? => ? = 3
[7,6,5,4,8,2,3,1] => {{1,3,5,7,8},{2,6},{4}}
=> ? => ? => ? = 2
[8,6,7,5,3,2,4,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ? => ? = 3
[8,7,5,6,3,2,4,1] => {{1,8},{2,4,6,7},{3,5}}
=> ? => ? => ? = 3
[8,5,6,7,3,2,4,1] => {{1,8},{2,3,5,6},{4,7}}
=> ? => ? => ? = 3
[8,7,6,5,2,3,4,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ? => ? = 3
[8,6,5,7,2,3,4,1] => {{1,8},{2,3,5,6},{4,7}}
=> ? => ? => ? = 3
[7,6,8,4,3,2,5,1] => {{1,3,5,7,8},{2,6},{4}}
=> ? => ? => ? = 2
[8,6,7,3,4,2,5,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ? => ? = 3
[8,7,6,3,2,4,5,1] => {{1,8},{2,5,7},{3,4,6}}
=> ? => ? => ? = 3
[8,6,7,2,3,4,5,1] => {{1,8},{2,4,6},{3,5,7}}
=> ? => ? => ? = 3
[8,7,5,3,4,2,6,1] => {{1,8},{2,6,7},{3,4,5}}
=> ? => ? => ? = 3
[7,8,4,3,5,2,6,1] => {{1,2,6,7,8},{3,4},{5}}
=> ? => ? => ? = 2
[8,7,4,3,2,5,6,1] => {{1,8},{2,5,6,7},{3,4}}
=> ? => ? => ? = 3
[8,6,5,4,3,2,7,1] => {{1,8},{2,6},{3,5},{4},{7}}
=> ? => ? => ? = 3
[8,6,3,4,5,2,7,1] => {{1,8},{2,6},{3},{4},{5},{7}}
=> ? => ? => ? = 2
[8,5,4,3,2,6,7,1] => {{1,8},{2,5},{3,4},{6},{7}}
=> ? => ? => ? = 3
[8,3,4,5,2,6,7,1] => {{1,8},{2,3,4,5},{6},{7}}
=> ? => ? => ? = 2
[8,5,4,2,3,6,7,1] => {{1,8},{2,3,4,5},{6},{7}}
=> ? => ? => ? = 2
[8,4,5,2,3,6,7,1] => {{1,8},{2,4},{3,5},{6},{7}}
=> ? => ? => ? = 3
[7,5,6,4,3,2,8,1] => {{1,7,8},{2,3,5,6},{4}}
=> ? => ? => ? = 2
[5,6,7,4,3,2,8,1] => {{1,3,5,7,8},{2,6},{4}}
=> ? => ? => ? = 2
[5,4,3,6,7,2,8,1] => {{1,5,7,8},{2,4,6},{3}}
=> ? => ? => ? = 2
[7,6,5,4,2,3,8,1] => {{1,7,8},{2,3,5,6},{4}}
=> ? => ? => ? = 2
[7,5,4,2,3,6,8,1] => {{1,7,8},{2,3,4,5},{6}}
=> ? => ? => ? = 2
[7,5,2,3,4,6,8,1] => {{1,7,8},{2,3,4,5},{6}}
=> ? => ? => ? = 2
[7,4,3,2,5,6,8,1] => {{1,7,8},{2,4},{3},{5},{6}}
=> ? => ? => ? = 2
[7,3,2,4,5,6,8,1] => {{1,7,8},{2,3},{4},{5},{6}}
=> ? => ? => ? = 2
[6,4,5,2,3,7,8,1] => {{1,6,7,8},{2,4},{3,5}}
=> ? => ? => ? = 3
[6,5,3,2,4,7,8,1] => {{1,6,7,8},{2,4,5},{3}}
=> ? => ? => ? = 2
Description
The number of descents of a binary word.
Matching statistic: St001418
Mp00108: Permutations cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
St001418: Dyck paths ⟶ ℤResult quality: 59% values known / values provided: 59%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> ? = 0
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 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
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 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
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 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
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 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
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,4,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,1,2,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 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
[4,3,1,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[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
[1,2,3,5,4] => [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
[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
[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]
=> 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]
=> 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
[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
[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
[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]
=> 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]
=> 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]
=> 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]
=> 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]
=> 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]
=> 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
[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]
=> 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
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[7,6,5,8,4,3,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,5,7,8,4,3,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,6,8,4,5,3,2,1] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,6,5,4,8,3,2,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[6,5,7,4,8,3,2,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,6,4,5,8,3,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,4,5,6,8,3,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,5,4,7,8,3,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,6,8,5,3,4,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,6,7,8,3,4,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,6,8,4,3,5,2,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,6,8,3,4,5,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,3,4,5,6,2,1] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7,5,6,4,3,8,2,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[5,6,7,4,3,8,2,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,5,4,6,3,8,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[4,5,6,7,3,8,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,6,3,4,5,8,2,1] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[7,5,3,4,6,8,2,1] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,3,4,5,6,8,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,4,3,6,7,8,2,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[5,3,4,6,7,8,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[3,4,5,6,7,8,2,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,5,6,8,4,2,3,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,4,5,6,2,3,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,5,6,4,8,2,3,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[4,5,6,7,8,2,3,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,6,5,3,2,4,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,8,5,3,2,4,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,6,8,5,2,3,4,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,5,6,2,3,4,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,5,8,2,3,4,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,6,7,8,2,3,4,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,6,4,3,2,5,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[6,7,8,4,3,2,5,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,8,6,3,4,2,5,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,8,3,4,2,5,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,6,8,4,2,3,5,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,8,6,2,3,4,5,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,8,2,3,4,5,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,3,4,2,5,6,1] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,8,4,2,3,5,6,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,3,2,4,5,6,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,8,2,3,4,5,6,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,2,3,4,5,6,7,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[6,5,7,4,3,2,8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[6,5,4,7,3,2,8,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,5,7,3,4,2,8,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,4,5,3,7,2,8,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Mp00108: Permutations cycle typeInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000251: Set partitions ⟶ ℤResult quality: 44% values known / values provided: 44%distinct values known / distinct values provided: 57%
Values
[1] => [1]
=> [[1]]
=> {{1}}
=> ? = 0
[1,2] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[2,1] => [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,3,1] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 1
[3,1,2] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 1
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 2
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 2
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[8,7,6,5,4,3,2,1] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 4
[7,8,6,5,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[8,6,7,5,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[6,7,8,5,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[8,7,5,6,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[7,8,5,6,4,3,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[8,5,6,7,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[5,6,7,8,4,3,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[8,7,6,4,5,3,2,1] => [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 3
[7,8,6,4,5,3,2,1] => [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 2
[8,6,7,4,5,3,2,1] => [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 2
[8,7,5,4,6,3,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 3
[8,7,4,5,6,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[8,6,5,4,7,3,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 2
[8,5,6,4,7,3,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 3
[6,7,5,4,8,3,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 2
[7,5,6,4,8,3,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 2
[5,6,7,4,8,3,2,1] => [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 2
[7,5,4,6,8,3,2,1] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[5,6,4,7,8,3,2,1] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[6,4,5,7,8,3,2,1] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[5,4,6,7,8,3,2,1] => [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? = 3
[8,7,6,5,3,4,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[7,8,6,5,3,4,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[8,7,5,6,3,4,2,1] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 4
[7,8,5,6,3,4,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[6,5,7,8,3,4,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[8,7,6,4,3,5,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 3
[7,8,6,4,3,5,2,1] => [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 2
[8,6,7,4,3,5,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 2
[6,7,8,4,3,5,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 2
[8,7,6,3,4,5,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[7,8,6,3,4,5,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[8,7,5,4,3,6,2,1] => [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 3
[7,8,5,4,3,6,2,1] => [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 2
[8,7,4,5,3,6,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 3
[7,8,5,3,4,6,2,1] => [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 2
[8,7,4,3,5,6,2,1] => [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 3
[8,7,3,4,5,6,2,1] => [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> ? = 2
[7,8,3,4,5,6,2,1] => [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 1
[8,6,5,4,3,7,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 3
[8,6,4,3,5,7,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 3
[8,5,4,3,6,7,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 3
[8,5,3,4,6,7,2,1] => [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 2
[8,4,3,5,6,7,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 2
[7,6,5,4,3,8,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 2
[6,7,5,4,3,8,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 3
[6,5,7,4,3,8,2,1] => [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 2
[6,7,4,5,3,8,2,1] => [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? = 3
Description
The number of nonsingleton blocks of a set partition.
Mp00151: Permutations to cycle typeSet partitions
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000659: Dyck paths ⟶ ℤResult quality: 29% values known / values provided: 29%distinct values known / distinct values provided: 71%
Values
[1] => {{1}}
=> [1] => [1,0]
=> ? = 0
[1,2] => {{1},{2}}
=> [1,1] => [1,0,1,0]
=> 0
[2,1] => {{1,2}}
=> [2] => [1,1,0,0]
=> 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => {{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => {{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 1
[3,1,2] => {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 1
[3,2,1] => {{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,2,3] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,3,1,2] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[7,8,6,5,4,3,2,1] => {{1,2,7,8},{3,6},{4,5}}
=> ? => ?
=> ? = 3
[7,6,8,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[6,7,8,5,4,3,2,1] => {{1,3,6,8},{2,7},{4,5}}
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 3
[7,8,5,6,4,3,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ?
=> ? = 3
[7,6,5,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,5,8,4,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[7,5,6,8,4,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[6,5,7,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,6,7,8,4,3,2,1] => {{1,4,5,8},{2,3,6,7}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? => ?
=> ? = 3
[7,8,6,4,5,3,2,1] => {{1,2,7,8},{3,6},{4},{5}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[7,6,8,4,5,3,2,1] => {{1,2,3,6,7,8},{4},{5}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 3
[7,6,5,4,8,3,2,1] => {{1,2,3,5,6,7,8},{4}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[6,7,5,4,8,3,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[7,5,6,4,8,3,2,1] => {{1,2,5,7,8},{3,6},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[6,5,7,4,8,3,2,1] => {{1,2,3,5,6,7,8},{4}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[5,6,7,4,8,3,2,1] => {{1,5,8},{2,3,6,7},{4}}
=> ? => ?
=> ? = 2
[7,6,4,5,8,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,4,5,8,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[7,5,4,6,8,3,2,1] => {{1,2,5,7,8},{3,4,6}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[7,4,5,6,8,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,5,4,7,8,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,6,4,7,8,3,2,1] => {{1,5,8},{2,3,4,6,7}}
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[6,4,5,7,8,3,2,1] => {{1,3,5,6,8},{2,4,7}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[5,4,6,7,8,3,2,1] => {{1,5,8},{2,4,7},{3,6}}
=> ? => ?
=> ? = 3
[4,5,6,7,8,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 3
[7,8,6,5,3,4,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[7,6,8,5,3,4,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,5,6,3,4,2,1] => {{1,2,7,8},{3,5},{4,6}}
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[7,6,5,8,3,4,2,1] => {{1,2,4,6,7,8},{3,5}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[6,5,7,8,3,4,2,1] => {{1,4,6,8},{2,3,5,7}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[5,6,7,8,3,4,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,6,4,3,5,2,1] => {{1,2,7,8},{3,5,6},{4}}
=> ? => ?
=> ? = 2
[7,6,8,4,3,5,2,1] => {{1,2,3,5,6,7,8},{4}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[6,7,8,4,3,5,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[8,7,6,3,4,5,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 3
[7,8,6,3,4,5,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[7,6,8,3,4,5,2,1] => {{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,8,3,4,5,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[8,7,5,4,3,6,2,1] => {{1,8},{2,7},{3,5},{4},{6}}
=> ? => ?
=> ? = 3
[7,8,5,4,3,6,2,1] => {{1,2,7,8},{3,5},{4},{6}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[7,8,5,3,4,6,2,1] => {{1,2,7,8},{3,4,5},{6}}
=> ? => ?
=> ? = 2
[7,8,3,4,5,6,2,1] => {{1,2,7,8},{3},{4},{5},{6}}
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,5,4,3,6,7,2,1] => {{1,8},{2,5,6,7},{3,4}}
=> ? => ?
=> ? = 3
[7,6,5,4,3,8,2,1] => {{1,2,6,7,8},{3,5},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000884
Mp00151: Permutations to cycle typeSet partitions
Mp00080: Set partitions to permutationPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
St000884: Permutations ⟶ ℤResult quality: 28% values known / values provided: 28%distinct values known / distinct values provided: 86%
Values
[1] => {{1}}
=> [1] => [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 1
[3,1,2] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 1
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => [2,3,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 1
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 2
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 1
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => 2
[1,3,4,5,2,6,7] => {{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,3,4,5,6,2,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,3,4,6,2,5,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,3,4,6,5,2,7] => {{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ? = 1
[1,3,4,7,6,5,2] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,3,5,2,4,6,7] => {{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,3,5,2,6,4,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,3,5,6,2,4,7] => {{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => [1,5,2,3,6,4,7] => ? = 2
[1,3,5,6,2,7,4] => {{1},{2,3,5},{4,6,7}}
=> [1,3,5,6,2,7,4] => [1,5,2,3,7,4,6] => ? = 2
[1,3,5,6,4,2,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,3,5,6,7,4,2] => {{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => ? = 2
[1,3,5,7,2,4,6] => {{1},{2,3,5},{4,6,7}}
=> [1,3,5,6,2,7,4] => [1,5,2,3,7,4,6] => ? = 2
[1,3,5,7,2,6,4] => {{1},{2,3,5},{4,7},{6}}
=> [1,3,5,7,2,6,4] => [1,5,2,3,6,7,4] => ? = 2
[1,3,5,7,6,2,4] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,3,6,2,4,5,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,3,6,2,5,4,7] => {{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ? = 1
[1,3,6,5,2,4,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,3,6,5,4,2,7] => {{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => [1,5,4,6,2,3,7] => ? = 2
[1,3,6,5,4,7,2] => {{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => ? = 2
[1,3,6,7,2,5,4] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,3,6,7,5,2,4] => {{1},{2,3,6},{4,7},{5}}
=> [1,3,6,7,5,2,4] => [1,5,6,2,3,7,4] => ? = 2
[1,3,7,2,6,5,4] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,3,7,4,6,5,2] => {{1},{2,3,7},{4},{5,6}}
=> [1,3,7,4,6,5,2] => [1,4,6,5,7,2,3] => ? = 2
[1,3,7,5,4,2,6] => {{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => ? = 2
[1,3,7,5,4,6,2] => {{1},{2,3,7},{4,5},{6}}
=> [1,3,7,5,4,6,2] => [1,5,4,6,7,2,3] => ? = 2
[1,3,7,5,6,4,2] => {{1},{2,3,7},{4,5,6}}
=> [1,3,7,5,6,4,2] => [1,6,4,5,7,2,3] => ? = 2
[1,3,7,6,2,4,5] => {{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => ? = 2
[1,3,7,6,4,5,2] => {{1},{2,3,7},{4,5,6}}
=> [1,3,7,5,6,4,2] => [1,6,4,5,7,2,3] => ? = 2
[1,3,7,6,5,4,2] => {{1},{2,3,7},{4,6},{5}}
=> [1,3,7,6,5,4,2] => [1,5,6,4,7,2,3] => ? = 2
[1,4,2,5,3,6,7] => {{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,4,2,5,6,3,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,4,2,6,3,5,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,4,2,6,5,3,7] => {{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ? = 1
[1,4,2,7,6,5,3] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,4,5,3,2,6,7] => {{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,4,5,3,6,2,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,4,5,6,2,3,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,4,5,6,3,2,7] => {{1},{2,4,6},{3,5},{7}}
=> [1,4,5,6,3,2,7] => [1,5,3,6,2,4,7] => ? = 2
[1,4,5,6,3,7,2] => {{1},{2,4,6,7},{3,5}}
=> [1,4,5,6,3,7,2] => [1,5,3,7,2,4,6] => ? = 2
[1,4,5,6,7,2,3] => {{1},{2,4,6},{3,5,7}}
=> [1,4,5,6,7,2,3] => [1,6,2,4,7,3,5] => ? = 2
[1,4,5,7,3,2,6] => {{1},{2,4,6,7},{3,5}}
=> [1,4,5,6,3,7,2] => [1,5,3,7,2,4,6] => ? = 2
[1,4,5,7,3,6,2] => {{1},{2,4,7},{3,5},{6}}
=> [1,4,5,7,3,6,2] => [1,5,3,6,7,2,4] => ? = 2
[1,4,5,7,6,3,2] => {{1},{2,4,7},{3,5,6}}
=> [1,4,5,7,6,3,2] => [1,6,3,5,7,2,4] => ? = 2
[1,4,6,3,2,5,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,4,6,3,5,2,7] => {{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ? = 1
[1,4,6,5,2,3,7] => {{1},{2,4,5},{3,6},{7}}
=> [1,4,6,5,2,3,7] => [1,5,2,4,6,3,7] => ? = 2
[1,4,6,5,2,7,3] => {{1},{2,4,5},{3,6,7}}
=> [1,4,6,5,2,7,3] => [1,5,2,4,7,3,6] => ? = 2
[1,4,6,5,3,2,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,4,6,5,7,3,2] => {{1},{2,4,5,7},{3,6}}
=> [1,4,6,5,7,3,2] => [1,6,3,7,2,4,5] => ? = 2
[1,4,6,7,2,3,5] => {{1},{2,4,5,7},{3,6}}
=> [1,4,6,5,7,3,2] => [1,6,3,7,2,4,5] => ? = 2
Description
The number of isolated descents of a permutation. A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000035The number of left outer peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000703The number of deficiencies of a permutation. St000245The number of ascents of a permutation. St001737The number of descents of type 2 in a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000325The width of the tree associated to a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001905The number of preferred parking spots in a parking function less than the index of the car. St001597The Frobenius rank of a skew partition. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice.