Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St000057
(load all 5 compositions to match this statistic)
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => [[1]]
 => 0
{{1,2}}
 => {{1},{2}}
 => [[1],[2]]
 => 0
{{1},{2}}
 => {{1,2}}
 => [[1,2]]
 => 0
{{1,2,3}}
 => {{1},{2},{3}}
 => [[1],[2],[3]]
 => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => [[1,2],[3]]
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [[1,3],[2]]
 => 0
{{1},{2,3}}
 => {{1,3},{2}}
 => [[1,3],[2]]
 => 0
{{1},{2},{3}}
 => {{1,2,3}}
 => [[1,2,3]]
 => 0
{{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 0
{{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 2
{{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 1
{{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 1
{{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 2
{{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 0
{{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => [[1,2],[3,4]]
 => 1
{{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => 0
{{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 1
{{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => [[1,3],[2,4]]
 => 0
{{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 0
{{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => [[1,2,3,4]]
 => 0
{{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => [[1],[2],[3],[4],[5]]
 => 0
{{1,2,3,4},{5}}
 => {{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => 3
{{1,2,3},{4,5}}
 => {{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => 2
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => 4
{{1,2},{3,4,5}}
 => {{1,4},{2},{3},{5}}
 => [[1,4],[2],[3],[5]]
 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => 3
{{1,2},{3},{4,5}}
 => {{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => 2
{{1,2},{3},{4},{5}}
 => {{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => 3
{{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => [[1,5],[2],[3],[4]]
 => 0
{{1},{2,3,4},{5}}
 => {{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => 2
{{1},{2,3},{4,5}}
 => {{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => 1
{{1},{2,3},{4},{5}}
 => {{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => 2
{{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => [[1,4,5],[2],[3]]
 => 0
{{1},{2},{3,4},{5}}
 => {{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => 1
{{1},{2},{3},{4,5}}
 => {{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => 0
{{1},{2},{3},{4},{5}}
 => {{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => 0
Description
The Shynar inversion number of a standard tableau. Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Matching statistic: St000496
(load all 9 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => {{1}}
 => 0
{{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 1
{{1,3},{2}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 0
{{1},{2,3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 2
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => {{1,4},{2,3}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => {{1,3,4,5},{2}}
 => 3
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => 2
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 4
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => {{1,3,5},{2},{4}}
 => 3
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => 2
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 3
{{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 0
{{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => {{1,3,4},{2},{5}}
 => 2
{{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => 1
{{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 2
{{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 0
{{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => 1
{{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The rcs statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St001438
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ 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] => [1,1,0,0]
 => [[2],[]]
 => 0
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [[1,1],[]]
 => 0
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [[3],[]]
 => 0
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => 1
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [[2,2],[]]
 => 0
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [[2,1],[]]
 => 0
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => 4
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => 3
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => 3
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => 0
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => 2
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => 2
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => 0
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => 0
Description
The number of missing boxes of a skew partition.
Matching statistic: St001843
(load all 3 compositions to match this statistic)
Mapping
Mp00215: Set partitions —Wachs-White⟶ Set partitions
Mp00219: Set partitions —inverse Yip⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St001843: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => {{1}}
 => {{1}}
 => 0
{{1,2}}
 => {{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => {{1,3},{2}}
 => 1
{{1,3},{2}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => {{1,2},{3}}
 => 0
{{1},{2,3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => {{1},{2,3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 2
{{1,2,4},{3}}
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => {{1,4},{2,3}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => {{1,3,4},{2}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => 2
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => {{1,3,4,5},{2}}
 => {{1,2,3,5},{4}}
 => 3
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => {{1,2,4},{3,5}}
 => {{1,3},{2,4,5}}
 => 2
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => {{1,2,5},{3},{4}}
 => 4
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => {{1,3,4,5},{2}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => {{1,3},{2,5},{4}}
 => {{1,4},{2},{3,5}}
 => 3
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => {{1,4,5},{2},{3}}
 => 2
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => 3
{{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => 0
{{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => {{1,3,4},{2},{5}}
 => {{1},{2,3,5},{4}}
 => 2
{{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => {{1},{2,4,5},{3}}
 => 1
{{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => {{1},{2,5},{3},{4}}
 => 2
{{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => 0
{{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => 1
{{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The Z-index of a set partition. The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$. The Z-index of $w$ equals $$ \sum_{i < j} w_{i,j}, $$ where $w_{i,j}$ is the word obtained from $w$ by removing all letters different from $i$ and $j$.
Matching statistic: St000589
(load all 56 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => ? = 0
{{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => 1
{{1,3},{2}}
 => {{1,3},{2}}
 => 0
{{1},{2,3}}
 => {{1,2},{3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 2
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => 2
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => 3
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => 2
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => 4
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => 3
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => 2
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => 3
{{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => 0
{{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => 2
{{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => 1
{{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => 2
{{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => 0
{{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => 1
{{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000609
(load all 13 compositions to match this statistic)
Mapping
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => ? = 0
{{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => 1
{{1,3},{2}}
 => {{1,3},{2}}
 => 0
{{1},{2,3}}
 => {{1,2},{3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 2
{{1,2,4},{3}}
 => {{1,3},{2,4}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => 2
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => 3
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => 2
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => 4
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => 3
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => 2
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => 3
{{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => 0
{{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => 2
{{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => 1
{{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => 2
{{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => 0
{{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => 1
{{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000612
(load all 12 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => ? = 0
{{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => 1
{{1,3},{2}}
 => {{1,3},{2}}
 => 0
{{1},{2,3}}
 => {{1,2},{3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 2
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => 2
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => 3
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => 2
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => 4
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => 3
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => 2
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => 3
{{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => 0
{{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => 2
{{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => 1
{{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => 2
{{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => 0
{{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => 1
{{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.
Matching statistic: St000491
(load all 14 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => {{1}}
 => ? = 0
{{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 1
{{1,3},{2}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 0
{{1},{2,3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 2
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => {{1,4},{2,3}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => {{1,3,4,5},{2}}
 => 3
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => 2
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 4
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => {{1,3,5},{2},{4}}
 => 3
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => 2
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 3
{{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 0
{{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => {{1,3,4},{2},{5}}
 => 2
{{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => 1
{{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 2
{{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 0
{{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => 1
{{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The number of inversions of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$. This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000581
(load all 13 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000581: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => {{1}}
 => ? = 0
{{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 1
{{1,3},{2}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 0
{{1},{2,3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 2
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => {{1,4},{2,3}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => {{1,3,4,5},{2}}
 => 3
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => 2
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 4
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => {{1,3,5},{2},{4}}
 => 3
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => 2
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 3
{{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 0
{{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => {{1,3,4},{2},{5}}
 => 2
{{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => 1
{{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 2
{{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 0
{{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => 1
{{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
Mapping
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [[1]]
 => => => ? = 0
{{1,2}}
 => [[1,2]]
 => 0 => 0 => 0
{{1},{2}}
 => [[1],[2]]
 => 1 => 1 => 0
{{1,2,3}}
 => [[1,2,3]]
 => 00 => 00 => 0
{{1,2},{3}}
 => [[1,2],[3]]
 => 01 => 10 => 1
{{1,3},{2}}
 => [[1,3],[2]]
 => 10 => 01 => 0
{{1},{2,3}}
 => [[1,3],[2]]
 => 10 => 01 => 0
{{1},{2},{3}}
 => [[1],[2],[3]]
 => 11 => 11 => 0
{{1,2,3,4}}
 => [[1,2,3,4]]
 => 000 => 000 => 0
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 001 => 100 => 2
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 010 => 010 => 1
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => 010 => 010 => 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 011 => 110 => 2
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 100 => 001 => 0
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 101 => 101 => 1
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => 100 => 001 => 0
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 101 => 101 => 1
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => 110 => 011 => 0
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 110 => 011 => 0
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 111 => 111 => 0
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => 0000 => 0000 => 0
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => 0001 => 1000 => 3
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => 0010 => 0100 => 2
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => 0011 => 1100 => 4
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => 0100 => 0010 => 1
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => 0101 => 1010 => 3
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => 0110 => 0110 => 2
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => 0111 => 1110 => 3
{{1},{2,3,4,5}}
 => [[1,3,4,5],[2]]
 => 1000 => 0001 => 0
{{1},{2,3,4},{5}}
 => [[1,3,4],[2],[5]]
 => 1001 => 1001 => 2
{{1},{2,3},{4,5}}
 => [[1,3],[2,5],[4]]
 => 1010 => 0101 => 1
{{1},{2,3},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => 1011 => 1101 => 2
{{1},{2},{3,4,5}}
 => [[1,4,5],[2],[3]]
 => 1100 => 0011 => 0
{{1},{2},{3,4},{5}}
 => [[1,4],[2],[3],[5]]
 => 1101 => 1011 => 1
{{1},{2},{3},{4,5}}
 => [[1,5],[2],[3],[4]]
 => 1110 => 0111 => 0
{{1},{2},{3},{4},{5}}
 => [[1],[2],[3],[4],[5]]
 => 1111 => 1111 => 0
Description
The number of inversions of a binary word.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St001868The number of alignments of type NE of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000102The charge of a semistandard tableau. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.