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Matching statistic: St001841
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
St001841: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001841: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{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}}
=> 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
{{1,3,4},{2}}
=> {{1,3},{2,4}}
=> 1
{{1,3},{2,4}}
=> {{1,2,4},{3}}
=> 2
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
{{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}}
=> 0
{{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 1
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
{{1,2,4,5},{3}}
=> {{1,3},{2,4,5}}
=> 1
{{1,2,4},{3,5}}
=> {{1,2,4,5},{3}}
=> 2
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 1
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 2
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,3},{2},{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
{{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> 3
{{1,3,4},{2,5}}
=> {{1,2,4},{3,5}}
=> 1
{{1,3,4},{2},{5}}
=> {{1},{2,4},{3,5}}
=> 1
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> 2
{{1,3},{2,4,5}}
=> {{1,2,3,5},{4}}
=> 3
{{1,3},{2,4},{5}}
=> {{1},{2,3,5},{4}}
=> 2
{{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> 2
{{1,3},{2,5},{4}}
=> {{1,3,5},{2},{4}}
=> 3
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 1
{{1,4,5},{2,3}}
=> {{1,3,4},{2,5}}
=> 2
{{1,4},{2,3,5}}
=> {{1,2,5},{3,4}}
=> 4
Description
The number of inversions 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 $i$.
A pair $(i,j)$ is an inversion of the word $w$ if $w_i > w_j$.
Matching statistic: St001207
Mp00215: Set partitions —Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 27%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 27%
Values
{{1}}
=> {{1}}
=> [1] => [1] => ? = 0
{{1,2}}
=> {{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [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}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,3},{2,4}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [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}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => ? = 0
{{1,2,3,5},{4}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ? = 1
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => ? = 0
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => ? = 0
{{1,2,4,5},{3}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ? = 1
{{1,2,4},{3,5}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => ? = 2
{{1,2,4},{3},{5}}
=> {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => ? = 1
{{1,2,5},{3,4}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => ? = 2
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => ? = 0
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => ? = 0
{{1,2,5},{3},{4}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => ? = 2
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ? = 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => ? = 0
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,4,5] => ? = 0
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ? = 3
{{1,3,4},{2,5}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ? = 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => ? = 1
{{1,3,5},{2,4}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ? = 2
{{1,3},{2,4,5}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => ? = 3
{{1,3},{2,4},{5}}
=> {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,4,5,3] => ? = 2
{{1,3,5},{2},{4}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ? = 2
{{1,3},{2,5},{4}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => ? = 3
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ? = 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => ? = 1
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ? = 2
{{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => ? = 4
{{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,2,5,3,4] => ? = 2
{{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => ? = 3
{{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => ? = 0
{{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => ? = 0
{{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => ? = 3
{{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ? = 1
{{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => ? = 0
{{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => ? = 0
{{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ? = 3
{{1,4},{2,5},{3}}
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => ? = 2
{{1,4},{2},{3,5}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => ? = 4
{{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,2,5,3,4] => ? = 2
{{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => ? = 4
{{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ? = 1
{{1},{2,4},{3,5}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => ? = 2
{{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => ? = 1
{{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => ? = 3
{{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => ? = 2
{{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => ? = 0
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => ? = 3
{{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,4,2,3,5] => ? = 2
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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