Your data matches 17 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000059
Mp00079: Set partitions shapeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000059: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [[1]]
=> 0
{{1,2}}
=> [2]
=> [[1,2]]
=> 0
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> 1
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> 0
{{1,2},{3}}
=> [2,1]
=> [[1,2],[3]]
=> 2
{{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> 2
{{1},{2,3}}
=> [2,1]
=> [[1,2],[3]]
=> 2
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
{{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 4
{{1,2},{3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 5
{{1,3,4},{2}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 4
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 5
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> 4
{{1},{2,3,4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 5
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 5
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 5
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 6
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 6
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 6
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 6
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 6
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 8
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 8
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 9
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 6
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 6
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 6
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 8
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 8
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 9
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 6
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 6
Description
The inversion number of a standard tableau as defined by Haglund and Stevens. Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].
Matching statistic: St001541
Mp00079: Set partitions shapeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St001541: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1]
=> 0
{{1,2}}
=> [2]
=> [1,1]
=> 0
{{1},{2}}
=> [1,1]
=> [2]
=> 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> 0
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> 2
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> 2
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> 2
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 3
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 3
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 3
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 4
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 5
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 3
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> 4
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 5
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 4
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 3
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 5
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 5
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 5
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 6
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> 4
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> 4
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> 6
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 7
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> 4
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> 6
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> 7
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> 6
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 6
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> 8
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> 7
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> 8
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 9
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> 4
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> 6
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> 7
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> 6
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> 6
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> 8
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> 7
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> 8
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 9
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> 6
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> 6
Description
The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.
Matching statistic: St000009
Mp00079: Set partitions shapeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1]
=> [[1]]
=> 0
{{1,2}}
=> [2]
=> [1,1]
=> [[1],[2]]
=> 0
{{1},{2}}
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 4
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 5
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 4
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 5
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 4
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 5
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 5
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 5
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 6
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 6
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 6
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 6
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 6
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 8
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 8
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 9
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 6
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 6
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 6
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 8
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 7
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 8
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 9
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 6
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 6
Description
The charge of a standard tableau.
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000018: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 9
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 9
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000228
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> []
=> 0
{{1,2}}
=> [2] => [1,1,0,0]
=> []
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1]
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> []
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2]
=> 2
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2]
=> 2
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> []
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 4
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 4
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 4
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 5
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 5
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 7
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 7
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 6
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 8
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 7
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 8
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 9
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 7
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 6
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 8
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 7
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 8
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 9
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 6
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000246
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
St000246: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 6
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 7
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 6
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 7
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 6
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 6
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 8
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 7
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 8
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 9
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 6
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 7
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 6
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 6
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 8
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 7
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 8
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 9
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 6
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 6
Description
The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000426
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000426: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [2,1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,3,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [3,1,2] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 5
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 5
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 5
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 6
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 8
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 8
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 9
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 6
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 8
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 8
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 9
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 6
Description
The number of occurrences of the pattern 132 or of the pattern 312 in a permutation.
Matching statistic: St000434
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000434: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [2,1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,3,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [3,1,2] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 5
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 5
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 5
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 6
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 8
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 8
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 9
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 6
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 8
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 7
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 8
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 9
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 6
Description
The number of occurrences of the pattern 213 or of the pattern 312 in a permutation.
Matching statistic: St000719
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
St000719: Perfect matchings ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [(1,2)]
=> 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 4
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 5
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 4
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 5
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 4
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 5
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 5
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 5
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 6
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 7
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 6
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 7
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 6
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 6
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 8
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 7
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 8
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 9
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 6
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 7
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 6
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 6
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 8
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 7
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 8
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 9
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 6
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 6
Description
The number of alignments in a perfect matching. An alignment is a pair of edges $(i,j)$, $(k,l)$ such that $i < j < k < l$. Since any two edges in a perfect matching are either nesting ([[St000041]]), crossing ([[St000042]]) or form an alignment, the sum of these numbers in a perfect matching with $n$ edges is $\binom{n}{2}$.
Mp00128: Set partitions to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St001759: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 9
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 9
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
Description
The Rajchgot index of a permutation. The '''Rajchgot index''' of a permutation $\sigma$ is the degree of the ''Grothendieck polynomial'' of $\sigma$. This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index [[St000004]] of the permutations smaller than or equal to $\sigma$ in the right ''weak Bruhat order''.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000567The sum of the products of all pairs of parts. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph.