Identifier
Values
[(1,2)] => {{1,2}} => 0
[(1,2),(3,4)] => {{1,2},{3,4}} => 0
[(1,3),(2,4)] => {{1,3},{2,4}} => 2
[(1,4),(2,3)] => {{1,4},{2,3}} => 1
[(1,2),(3,4),(5,6)] => {{1,2},{3,4},{5,6}} => 0
[(1,3),(2,4),(5,6)] => {{1,3},{2,4},{5,6}} => 2
[(1,4),(2,3),(5,6)] => {{1,4},{2,3},{5,6}} => 1
[(1,5),(2,3),(4,6)] => {{1,5},{2,3},{4,6}} => 3
[(1,6),(2,3),(4,5)] => {{1,6},{2,3},{4,5}} => 2
[(1,6),(2,4),(3,5)] => {{1,6},{2,4},{3,5}} => 4
[(1,5),(2,4),(3,6)] => {{1,5},{2,4},{3,6}} => 5
[(1,4),(2,5),(3,6)] => {{1,4},{2,5},{3,6}} => 6
[(1,3),(2,5),(4,6)] => {{1,3},{2,5},{4,6}} => 4
[(1,2),(3,5),(4,6)] => {{1,2},{3,5},{4,6}} => 2
[(1,2),(3,6),(4,5)] => {{1,2},{3,6},{4,5}} => 1
[(1,3),(2,6),(4,5)] => {{1,3},{2,6},{4,5}} => 3
[(1,4),(2,6),(3,5)] => {{1,4},{2,6},{3,5}} => 5
[(1,5),(2,6),(3,4)] => {{1,5},{2,6},{3,4}} => 4
[(1,6),(2,5),(3,4)] => {{1,6},{2,5},{3,4}} => 3
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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 $j$.
Map
to set partition
Description
Return the set partition corresponding to the perfect matching.