Identifier
-
Mp00092:
Perfect matchings
—to set partition⟶
Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000232: 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}} => 0
[(1,4),(2,3)] => {{1,4},{2,3}} => {{1,3},{2,4}} => 1
[(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}} => 0
[(1,4),(2,3),(5,6)] => {{1,4},{2,3},{5,6}} => {{1,3},{2,4},{5,6}} => 1
[(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}} => 2
[(1,6),(2,4),(3,5)] => {{1,6},{2,4},{3,5}} => {{1,5},{2,4},{3,6}} => 2
[(1,5),(2,4),(3,6)] => {{1,5},{2,4},{3,6}} => {{1,6},{2,4},{3,5}} => 1
[(1,4),(2,5),(3,6)] => {{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => 0
[(1,3),(2,5),(4,6)] => {{1,3},{2,5},{4,6}} => {{1,6},{2,3},{4,5}} => 0
[(1,2),(3,5),(4,6)] => {{1,2},{3,5},{4,6}} => {{1,2},{3,6},{4,5}} => 0
[(1,2),(3,6),(4,5)] => {{1,2},{3,6},{4,5}} => {{1,2},{3,5},{4,6}} => 1
[(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}} => 1
[(1,5),(2,6),(3,4)] => {{1,5},{2,6},{3,4}} => {{1,4},{2,6},{3,5}} => 2
[(1,6),(2,5),(3,4)] => {{1,6},{2,5},{3,4}} => {{1,4},{2,5},{3,6}} => 3
[(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}} => 0
[(1,4),(2,3),(5,6),(7,8)] => {{1,4},{2,3},{5,6},{7,8}} => {{1,3},{2,4},{5,6},{7,8}} => 1
[(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}} => 2
[(1,7),(2,3),(4,5),(6,8)] => {{1,7},{2,3},{4,5},{6,8}} => {{1,3},{2,5},{4,8},{6,7}} => 2
[(1,8),(2,3),(4,5),(6,7)] => {{1,8},{2,3},{4,5},{6,7}} => {{1,3},{2,5},{4,7},{6,8}} => 3
[(1,8),(2,4),(3,5),(6,7)] => {{1,8},{2,4},{3,5},{6,7}} => {{1,5},{2,4},{3,7},{6,8}} => 3
[(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}} => 2
[(1,5),(2,4),(3,6),(7,8)] => {{1,5},{2,4},{3,6},{7,8}} => {{1,6},{2,4},{3,5},{7,8}} => 1
[(1,4),(2,5),(3,6),(7,8)] => {{1,4},{2,5},{3,6},{7,8}} => {{1,6},{2,5},{3,4},{7,8}} => 0
[(1,3),(2,5),(4,6),(7,8)] => {{1,3},{2,5},{4,6},{7,8}} => {{1,6},{2,3},{4,5},{7,8}} => 0
[(1,2),(3,5),(4,6),(7,8)] => {{1,2},{3,5},{4,6},{7,8}} => {{1,2},{3,6},{4,5},{7,8}} => 0
[(1,2),(3,6),(4,5),(7,8)] => {{1,2},{3,6},{4,5},{7,8}} => {{1,2},{3,5},{4,6},{7,8}} => 1
[(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}} => 1
[(1,5),(2,6),(3,4),(7,8)] => {{1,5},{2,6},{3,4},{7,8}} => {{1,4},{2,6},{3,5},{7,8}} => 2
[(1,6),(2,5),(3,4),(7,8)] => {{1,6},{2,5},{3,4},{7,8}} => {{1,4},{2,5},{3,6},{7,8}} => 3
[(1,7),(2,5),(3,4),(6,8)] => {{1,7},{2,5},{3,4},{6,8}} => {{1,4},{2,5},{3,8},{6,7}} => 3
[(1,8),(2,5),(3,4),(6,7)] => {{1,8},{2,5},{3,4},{6,7}} => {{1,4},{2,5},{3,7},{6,8}} => 4
[(1,8),(2,6),(3,4),(5,7)] => {{1,8},{2,6},{3,4},{5,7}} => {{1,4},{2,7},{3,6},{5,8}} => 4
[(1,7),(2,6),(3,4),(5,8)] => {{1,7},{2,6},{3,4},{5,8}} => {{1,4},{2,8},{3,6},{5,7}} => 3
[(1,6),(2,7),(3,4),(5,8)] => {{1,6},{2,7},{3,4},{5,8}} => {{1,4},{2,8},{3,7},{5,6}} => 2
[(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}} => 1
[(1,3),(2,7),(4,5),(6,8)] => {{1,3},{2,7},{4,5},{6,8}} => {{1,5},{2,3},{4,8},{6,7}} => 1
[(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}} => 2
[(1,3),(2,8),(4,5),(6,7)] => {{1,3},{2,8},{4,5},{6,7}} => {{1,5},{2,3},{4,7},{6,8}} => 2
[(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}} => 3
[(1,6),(2,8),(3,4),(5,7)] => {{1,6},{2,8},{3,4},{5,7}} => {{1,4},{2,7},{3,8},{5,6}} => 3
[(1,7),(2,8),(3,4),(5,6)] => {{1,7},{2,8},{3,4},{5,6}} => {{1,4},{2,6},{3,8},{5,7}} => 4
[(1,8),(2,7),(3,4),(5,6)] => {{1,8},{2,7},{3,4},{5,6}} => {{1,4},{2,6},{3,7},{5,8}} => 5
[(1,8),(2,7),(3,5),(4,6)] => {{1,8},{2,7},{3,5},{4,6}} => {{1,6},{2,5},{3,7},{4,8}} => 5
[(1,7),(2,8),(3,5),(4,6)] => {{1,7},{2,8},{3,5},{4,6}} => {{1,6},{2,5},{3,8},{4,7}} => 4
[(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}} => 2
[(1,4),(2,8),(3,6),(5,7)] => {{1,4},{2,8},{3,6},{5,7}} => {{1,7},{2,6},{3,4},{5,8}} => 2
[(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}} => 2
[(1,2),(3,7),(4,6),(5,8)] => {{1,2},{3,7},{4,6},{5,8}} => {{1,2},{3,8},{4,6},{5,7}} => 1
[(1,3),(2,7),(4,6),(5,8)] => {{1,3},{2,7},{4,6},{5,8}} => {{1,8},{2,3},{4,6},{5,7}} => 1
[(1,4),(2,7),(3,6),(5,8)] => {{1,4},{2,7},{3,6},{5,8}} => {{1,8},{2,6},{3,4},{5,7}} => 1
[(1,5),(2,7),(3,6),(4,8)] => {{1,5},{2,7},{3,6},{4,8}} => {{1,8},{2,6},{3,7},{4,5}} => 1
[(1,6),(2,7),(3,5),(4,8)] => {{1,6},{2,7},{3,5},{4,8}} => {{1,8},{2,5},{3,7},{4,6}} => 2
[(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}} => 4
[(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}} => 2
[(1,6),(2,5),(3,7),(4,8)] => {{1,6},{2,5},{3,7},{4,8}} => {{1,8},{2,7},{3,5},{4,6}} => 1
[(1,5),(2,6),(3,7),(4,8)] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,4),(2,6),(3,7),(5,8)] => {{1,4},{2,6},{3,7},{5,8}} => {{1,8},{2,7},{3,4},{5,6}} => 0
[(1,3),(2,6),(4,7),(5,8)] => {{1,3},{2,6},{4,7},{5,8}} => {{1,8},{2,3},{4,7},{5,6}} => 0
[(1,2),(3,6),(4,7),(5,8)] => {{1,2},{3,6},{4,7},{5,8}} => {{1,2},{3,8},{4,7},{5,6}} => 0
[(1,2),(3,5),(4,7),(6,8)] => {{1,2},{3,5},{4,7},{6,8}} => {{1,2},{3,8},{4,5},{6,7}} => 0
[(1,3),(2,5),(4,7),(6,8)] => {{1,3},{2,5},{4,7},{6,8}} => {{1,8},{2,3},{4,5},{6,7}} => 0
[(1,4),(2,5),(3,7),(6,8)] => {{1,4},{2,5},{3,7},{6,8}} => {{1,8},{2,5},{3,4},{6,7}} => 0
[(1,5),(2,4),(3,7),(6,8)] => {{1,5},{2,4},{3,7},{6,8}} => {{1,8},{2,4},{3,5},{6,7}} => 1
[(1,6),(2,4),(3,7),(5,8)] => {{1,6},{2,4},{3,7},{5,8}} => {{1,8},{2,4},{3,7},{5,6}} => 1
[(1,7),(2,4),(3,6),(5,8)] => {{1,7},{2,4},{3,6},{5,8}} => {{1,8},{2,4},{3,6},{5,7}} => 2
[(1,8),(2,4),(3,6),(5,7)] => {{1,8},{2,4},{3,6},{5,7}} => {{1,7},{2,4},{3,6},{5,8}} => 3
[(1,8),(2,3),(4,6),(5,7)] => {{1,8},{2,3},{4,6},{5,7}} => {{1,3},{2,7},{4,6},{5,8}} => 3
[(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}} => 1
[(1,5),(2,3),(4,7),(6,8)] => {{1,5},{2,3},{4,7},{6,8}} => {{1,3},{2,8},{4,5},{6,7}} => 1
[(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}} => 0
[(1,2),(3,4),(5,7),(6,8)] => {{1,2},{3,4},{5,7},{6,8}} => {{1,2},{3,4},{5,8},{6,7}} => 0
[(1,2),(3,4),(5,8),(6,7)] => {{1,2},{3,4},{5,8},{6,7}} => {{1,2},{3,4},{5,7},{6,8}} => 1
[(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}} => 2
[(1,5),(2,3),(4,8),(6,7)] => {{1,5},{2,3},{4,8},{6,7}} => {{1,3},{2,7},{4,5},{6,8}} => 2
[(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}} => 3
[(1,8),(2,3),(4,7),(5,6)] => {{1,8},{2,3},{4,7},{5,6}} => {{1,3},{2,6},{4,7},{5,8}} => 4
[(1,8),(2,4),(3,7),(5,6)] => {{1,8},{2,4},{3,7},{5,6}} => {{1,6},{2,4},{3,7},{5,8}} => 4
[(1,7),(2,4),(3,8),(5,6)] => {{1,7},{2,4},{3,8},{5,6}} => {{1,6},{2,4},{3,8},{5,7}} => 3
[(1,6),(2,4),(3,8),(5,7)] => {{1,6},{2,4},{3,8},{5,7}} => {{1,7},{2,4},{3,8},{5,6}} => 2
[(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}} => 1
>>> Load all 127 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
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.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!