Identifier
-
Mp00092:
Perfect matchings
—to set partition⟶
Set partitions
St000497: Set partitions ⟶ ℤ
Values
[(1,2)] => {{1,2}} => 0
[(1,2),(3,4)] => {{1,2},{3,4}} => 0
[(1,3),(2,4)] => {{1,3},{2,4}} => 1
[(1,4),(2,3)] => {{1,4},{2,3}} => 2
[(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}} => 1
[(1,4),(2,3),(5,6)] => {{1,4},{2,3},{5,6}} => 2
[(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}} => 4
[(1,6),(2,4),(3,5)] => {{1,6},{2,4},{3,5}} => 5
[(1,5),(2,4),(3,6)] => {{1,5},{2,4},{3,6}} => 4
[(1,4),(2,5),(3,6)] => {{1,4},{2,5},{3,6}} => 3
[(1,3),(2,5),(4,6)] => {{1,3},{2,5},{4,6}} => 2
[(1,2),(3,5),(4,6)] => {{1,2},{3,5},{4,6}} => 1
[(1,2),(3,6),(4,5)] => {{1,2},{3,6},{4,5}} => 2
[(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}} => 4
[(1,5),(2,6),(3,4)] => {{1,5},{2,6},{3,4}} => 5
[(1,6),(2,5),(3,4)] => {{1,6},{2,5},{3,4}} => 6
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Description
The lcb 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 lcb (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
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 lcb (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Map
to set partition
Description
Return the set partition corresponding to the perfect matching.
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