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Matching statistic: St001843

St001843: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => 0
{{1,2}}
 => 0
{{1},{2}}
 => 0
{{1,2,3}}
 => 0
{{1,2},{3}}
 => 0
{{1,3},{2}}
 => 1
{{1},{2,3}}
 => 0
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => 2
{{1,2},{3,4}}
 => 0
{{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => 1
{{1,3},{2,4}}
 => 2
{{1,3},{2},{4}}
 => 1
{{1,4},{2,3}}
 => 1
{{1},{2,3,4}}
 => 0
{{1},{2,3},{4}}
 => 0
{{1,4},{2},{3}}
 => 2
{{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => 0
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => 0
{{1,2,3,5},{4}}
 => 3
{{1,2,3},{4,5}}
 => 0
{{1,2,3},{4},{5}}
 => 0
{{1,2,4,5},{3}}
 => 2
{{1,2,4},{3,5}}
 => 3
{{1,2,4},{3},{5}}
 => 2
{{1,2,5},{3,4}}
 => 2
{{1,2},{3,4,5}}
 => 0
{{1,2},{3,4},{5}}
 => 0
{{1,2,5},{3},{4}}
 => 4
{{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => 0
{{1,2},{3},{4},{5}}
 => 0
{{1,3,4,5},{2}}
 => 1
{{1,3,4},{2,5}}
 => 2
{{1,3,4},{2},{5}}
 => 1
{{1,3,5},{2,4}}
 => 4
{{1,3},{2,4,5}}
 => 2
{{1,3},{2,4},{5}}
 => 2
{{1,3,5},{2},{4}}
 => 3
{{1,3},{2,5},{4}}
 => 3
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 1
{{1,4,5},{2,3}}
 => 1
{{1,4},{2,3,5}}
 => 3

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

$j$ .