searching the database
Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000564
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
St000564: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 1
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 2
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 3
{{1,2},{3,4}}
=> 4
{{1,2},{3},{4}}
=> 5
{{1,3,4},{2}}
=> 3
{{1,3},{2,4}}
=> 4
{{1,3},{2},{4}}
=> 5
{{1,4},{2,3}}
=> 4
{{1},{2,3,4}}
=> 3
{{1},{2,3},{4}}
=> 5
{{1,4},{2},{3}}
=> 5
{{1},{2,4},{3}}
=> 5
{{1},{2},{3,4}}
=> 5
{{1},{2},{3},{4}}
=> 6
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 4
{{1,2,3},{4,5}}
=> 6
{{1,2,3},{4},{5}}
=> 7
{{1,2,4,5},{3}}
=> 4
{{1,2,4},{3,5}}
=> 6
{{1,2,4},{3},{5}}
=> 7
{{1,2,5},{3,4}}
=> 6
{{1,2},{3,4,5}}
=> 6
{{1,2},{3,4},{5}}
=> 8
{{1,2,5},{3},{4}}
=> 7
{{1,2},{3,5},{4}}
=> 8
{{1,2},{3},{4,5}}
=> 8
{{1,2},{3},{4},{5}}
=> 9
{{1,3,4,5},{2}}
=> 4
{{1,3,4},{2,5}}
=> 6
{{1,3,4},{2},{5}}
=> 7
{{1,3,5},{2,4}}
=> 6
{{1,3},{2,4,5}}
=> 6
{{1,3},{2,4},{5}}
=> 8
{{1,3,5},{2},{4}}
=> 7
{{1,3},{2,5},{4}}
=> 8
{{1,3},{2},{4,5}}
=> 8
{{1,3},{2},{4},{5}}
=> 9
{{1,4,5},{2,3}}
=> 6
{{1,4},{2,3,5}}
=> 6
{{1,4},{2,3},{5}}
=> 8
Description
The number of occurrences of the pattern {{1},{2}} in a set partition.
Matching statistic: St000567
Mp00079: Set partitions —shape⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
St000567: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> 0
{{1},{2}}
=> [1,1]
=> 1
{{1,2,3}}
=> [3]
=> 0
{{1,2},{3}}
=> [2,1]
=> 2
{{1,3},{2}}
=> [2,1]
=> 2
{{1},{2,3}}
=> [2,1]
=> 2
{{1},{2},{3}}
=> [1,1,1]
=> 3
{{1,2,3,4}}
=> [4]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> 3
{{1,2,4},{3}}
=> [3,1]
=> 3
{{1,2},{3,4}}
=> [2,2]
=> 4
{{1,2},{3},{4}}
=> [2,1,1]
=> 5
{{1,3,4},{2}}
=> [3,1]
=> 3
{{1,3},{2,4}}
=> [2,2]
=> 4
{{1,3},{2},{4}}
=> [2,1,1]
=> 5
{{1,4},{2,3}}
=> [2,2]
=> 4
{{1},{2,3,4}}
=> [3,1]
=> 3
{{1},{2,3},{4}}
=> [2,1,1]
=> 5
{{1,4},{2},{3}}
=> [2,1,1]
=> 5
{{1},{2,4},{3}}
=> [2,1,1]
=> 5
{{1},{2},{3,4}}
=> [2,1,1]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 6
{{1,2,3,4,5}}
=> [5]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> 4
{{1,2,3,5},{4}}
=> [4,1]
=> 4
{{1,2,3},{4,5}}
=> [3,2]
=> 6
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 7
{{1,2,4,5},{3}}
=> [4,1]
=> 4
{{1,2,4},{3,5}}
=> [3,2]
=> 6
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 7
{{1,2,5},{3,4}}
=> [3,2]
=> 6
{{1,2},{3,4,5}}
=> [3,2]
=> 6
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 8
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 7
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 8
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 9
{{1,3,4,5},{2}}
=> [4,1]
=> 4
{{1,3,4},{2,5}}
=> [3,2]
=> 6
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 7
{{1,3,5},{2,4}}
=> [3,2]
=> 6
{{1,3},{2,4,5}}
=> [3,2]
=> 6
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 8
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 7
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 8
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 9
{{1,4,5},{2,3}}
=> [3,2]
=> 6
{{1,4},{2,3,5}}
=> [3,2]
=> 6
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 8
{{1},{2,3,5,6,7,8},{4}}
=> ?
=> ? = 13
{{1},{2,3,4,5,6,7},{8}}
=> ?
=> ? = 13
{{1,4,5,6,7,8},{2},{3}}
=> ?
=> ? = 13
{{1,3,5,6,7,8},{2},{4}}
=> ?
=> ? = 13
{{1,2,5,6,7,8},{3,4}}
=> ?
=> ? = 12
{{1,2,3,4,7,8},{5,6}}
=> ?
=> ? = 12
{{1,3,4,5,7,8},{2,6}}
=> ?
=> ? = 12
{{1,3,4,5,6,8},{2,7}}
=> ?
=> ? = 12
{{1,3,4,5,6,7},{2,8}}
=> ?
=> ? = 12
{{1,2,3,5,6,7},{4,8}}
=> ?
=> ? = 12
{{1,2,3,4,6,7},{5,8}}
=> ?
=> ? = 12
{{1,2,3,4,5,7},{6,8}}
=> ?
=> ? = 12
{{1,7},{2,3,4,5,6,8}}
=> ?
=> ? = 12
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000059
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000059: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000059: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{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
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 8
{{1},{2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ? = 13
{{1},{2,3,4,5,6,7},{8}}
=> ?
=> ?
=> ? = 13
{{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 13
{{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 13
{{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 12
{{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 12
{{1,3,4,5,7,8},{2,6}}
=> ?
=> ?
=> ? = 12
{{1,3,4,5,6,8},{2,7}}
=> ?
=> ?
=> ? = 12
{{1,3,4,5,6,7},{2,8}}
=> ?
=> ?
=> ? = 12
{{1,2,3,5,6,7},{4,8}}
=> ?
=> ?
=> ? = 12
{{1,2,3,4,6,7},{5,8}}
=> ?
=> ?
=> ? = 12
{{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ? = 12
{{1,7},{2,3,4,5,6,8}}
=> ?
=> ?
=> ? = 12
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 —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001541: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001541: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{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
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> 8
{{1},{2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ? = 13
{{1},{2,3,4,5,6,7},{8}}
=> ?
=> ?
=> ? = 13
{{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 13
{{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 13
{{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 12
{{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 12
{{1,3,4,5,7,8},{2,6}}
=> ?
=> ?
=> ? = 12
{{1,3,4,5,6,8},{2,7}}
=> ?
=> ?
=> ? = 12
{{1,3,4,5,6,7},{2,8}}
=> ?
=> ?
=> ? = 12
{{1,2,3,5,6,7},{4,8}}
=> ?
=> ?
=> ? = 12
{{1,2,3,4,6,7},{5,8}}
=> ?
=> ?
=> ? = 12
{{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ? = 12
{{1,7},{2,3,4,5,6,8}}
=> ?
=> ?
=> ? = 12
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 —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{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
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 8
{{1},{2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ?
=> ? = 13
{{1},{2,3,4,5,6,7},{8}}
=> ?
=> ?
=> ?
=> ? = 13
{{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ?
=> ? = 13
{{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ?
=> ? = 13
{{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ?
=> ? = 12
{{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ?
=> ? = 12
{{1,3,4,5,7,8},{2,6}}
=> ?
=> ?
=> ?
=> ? = 12
{{1,3,4,5,6,8},{2,7}}
=> ?
=> ?
=> ?
=> ? = 12
{{1,3,4,5,6,7},{2,8}}
=> ?
=> ?
=> ?
=> ? = 12
{{1,2,3,5,6,7},{4,8}}
=> ?
=> ?
=> ?
=> ? = 12
{{1,2,3,4,6,7},{5,8}}
=> ?
=> ?
=> ?
=> ? = 12
{{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ?
=> ? = 12
{{1,7},{2,3,4,5,6,8}}
=> ?
=> ?
=> ?
=> ? = 12
Description
The charge of a standard tableau.
Matching statistic: St000018
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [[1,2]]
=> [1,2] => 0
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
{{1,2},{3}}
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
{{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
{{1},{2,3}}
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
{{1,2,3},{4}}
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 3
{{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 3
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 4
{{1,2},{3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 5
{{1,3,4},{2}}
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 3
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 4
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 5
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 4
{{1},{2,3,4}}
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 3
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 5
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 5
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 5
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 6
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 6
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 7
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 6
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 7
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 6
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 6
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 8
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 7
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 8
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 8
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 9
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 6
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 7
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 6
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 6
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 8
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 7
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 8
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 8
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 9
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 6
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 6
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 8
{{1},{2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ? => ? = 13
{{1},{2,3,4,5,6,7},{8}}
=> ?
=> ?
=> ? => ? = 13
{{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? => ? = 13
{{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? => ? = 13
{{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? => ? = 12
{{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? => ? = 12
{{1,3,4,5,7,8},{2,6}}
=> ?
=> ?
=> ? => ? = 12
{{1,3,4,5,6,8},{2,7}}
=> ?
=> ?
=> ? => ? = 12
{{1,3,4,5,6,7},{2,8}}
=> ?
=> ?
=> ? => ? = 12
{{1,2,3,5,6,7},{4,8}}
=> ?
=> ?
=> ? => ? = 12
{{1,2,3,4,6,7},{5,8}}
=> ?
=> ?
=> ? => ? = 12
{{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ? => ? = 12
{{1,7},{2,3,4,5,6,8}}
=> ?
=> ?
=> ? => ? = 12
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: St000246
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 86%
Values
{{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
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 8
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 12
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 14
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 12
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 14
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 12
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 12
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 12
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 15
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 15
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 16
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 17
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 14
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 15
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 16
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 17
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 14
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 16
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 15
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 17
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 17
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 17
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => ? = 17
{{1,2,4,5},{3,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 12
{{1,2,4,5},{3,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,4,5},{3,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,4,5},{3},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 14
{{1,2,4,6},{3,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 12
{{1,2,4,6},{3,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,4,7},{3,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 12
{{1,2,4},{3,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 12
{{1,2,4},{3,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 15
{{1,2,4,7},{3,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,4},{3,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 15
{{1,2,4},{3,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 16
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 17
{{1,2,4,6},{3,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,4,6},{3},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 14
{{1,2,4,7},{3,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 14
{{1,2,4},{3,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 15
{{1,2,4},{3,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 16
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 17
{{1,2,4,7},{3},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 14
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: St001759
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 86%
Values
{{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
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
{{1,2,3,4,5,6},{7}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 6
{{1,2,3,4,5,7},{6}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 6
{{1,2,3,4,5},{6,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 11
{{1,2,3,4,6,7},{5}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 6
{{1,2,3,4,6},{5,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 11
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 14
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 11
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 14
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 14
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 15
{{1,2,3,5,6,7},{4}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 6
{{1,2,3,5,6},{4,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 11
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 14
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 11
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 14
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 14
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 15
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 14
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ? = 15
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 14
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ? = 15
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ? = 16
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ? = 17
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 11
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 14
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 14
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 15
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 14
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ? = 15
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ? = 16
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ? = 17
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 14
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ? = 16
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,1,2,3] => ? = 17
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 15
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ? = 17
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,1,2,3] => ? = 17
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,1,2,3] => ? = 17
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1,2,3] => ? = 18
{{1,2,4,5,6,7},{3}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 6
{{1,2,4,5,6},{3,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,4,5,6},{3},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 11
{{1,2,4,5,7},{3,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,4,5},{3,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 14
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''.
Matching statistic: St000228
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Values
{{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
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 8
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 11
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 12
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,2,4},{3},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 11
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 12
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 11
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 11
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 12
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 13
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,2,5},{3},{4,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 11
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 12
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 11
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 12
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 13
{{1,2,6},{3},{4,5}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 11
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 12
{{1,2},{3},{4,5,6}}
=> [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 11
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 13
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 12
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 13
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 13
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 13
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 14
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,3,4},{2},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 11
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 12
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 11
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 11
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 12
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 13
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,3,5},{2},{4,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 11
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 12
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 11
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 11
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 12
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 13
{{1,3,6},{2},{4,5}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 11
{{1,3},{2,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 12
{{1,3},{2},{4,5,6}}
=> [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 11
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000719
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000719: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 68%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000719: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 68%
Values
{{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
{{1,4},{2,3},{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
{{1,2,3,4,5,6}}
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> ? = 0
{{1,2,3,4,5},{6}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 5
{{1,2,3,4,6},{5}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 5
{{1,2,3,4},{5,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 8
{{1,2,3,4},{5},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 9
{{1,2,3,5,6},{4}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 5
{{1,2,3,5},{4,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 8
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 9
{{1,2,3,6},{4,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 8
{{1,2,3},{4,5,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 9
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 11
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 9
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 11
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> ? = 11
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 12
{{1,2,4,5,6},{3}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 5
{{1,2,4,5},{3,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 8
{{1,2,4,5},{3},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 9
{{1,2,4,6},{3,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 8
{{1,2,4},{3,5,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 9
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 11
{{1,2,4,6},{3},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 9
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 11
{{1,2,4},{3},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> ? = 11
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 12
{{1,2,5,6},{3,4}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 8
{{1,2,5},{3,4,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 9
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 11
{{1,2,6},{3,4,5}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 9
{{1,2},{3,4,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 8
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 11
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 11
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 11
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 12
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 13
{{1,2,5,6},{3},{4}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 9
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 11
{{1,2,5},{3},{4,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> ? = 11
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 12
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 11
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 11
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 12
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 13
{{1,2,6},{3},{4,5}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> ? = 11
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 12
{{1,2},{3},{4,5,6}}
=> [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> ? = 11
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 13
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 12
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 13
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 13
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 $\binom{n}{2}$.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!