Identifier
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Images
=>
Cc0012;cc-rep-0Cc0009;cc-rep-1Cc0009;cc-rep-2Cc0002;cc-rep-3
[(1,2)]=>{{1,2}}=>{{1,2}}=>[2] [(1,2),(3,4)]=>{{1,2},{3,4}}=>{{1,2},{3,4}}=>[2,2] [(1,3),(2,4)]=>{{1,3},{2,4}}=>{{1,4},{2,3}}=>[2,2] [(1,4),(2,3)]=>{{1,4},{2,3}}=>{{1,3},{2,4}}=>[2,2] [(1,2),(3,4),(5,6)]=>{{1,2},{3,4},{5,6}}=>{{1,2},{3,4},{5,6}}=>[2,2,2] [(1,3),(2,4),(5,6)]=>{{1,3},{2,4},{5,6}}=>{{1,4},{2,3},{5,6}}=>[2,2,2] [(1,4),(2,3),(5,6)]=>{{1,4},{2,3},{5,6}}=>{{1,3},{2,4},{5,6}}=>[2,2,2] [(1,5),(2,3),(4,6)]=>{{1,5},{2,3},{4,6}}=>{{1,3},{2,6},{4,5}}=>[2,2,2] [(1,6),(2,3),(4,5)]=>{{1,6},{2,3},{4,5}}=>{{1,3},{2,5},{4,6}}=>[2,2,2] [(1,6),(2,4),(3,5)]=>{{1,6},{2,4},{3,5}}=>{{1,5},{2,4},{3,6}}=>[2,2,2] [(1,5),(2,4),(3,6)]=>{{1,5},{2,4},{3,6}}=>{{1,6},{2,4},{3,5}}=>[2,2,2] [(1,4),(2,5),(3,6)]=>{{1,4},{2,5},{3,6}}=>{{1,6},{2,5},{3,4}}=>[2,2,2] [(1,3),(2,5),(4,6)]=>{{1,3},{2,5},{4,6}}=>{{1,6},{2,3},{4,5}}=>[2,2,2] [(1,2),(3,5),(4,6)]=>{{1,2},{3,5},{4,6}}=>{{1,2},{3,6},{4,5}}=>[2,2,2] [(1,2),(3,6),(4,5)]=>{{1,2},{3,6},{4,5}}=>{{1,2},{3,5},{4,6}}=>[2,2,2] [(1,3),(2,6),(4,5)]=>{{1,3},{2,6},{4,5}}=>{{1,5},{2,3},{4,6}}=>[2,2,2] [(1,4),(2,6),(3,5)]=>{{1,4},{2,6},{3,5}}=>{{1,5},{2,6},{3,4}}=>[2,2,2] [(1,5),(2,6),(3,4)]=>{{1,5},{2,6},{3,4}}=>{{1,4},{2,6},{3,5}}=>[2,2,2] [(1,6),(2,5),(3,4)]=>{{1,6},{2,5},{3,4}}=>{{1,4},{2,5},{3,6}}=>[2,2,2] [(1,2),(3,4),(5,6),(7,8)]=>{{1,2},{3,4},{5,6},{7,8}}=>{{1,2},{3,4},{5,6},{7,8}}=>[2,2,2,2] [(1,3),(2,4),(5,6),(7,8)]=>{{1,3},{2,4},{5,6},{7,8}}=>{{1,4},{2,3},{5,6},{7,8}}=>[2,2,2,2] [(1,4),(2,3),(5,6),(7,8)]=>{{1,4},{2,3},{5,6},{7,8}}=>{{1,3},{2,4},{5,6},{7,8}}=>[2,2,2,2] [(1,5),(2,3),(4,6),(7,8)]=>{{1,5},{2,3},{4,6},{7,8}}=>{{1,3},{2,6},{4,5},{7,8}}=>[2,2,2,2] [(1,6),(2,3),(4,5),(7,8)]=>{{1,6},{2,3},{4,5},{7,8}}=>{{1,3},{2,5},{4,6},{7,8}}=>[2,2,2,2] [(1,7),(2,3),(4,5),(6,8)]=>{{1,7},{2,3},{4,5},{6,8}}=>{{1,3},{2,5},{4,8},{6,7}}=>[2,2,2,2] [(1,8),(2,3),(4,5),(6,7)]=>{{1,8},{2,3},{4,5},{6,7}}=>{{1,3},{2,5},{4,7},{6,8}}=>[2,2,2,2] [(1,8),(2,4),(3,5),(6,7)]=>{{1,8},{2,4},{3,5},{6,7}}=>{{1,5},{2,4},{3,7},{6,8}}=>[2,2,2,2] [(1,7),(2,4),(3,5),(6,8)]=>{{1,7},{2,4},{3,5},{6,8}}=>{{1,5},{2,4},{3,8},{6,7}}=>[2,2,2,2] [(1,6),(2,4),(3,5),(7,8)]=>{{1,6},{2,4},{3,5},{7,8}}=>{{1,5},{2,4},{3,6},{7,8}}=>[2,2,2,2] [(1,5),(2,4),(3,6),(7,8)]=>{{1,5},{2,4},{3,6},{7,8}}=>{{1,6},{2,4},{3,5},{7,8}}=>[2,2,2,2] [(1,4),(2,5),(3,6),(7,8)]=>{{1,4},{2,5},{3,6},{7,8}}=>{{1,6},{2,5},{3,4},{7,8}}=>[2,2,2,2] [(1,3),(2,5),(4,6),(7,8)]=>{{1,3},{2,5},{4,6},{7,8}}=>{{1,6},{2,3},{4,5},{7,8}}=>[2,2,2,2] [(1,2),(3,5),(4,6),(7,8)]=>{{1,2},{3,5},{4,6},{7,8}}=>{{1,2},{3,6},{4,5},{7,8}}=>[2,2,2,2] [(1,2),(3,6),(4,5),(7,8)]=>{{1,2},{3,6},{4,5},{7,8}}=>{{1,2},{3,5},{4,6},{7,8}}=>[2,2,2,2] [(1,3),(2,6),(4,5),(7,8)]=>{{1,3},{2,6},{4,5},{7,8}}=>{{1,5},{2,3},{4,6},{7,8}}=>[2,2,2,2] [(1,4),(2,6),(3,5),(7,8)]=>{{1,4},{2,6},{3,5},{7,8}}=>{{1,5},{2,6},{3,4},{7,8}}=>[2,2,2,2] [(1,5),(2,6),(3,4),(7,8)]=>{{1,5},{2,6},{3,4},{7,8}}=>{{1,4},{2,6},{3,5},{7,8}}=>[2,2,2,2] [(1,6),(2,5),(3,4),(7,8)]=>{{1,6},{2,5},{3,4},{7,8}}=>{{1,4},{2,5},{3,6},{7,8}}=>[2,2,2,2] [(1,7),(2,5),(3,4),(6,8)]=>{{1,7},{2,5},{3,4},{6,8}}=>{{1,4},{2,5},{3,8},{6,7}}=>[2,2,2,2] [(1,8),(2,5),(3,4),(6,7)]=>{{1,8},{2,5},{3,4},{6,7}}=>{{1,4},{2,5},{3,7},{6,8}}=>[2,2,2,2] [(1,8),(2,6),(3,4),(5,7)]=>{{1,8},{2,6},{3,4},{5,7}}=>{{1,4},{2,7},{3,6},{5,8}}=>[2,2,2,2] [(1,7),(2,6),(3,4),(5,8)]=>{{1,7},{2,6},{3,4},{5,8}}=>{{1,4},{2,8},{3,6},{5,7}}=>[2,2,2,2] [(1,6),(2,7),(3,4),(5,8)]=>{{1,6},{2,7},{3,4},{5,8}}=>{{1,4},{2,8},{3,7},{5,6}}=>[2,2,2,2] [(1,5),(2,7),(3,4),(6,8)]=>{{1,5},{2,7},{3,4},{6,8}}=>{{1,4},{2,8},{3,5},{6,7}}=>[2,2,2,2] [(1,4),(2,7),(3,5),(6,8)]=>{{1,4},{2,7},{3,5},{6,8}}=>{{1,5},{2,8},{3,4},{6,7}}=>[2,2,2,2] [(1,3),(2,7),(4,5),(6,8)]=>{{1,3},{2,7},{4,5},{6,8}}=>{{1,5},{2,3},{4,8},{6,7}}=>[2,2,2,2] [(1,2),(3,7),(4,5),(6,8)]=>{{1,2},{3,7},{4,5},{6,8}}=>{{1,2},{3,5},{4,8},{6,7}}=>[2,2,2,2] [(1,2),(3,8),(4,5),(6,7)]=>{{1,2},{3,8},{4,5},{6,7}}=>{{1,2},{3,5},{4,7},{6,8}}=>[2,2,2,2] [(1,3),(2,8),(4,5),(6,7)]=>{{1,3},{2,8},{4,5},{6,7}}=>{{1,5},{2,3},{4,7},{6,8}}=>[2,2,2,2] [(1,4),(2,8),(3,5),(6,7)]=>{{1,4},{2,8},{3,5},{6,7}}=>{{1,5},{2,7},{3,4},{6,8}}=>[2,2,2,2] [(1,5),(2,8),(3,4),(6,7)]=>{{1,5},{2,8},{3,4},{6,7}}=>{{1,4},{2,7},{3,5},{6,8}}=>[2,2,2,2] [(1,6),(2,8),(3,4),(5,7)]=>{{1,6},{2,8},{3,4},{5,7}}=>{{1,4},{2,7},{3,8},{5,6}}=>[2,2,2,2] [(1,7),(2,8),(3,4),(5,6)]=>{{1,7},{2,8},{3,4},{5,6}}=>{{1,4},{2,6},{3,8},{5,7}}=>[2,2,2,2] [(1,8),(2,7),(3,4),(5,6)]=>{{1,8},{2,7},{3,4},{5,6}}=>{{1,4},{2,6},{3,7},{5,8}}=>[2,2,2,2] [(1,8),(2,7),(3,5),(4,6)]=>{{1,8},{2,7},{3,5},{4,6}}=>{{1,6},{2,5},{3,7},{4,8}}=>[2,2,2,2] [(1,7),(2,8),(3,5),(4,6)]=>{{1,7},{2,8},{3,5},{4,6}}=>{{1,6},{2,5},{3,8},{4,7}}=>[2,2,2,2] [(1,6),(2,8),(3,5),(4,7)]=>{{1,6},{2,8},{3,5},{4,7}}=>{{1,7},{2,5},{3,8},{4,6}}=>[2,2,2,2] [(1,5),(2,8),(3,6),(4,7)]=>{{1,5},{2,8},{3,6},{4,7}}=>{{1,7},{2,6},{3,8},{4,5}}=>[2,2,2,2] [(1,4),(2,8),(3,6),(5,7)]=>{{1,4},{2,8},{3,6},{5,7}}=>{{1,7},{2,6},{3,4},{5,8}}=>[2,2,2,2] [(1,3),(2,8),(4,6),(5,7)]=>{{1,3},{2,8},{4,6},{5,7}}=>{{1,7},{2,3},{4,6},{5,8}}=>[2,2,2,2] [(1,2),(3,8),(4,6),(5,7)]=>{{1,2},{3,8},{4,6},{5,7}}=>{{1,2},{3,7},{4,6},{5,8}}=>[2,2,2,2] [(1,2),(3,7),(4,6),(5,8)]=>{{1,2},{3,7},{4,6},{5,8}}=>{{1,2},{3,8},{4,6},{5,7}}=>[2,2,2,2] [(1,3),(2,7),(4,6),(5,8)]=>{{1,3},{2,7},{4,6},{5,8}}=>{{1,8},{2,3},{4,6},{5,7}}=>[2,2,2,2] [(1,4),(2,7),(3,6),(5,8)]=>{{1,4},{2,7},{3,6},{5,8}}=>{{1,8},{2,6},{3,4},{5,7}}=>[2,2,2,2] [(1,5),(2,7),(3,6),(4,8)]=>{{1,5},{2,7},{3,6},{4,8}}=>{{1,8},{2,6},{3,7},{4,5}}=>[2,2,2,2] [(1,6),(2,7),(3,5),(4,8)]=>{{1,6},{2,7},{3,5},{4,8}}=>{{1,8},{2,5},{3,7},{4,6}}=>[2,2,2,2] [(1,7),(2,6),(3,5),(4,8)]=>{{1,7},{2,6},{3,5},{4,8}}=>{{1,8},{2,5},{3,6},{4,7}}=>[2,2,2,2] [(1,8),(2,6),(3,5),(4,7)]=>{{1,8},{2,6},{3,5},{4,7}}=>{{1,7},{2,5},{3,6},{4,8}}=>[2,2,2,2] [(1,8),(2,5),(3,6),(4,7)]=>{{1,8},{2,5},{3,6},{4,7}}=>{{1,7},{2,6},{3,5},{4,8}}=>[2,2,2,2] [(1,7),(2,5),(3,6),(4,8)]=>{{1,7},{2,5},{3,6},{4,8}}=>{{1,8},{2,6},{3,5},{4,7}}=>[2,2,2,2] [(1,6),(2,5),(3,7),(4,8)]=>{{1,6},{2,5},{3,7},{4,8}}=>{{1,8},{2,7},{3,5},{4,6}}=>[2,2,2,2] [(1,5),(2,6),(3,7),(4,8)]=>{{1,5},{2,6},{3,7},{4,8}}=>{{1,8},{2,7},{3,6},{4,5}}=>[2,2,2,2] [(1,4),(2,6),(3,7),(5,8)]=>{{1,4},{2,6},{3,7},{5,8}}=>{{1,8},{2,7},{3,4},{5,6}}=>[2,2,2,2] [(1,3),(2,6),(4,7),(5,8)]=>{{1,3},{2,6},{4,7},{5,8}}=>{{1,8},{2,3},{4,7},{5,6}}=>[2,2,2,2] [(1,2),(3,6),(4,7),(5,8)]=>{{1,2},{3,6},{4,7},{5,8}}=>{{1,2},{3,8},{4,7},{5,6}}=>[2,2,2,2] [(1,2),(3,5),(4,7),(6,8)]=>{{1,2},{3,5},{4,7},{6,8}}=>{{1,2},{3,8},{4,5},{6,7}}=>[2,2,2,2] [(1,3),(2,5),(4,7),(6,8)]=>{{1,3},{2,5},{4,7},{6,8}}=>{{1,8},{2,3},{4,5},{6,7}}=>[2,2,2,2] [(1,4),(2,5),(3,7),(6,8)]=>{{1,4},{2,5},{3,7},{6,8}}=>{{1,8},{2,5},{3,4},{6,7}}=>[2,2,2,2] [(1,5),(2,4),(3,7),(6,8)]=>{{1,5},{2,4},{3,7},{6,8}}=>{{1,8},{2,4},{3,5},{6,7}}=>[2,2,2,2] [(1,6),(2,4),(3,7),(5,8)]=>{{1,6},{2,4},{3,7},{5,8}}=>{{1,8},{2,4},{3,7},{5,6}}=>[2,2,2,2] [(1,7),(2,4),(3,6),(5,8)]=>{{1,7},{2,4},{3,6},{5,8}}=>{{1,8},{2,4},{3,6},{5,7}}=>[2,2,2,2] [(1,8),(2,4),(3,6),(5,7)]=>{{1,8},{2,4},{3,6},{5,7}}=>{{1,7},{2,4},{3,6},{5,8}}=>[2,2,2,2] [(1,8),(2,3),(4,6),(5,7)]=>{{1,8},{2,3},{4,6},{5,7}}=>{{1,3},{2,7},{4,6},{5,8}}=>[2,2,2,2] [(1,7),(2,3),(4,6),(5,8)]=>{{1,7},{2,3},{4,6},{5,8}}=>{{1,3},{2,8},{4,6},{5,7}}=>[2,2,2,2] [(1,6),(2,3),(4,7),(5,8)]=>{{1,6},{2,3},{4,7},{5,8}}=>{{1,3},{2,8},{4,7},{5,6}}=>[2,2,2,2] [(1,5),(2,3),(4,7),(6,8)]=>{{1,5},{2,3},{4,7},{6,8}}=>{{1,3},{2,8},{4,5},{6,7}}=>[2,2,2,2] [(1,4),(2,3),(5,7),(6,8)]=>{{1,4},{2,3},{5,7},{6,8}}=>{{1,3},{2,4},{5,8},{6,7}}=>[2,2,2,2] [(1,3),(2,4),(5,7),(6,8)]=>{{1,3},{2,4},{5,7},{6,8}}=>{{1,4},{2,3},{5,8},{6,7}}=>[2,2,2,2] [(1,2),(3,4),(5,7),(6,8)]=>{{1,2},{3,4},{5,7},{6,8}}=>{{1,2},{3,4},{5,8},{6,7}}=>[2,2,2,2] [(1,2),(3,4),(5,8),(6,7)]=>{{1,2},{3,4},{5,8},{6,7}}=>{{1,2},{3,4},{5,7},{6,8}}=>[2,2,2,2] [(1,3),(2,4),(5,8),(6,7)]=>{{1,3},{2,4},{5,8},{6,7}}=>{{1,4},{2,3},{5,7},{6,8}}=>[2,2,2,2] [(1,4),(2,3),(5,8),(6,7)]=>{{1,4},{2,3},{5,8},{6,7}}=>{{1,3},{2,4},{5,7},{6,8}}=>[2,2,2,2] [(1,5),(2,3),(4,8),(6,7)]=>{{1,5},{2,3},{4,8},{6,7}}=>{{1,3},{2,7},{4,5},{6,8}}=>[2,2,2,2] [(1,6),(2,3),(4,8),(5,7)]=>{{1,6},{2,3},{4,8},{5,7}}=>{{1,3},{2,7},{4,8},{5,6}}=>[2,2,2,2] [(1,7),(2,3),(4,8),(5,6)]=>{{1,7},{2,3},{4,8},{5,6}}=>{{1,3},{2,6},{4,8},{5,7}}=>[2,2,2,2] [(1,8),(2,3),(4,7),(5,6)]=>{{1,8},{2,3},{4,7},{5,6}}=>{{1,3},{2,6},{4,7},{5,8}}=>[2,2,2,2] [(1,8),(2,4),(3,7),(5,6)]=>{{1,8},{2,4},{3,7},{5,6}}=>{{1,6},{2,4},{3,7},{5,8}}=>[2,2,2,2] [(1,7),(2,4),(3,8),(5,6)]=>{{1,7},{2,4},{3,8},{5,6}}=>{{1,6},{2,4},{3,8},{5,7}}=>[2,2,2,2] [(1,6),(2,4),(3,8),(5,7)]=>{{1,6},{2,4},{3,8},{5,7}}=>{{1,7},{2,4},{3,8},{5,6}}=>[2,2,2,2] [(1,5),(2,4),(3,8),(6,7)]=>{{1,5},{2,4},{3,8},{6,7}}=>{{1,7},{2,4},{3,5},{6,8}}=>[2,2,2,2] [(1,4),(2,5),(3,8),(6,7)]=>{{1,4},{2,5},{3,8},{6,7}}=>{{1,7},{2,5},{3,4},{6,8}}=>[2,2,2,2] [(1,3),(2,5),(4,8),(6,7)]=>{{1,3},{2,5},{4,8},{6,7}}=>{{1,7},{2,3},{4,5},{6,8}}=>[2,2,2,2] [(1,2),(3,5),(4,8),(6,7)]=>{{1,2},{3,5},{4,8},{6,7}}=>{{1,2},{3,7},{4,5},{6,8}}=>[2,2,2,2] [(1,2),(3,6),(4,8),(5,7)]=>{{1,2},{3,6},{4,8},{5,7}}=>{{1,2},{3,7},{4,8},{5,6}}=>[2,2,2,2] [(1,3),(2,6),(4,8),(5,7)]=>{{1,3},{2,6},{4,8},{5,7}}=>{{1,7},{2,3},{4,8},{5,6}}=>[2,2,2,2] [(1,4),(2,6),(3,8),(5,7)]=>{{1,4},{2,6},{3,8},{5,7}}=>{{1,7},{2,8},{3,4},{5,6}}=>[2,2,2,2] [(1,5),(2,6),(3,8),(4,7)]=>{{1,5},{2,6},{3,8},{4,7}}=>{{1,7},{2,8},{3,6},{4,5}}=>[2,2,2,2] [(1,6),(2,5),(3,8),(4,7)]=>{{1,6},{2,5},{3,8},{4,7}}=>{{1,7},{2,8},{3,5},{4,6}}=>[2,2,2,2] [(1,7),(2,5),(3,8),(4,6)]=>{{1,7},{2,5},{3,8},{4,6}}=>{{1,6},{2,8},{3,5},{4,7}}=>[2,2,2,2] [(1,8),(2,5),(3,7),(4,6)]=>{{1,8},{2,5},{3,7},{4,6}}=>{{1,6},{2,7},{3,5},{4,8}}=>[2,2,2,2] [(1,8),(2,6),(3,7),(4,5)]=>{{1,8},{2,6},{3,7},{4,5}}=>{{1,5},{2,7},{3,6},{4,8}}=>[2,2,2,2] [(1,7),(2,6),(3,8),(4,5)]=>{{1,7},{2,6},{3,8},{4,5}}=>{{1,5},{2,8},{3,6},{4,7}}=>[2,2,2,2] [(1,6),(2,7),(3,8),(4,5)]=>{{1,6},{2,7},{3,8},{4,5}}=>{{1,5},{2,8},{3,7},{4,6}}=>[2,2,2,2] [(1,5),(2,7),(3,8),(4,6)]=>{{1,5},{2,7},{3,8},{4,6}}=>{{1,6},{2,8},{3,7},{4,5}}=>[2,2,2,2] [(1,4),(2,7),(3,8),(5,6)]=>{{1,4},{2,7},{3,8},{5,6}}=>{{1,6},{2,8},{3,4},{5,7}}=>[2,2,2,2] [(1,3),(2,7),(4,8),(5,6)]=>{{1,3},{2,7},{4,8},{5,6}}=>{{1,6},{2,3},{4,8},{5,7}}=>[2,2,2,2] [(1,2),(3,7),(4,8),(5,6)]=>{{1,2},{3,7},{4,8},{5,6}}=>{{1,2},{3,6},{4,8},{5,7}}=>[2,2,2,2] [(1,2),(3,8),(4,7),(5,6)]=>{{1,2},{3,8},{4,7},{5,6}}=>{{1,2},{3,6},{4,7},{5,8}}=>[2,2,2,2] [(1,3),(2,8),(4,7),(5,6)]=>{{1,3},{2,8},{4,7},{5,6}}=>{{1,6},{2,3},{4,7},{5,8}}=>[2,2,2,2] [(1,4),(2,8),(3,7),(5,6)]=>{{1,4},{2,8},{3,7},{5,6}}=>{{1,6},{2,7},{3,4},{5,8}}=>[2,2,2,2] [(1,5),(2,8),(3,7),(4,6)]=>{{1,5},{2,8},{3,7},{4,6}}=>{{1,6},{2,7},{3,8},{4,5}}=>[2,2,2,2] [(1,6),(2,8),(3,7),(4,5)]=>{{1,6},{2,8},{3,7},{4,5}}=>{{1,5},{2,7},{3,8},{4,6}}=>[2,2,2,2] [(1,7),(2,8),(3,6),(4,5)]=>{{1,7},{2,8},{3,6},{4,5}}=>{{1,5},{2,6},{3,8},{4,7}}=>[2,2,2,2] [(1,8),(2,7),(3,6),(4,5)]=>{{1,8},{2,7},{3,6},{4,5}}=>{{1,5},{2,6},{3,7},{4,8}}=>[2,2,2,2]
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.
Map
shape
Description
Sends a set partition to the integer partition obtained by the sizes of the blocks.