Identifier
-
Mp00092:
Perfect matchings
—to set partition⟶
Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000496: Set partitions ⟶ ℤ
Values
[(1,2)] => {{1,2}} => {{1,2}} => 0
[(1,2),(3,4)] => {{1,2},{3,4}} => {{1,2},{3,4}} => 0
[(1,3),(2,4)] => {{1,3},{2,4}} => {{1,4},{2,3}} => 1
[(1,4),(2,3)] => {{1,4},{2,3}} => {{1,3},{2,4}} => 0
[(1,2),(3,4),(5,6)] => {{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}} => {{1,4},{2,3},{5,6}} => 1
[(1,4),(2,3),(5,6)] => {{1,4},{2,3},{5,6}} => {{1,3},{2,4},{5,6}} => 0
[(1,5),(2,3),(4,6)] => {{1,5},{2,3},{4,6}} => {{1,3},{2,6},{4,5}} => 1
[(1,6),(2,3),(4,5)] => {{1,6},{2,3},{4,5}} => {{1,3},{2,5},{4,6}} => 0
[(1,6),(2,4),(3,5)] => {{1,6},{2,4},{3,5}} => {{1,5},{2,4},{3,6}} => 1
[(1,5),(2,4),(3,6)] => {{1,5},{2,4},{3,6}} => {{1,6},{2,4},{3,5}} => 2
[(1,4),(2,5),(3,6)] => {{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => 3
[(1,3),(2,5),(4,6)] => {{1,3},{2,5},{4,6}} => {{1,6},{2,3},{4,5}} => 2
[(1,2),(3,5),(4,6)] => {{1,2},{3,5},{4,6}} => {{1,2},{3,6},{4,5}} => 1
[(1,2),(3,6),(4,5)] => {{1,2},{3,6},{4,5}} => {{1,2},{3,5},{4,6}} => 0
[(1,3),(2,6),(4,5)] => {{1,3},{2,6},{4,5}} => {{1,5},{2,3},{4,6}} => 1
[(1,4),(2,6),(3,5)] => {{1,4},{2,6},{3,5}} => {{1,5},{2,6},{3,4}} => 2
[(1,5),(2,6),(3,4)] => {{1,5},{2,6},{3,4}} => {{1,4},{2,6},{3,5}} => 1
[(1,6),(2,5),(3,4)] => {{1,6},{2,5},{3,4}} => {{1,4},{2,5},{3,6}} => 0
[(1,2),(3,4),(5,6),(7,8)] => {{1,2},{3,4},{5,6},{7,8}} => {{1,2},{3,4},{5,6},{7,8}} => 0
[(1,3),(2,4),(5,6),(7,8)] => {{1,3},{2,4},{5,6},{7,8}} => {{1,4},{2,3},{5,6},{7,8}} => 1
[(1,4),(2,3),(5,6),(7,8)] => {{1,4},{2,3},{5,6},{7,8}} => {{1,3},{2,4},{5,6},{7,8}} => 0
[(1,5),(2,3),(4,6),(7,8)] => {{1,5},{2,3},{4,6},{7,8}} => {{1,3},{2,6},{4,5},{7,8}} => 1
[(1,6),(2,3),(4,5),(7,8)] => {{1,6},{2,3},{4,5},{7,8}} => {{1,3},{2,5},{4,6},{7,8}} => 0
[(1,7),(2,3),(4,5),(6,8)] => {{1,7},{2,3},{4,5},{6,8}} => {{1,3},{2,5},{4,8},{6,7}} => 1
[(1,8),(2,3),(4,5),(6,7)] => {{1,8},{2,3},{4,5},{6,7}} => {{1,3},{2,5},{4,7},{6,8}} => 0
[(1,8),(2,4),(3,5),(6,7)] => {{1,8},{2,4},{3,5},{6,7}} => {{1,5},{2,4},{3,7},{6,8}} => 1
[(1,7),(2,4),(3,5),(6,8)] => {{1,7},{2,4},{3,5},{6,8}} => {{1,5},{2,4},{3,8},{6,7}} => 2
[(1,6),(2,4),(3,5),(7,8)] => {{1,6},{2,4},{3,5},{7,8}} => {{1,5},{2,4},{3,6},{7,8}} => 1
[(1,5),(2,4),(3,6),(7,8)] => {{1,5},{2,4},{3,6},{7,8}} => {{1,6},{2,4},{3,5},{7,8}} => 2
[(1,4),(2,5),(3,6),(7,8)] => {{1,4},{2,5},{3,6},{7,8}} => {{1,6},{2,5},{3,4},{7,8}} => 3
[(1,3),(2,5),(4,6),(7,8)] => {{1,3},{2,5},{4,6},{7,8}} => {{1,6},{2,3},{4,5},{7,8}} => 2
[(1,2),(3,5),(4,6),(7,8)] => {{1,2},{3,5},{4,6},{7,8}} => {{1,2},{3,6},{4,5},{7,8}} => 1
[(1,2),(3,6),(4,5),(7,8)] => {{1,2},{3,6},{4,5},{7,8}} => {{1,2},{3,5},{4,6},{7,8}} => 0
[(1,3),(2,6),(4,5),(7,8)] => {{1,3},{2,6},{4,5},{7,8}} => {{1,5},{2,3},{4,6},{7,8}} => 1
[(1,4),(2,6),(3,5),(7,8)] => {{1,4},{2,6},{3,5},{7,8}} => {{1,5},{2,6},{3,4},{7,8}} => 2
[(1,5),(2,6),(3,4),(7,8)] => {{1,5},{2,6},{3,4},{7,8}} => {{1,4},{2,6},{3,5},{7,8}} => 1
[(1,6),(2,5),(3,4),(7,8)] => {{1,6},{2,5},{3,4},{7,8}} => {{1,4},{2,5},{3,6},{7,8}} => 0
[(1,7),(2,5),(3,4),(6,8)] => {{1,7},{2,5},{3,4},{6,8}} => {{1,4},{2,5},{3,8},{6,7}} => 1
[(1,8),(2,5),(3,4),(6,7)] => {{1,8},{2,5},{3,4},{6,7}} => {{1,4},{2,5},{3,7},{6,8}} => 0
[(1,8),(2,6),(3,4),(5,7)] => {{1,8},{2,6},{3,4},{5,7}} => {{1,4},{2,7},{3,6},{5,8}} => 1
[(1,7),(2,6),(3,4),(5,8)] => {{1,7},{2,6},{3,4},{5,8}} => {{1,4},{2,8},{3,6},{5,7}} => 2
[(1,6),(2,7),(3,4),(5,8)] => {{1,6},{2,7},{3,4},{5,8}} => {{1,4},{2,8},{3,7},{5,6}} => 3
[(1,5),(2,7),(3,4),(6,8)] => {{1,5},{2,7},{3,4},{6,8}} => {{1,4},{2,8},{3,5},{6,7}} => 2
[(1,4),(2,7),(3,5),(6,8)] => {{1,4},{2,7},{3,5},{6,8}} => {{1,5},{2,8},{3,4},{6,7}} => 3
[(1,3),(2,7),(4,5),(6,8)] => {{1,3},{2,7},{4,5},{6,8}} => {{1,5},{2,3},{4,8},{6,7}} => 2
[(1,2),(3,7),(4,5),(6,8)] => {{1,2},{3,7},{4,5},{6,8}} => {{1,2},{3,5},{4,8},{6,7}} => 1
[(1,2),(3,8),(4,5),(6,7)] => {{1,2},{3,8},{4,5},{6,7}} => {{1,2},{3,5},{4,7},{6,8}} => 0
[(1,3),(2,8),(4,5),(6,7)] => {{1,3},{2,8},{4,5},{6,7}} => {{1,5},{2,3},{4,7},{6,8}} => 1
[(1,4),(2,8),(3,5),(6,7)] => {{1,4},{2,8},{3,5},{6,7}} => {{1,5},{2,7},{3,4},{6,8}} => 2
[(1,5),(2,8),(3,4),(6,7)] => {{1,5},{2,8},{3,4},{6,7}} => {{1,4},{2,7},{3,5},{6,8}} => 1
[(1,6),(2,8),(3,4),(5,7)] => {{1,6},{2,8},{3,4},{5,7}} => {{1,4},{2,7},{3,8},{5,6}} => 2
[(1,7),(2,8),(3,4),(5,6)] => {{1,7},{2,8},{3,4},{5,6}} => {{1,4},{2,6},{3,8},{5,7}} => 1
[(1,8),(2,7),(3,4),(5,6)] => {{1,8},{2,7},{3,4},{5,6}} => {{1,4},{2,6},{3,7},{5,8}} => 0
[(1,8),(2,7),(3,5),(4,6)] => {{1,8},{2,7},{3,5},{4,6}} => {{1,6},{2,5},{3,7},{4,8}} => 1
[(1,7),(2,8),(3,5),(4,6)] => {{1,7},{2,8},{3,5},{4,6}} => {{1,6},{2,5},{3,8},{4,7}} => 2
[(1,6),(2,8),(3,5),(4,7)] => {{1,6},{2,8},{3,5},{4,7}} => {{1,7},{2,5},{3,8},{4,6}} => 3
[(1,5),(2,8),(3,6),(4,7)] => {{1,5},{2,8},{3,6},{4,7}} => {{1,7},{2,6},{3,8},{4,5}} => 4
[(1,4),(2,8),(3,6),(5,7)] => {{1,4},{2,8},{3,6},{5,7}} => {{1,7},{2,6},{3,4},{5,8}} => 3
[(1,3),(2,8),(4,6),(5,7)] => {{1,3},{2,8},{4,6},{5,7}} => {{1,7},{2,3},{4,6},{5,8}} => 2
[(1,2),(3,8),(4,6),(5,7)] => {{1,2},{3,8},{4,6},{5,7}} => {{1,2},{3,7},{4,6},{5,8}} => 1
[(1,2),(3,7),(4,6),(5,8)] => {{1,2},{3,7},{4,6},{5,8}} => {{1,2},{3,8},{4,6},{5,7}} => 2
[(1,3),(2,7),(4,6),(5,8)] => {{1,3},{2,7},{4,6},{5,8}} => {{1,8},{2,3},{4,6},{5,7}} => 3
[(1,4),(2,7),(3,6),(5,8)] => {{1,4},{2,7},{3,6},{5,8}} => {{1,8},{2,6},{3,4},{5,7}} => 4
[(1,5),(2,7),(3,6),(4,8)] => {{1,5},{2,7},{3,6},{4,8}} => {{1,8},{2,6},{3,7},{4,5}} => 5
[(1,6),(2,7),(3,5),(4,8)] => {{1,6},{2,7},{3,5},{4,8}} => {{1,8},{2,5},{3,7},{4,6}} => 4
[(1,7),(2,6),(3,5),(4,8)] => {{1,7},{2,6},{3,5},{4,8}} => {{1,8},{2,5},{3,6},{4,7}} => 3
[(1,8),(2,6),(3,5),(4,7)] => {{1,8},{2,6},{3,5},{4,7}} => {{1,7},{2,5},{3,6},{4,8}} => 2
[(1,8),(2,5),(3,6),(4,7)] => {{1,8},{2,5},{3,6},{4,7}} => {{1,7},{2,6},{3,5},{4,8}} => 3
[(1,7),(2,5),(3,6),(4,8)] => {{1,7},{2,5},{3,6},{4,8}} => {{1,8},{2,6},{3,5},{4,7}} => 4
[(1,6),(2,5),(3,7),(4,8)] => {{1,6},{2,5},{3,7},{4,8}} => {{1,8},{2,7},{3,5},{4,6}} => 5
[(1,5),(2,6),(3,7),(4,8)] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 6
[(1,4),(2,6),(3,7),(5,8)] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}} => 5
[(1,3),(2,6),(4,7),(5,8)] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}} => 4
[(1,2),(3,6),(4,7),(5,8)] => {{1,2},{3,6},{4,7},{5,8}} => {{1,2},{3,8},{4,7},{5,6}} => 3
[(1,2),(3,5),(4,7),(6,8)] => {{1,2},{3,5},{4,7},{6,8}} => {{1,2},{3,8},{4,5},{6,7}} => 2
[(1,3),(2,5),(4,7),(6,8)] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}} => 3
[(1,4),(2,5),(3,7),(6,8)] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}} => 4
[(1,5),(2,4),(3,7),(6,8)] => {{1,5},{2,4},{3,7},{6,8}} => {{1,8},{2,4},{3,5},{6,7}} => 3
[(1,6),(2,4),(3,7),(5,8)] => {{1,6},{2,4},{3,7},{5,8}} => {{1,8},{2,4},{3,7},{5,6}} => 4
[(1,7),(2,4),(3,6),(5,8)] => {{1,7},{2,4},{3,6},{5,8}} => {{1,8},{2,4},{3,6},{5,7}} => 3
[(1,8),(2,4),(3,6),(5,7)] => {{1,8},{2,4},{3,6},{5,7}} => {{1,7},{2,4},{3,6},{5,8}} => 2
[(1,8),(2,3),(4,6),(5,7)] => {{1,8},{2,3},{4,6},{5,7}} => {{1,3},{2,7},{4,6},{5,8}} => 1
[(1,7),(2,3),(4,6),(5,8)] => {{1,7},{2,3},{4,6},{5,8}} => {{1,3},{2,8},{4,6},{5,7}} => 2
[(1,6),(2,3),(4,7),(5,8)] => {{1,6},{2,3},{4,7},{5,8}} => {{1,3},{2,8},{4,7},{5,6}} => 3
[(1,5),(2,3),(4,7),(6,8)] => {{1,5},{2,3},{4,7},{6,8}} => {{1,3},{2,8},{4,5},{6,7}} => 2
[(1,4),(2,3),(5,7),(6,8)] => {{1,4},{2,3},{5,7},{6,8}} => {{1,3},{2,4},{5,8},{6,7}} => 1
[(1,3),(2,4),(5,7),(6,8)] => {{1,3},{2,4},{5,7},{6,8}} => {{1,4},{2,3},{5,8},{6,7}} => 2
[(1,2),(3,4),(5,7),(6,8)] => {{1,2},{3,4},{5,7},{6,8}} => {{1,2},{3,4},{5,8},{6,7}} => 1
[(1,2),(3,4),(5,8),(6,7)] => {{1,2},{3,4},{5,8},{6,7}} => {{1,2},{3,4},{5,7},{6,8}} => 0
[(1,3),(2,4),(5,8),(6,7)] => {{1,3},{2,4},{5,8},{6,7}} => {{1,4},{2,3},{5,7},{6,8}} => 1
[(1,4),(2,3),(5,8),(6,7)] => {{1,4},{2,3},{5,8},{6,7}} => {{1,3},{2,4},{5,7},{6,8}} => 0
[(1,5),(2,3),(4,8),(6,7)] => {{1,5},{2,3},{4,8},{6,7}} => {{1,3},{2,7},{4,5},{6,8}} => 1
[(1,6),(2,3),(4,8),(5,7)] => {{1,6},{2,3},{4,8},{5,7}} => {{1,3},{2,7},{4,8},{5,6}} => 2
[(1,7),(2,3),(4,8),(5,6)] => {{1,7},{2,3},{4,8},{5,6}} => {{1,3},{2,6},{4,8},{5,7}} => 1
[(1,8),(2,3),(4,7),(5,6)] => {{1,8},{2,3},{4,7},{5,6}} => {{1,3},{2,6},{4,7},{5,8}} => 0
[(1,8),(2,4),(3,7),(5,6)] => {{1,8},{2,4},{3,7},{5,6}} => {{1,6},{2,4},{3,7},{5,8}} => 1
[(1,7),(2,4),(3,8),(5,6)] => {{1,7},{2,4},{3,8},{5,6}} => {{1,6},{2,4},{3,8},{5,7}} => 2
[(1,6),(2,4),(3,8),(5,7)] => {{1,6},{2,4},{3,8},{5,7}} => {{1,7},{2,4},{3,8},{5,6}} => 3
[(1,5),(2,4),(3,8),(6,7)] => {{1,5},{2,4},{3,8},{6,7}} => {{1,7},{2,4},{3,5},{6,8}} => 2
[(1,4),(2,5),(3,8),(6,7)] => {{1,4},{2,5},{3,8},{6,7}} => {{1,7},{2,5},{3,4},{6,8}} => 3
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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 $a < b$.
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 $a < b$.
Map
to set partition
Description
Return the set partition corresponding to the perfect matching.
Map
Kasraoui-Zeng
Description
The Kasraoui-Zeng involution.
This is defined in [1] and yields the set partition with the number of nestings and crossings exchanged.
This is defined in [1] and yields the set partition with the number of nestings and crossings exchanged.
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