- FindStat

Identifier

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00164: Set partitions —Chen Deng Du Stanley Yan⟶ Set partitions

Images

=>
            
              Cc0005;cc-rep-0Cc0012;cc-rep-1Cc0009;cc-rep-2Cc0009;cc-rep-3
            

            [1,0]=>[(1,2)]=>{{1,2}}=>{{1,2}}
[1,0,1,0]=>[(1,2),(3,4)]=>{{1,2},{3,4}}=>{{1,2},{3,4}}
[1,1,0,0]=>[(1,4),(2,3)]=>{{1,4},{2,3}}=>{{1,3},{2,4}}
[1,0,1,0,1,0]=>[(1,2),(3,4),(5,6)]=>{{1,2},{3,4},{5,6}}=>{{1,2},{3,4},{5,6}}
[1,0,1,1,0,0]=>[(1,2),(3,6),(4,5)]=>{{1,2},{3,6},{4,5}}=>{{1,2},{3,5},{4,6}}
[1,1,0,0,1,0]=>[(1,4),(2,3),(5,6)]=>{{1,4},{2,3},{5,6}}=>{{1,3},{2,4},{5,6}}
[1,1,0,1,0,0]=>[(1,6),(2,3),(4,5)]=>{{1,6},{2,3},{4,5}}=>{{1,3},{2,5},{4,6}}
[1,1,1,0,0,0]=>[(1,6),(2,5),(3,4)]=>{{1,6},{2,5},{3,4}}=>{{1,4},{2,5},{3,6}}
[1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8)]=>{{1,2},{3,4},{5,6},{7,8}}=>{{1,2},{3,4},{5,6},{7,8}}
[1,0,1,0,1,1,0,0]=>[(1,2),(3,4),(5,8),(6,7)]=>{{1,2},{3,4},{5,8},{6,7}}=>{{1,2},{3,4},{5,7},{6,8}}
[1,0,1,1,0,0,1,0]=>[(1,2),(3,6),(4,5),(7,8)]=>{{1,2},{3,6},{4,5},{7,8}}=>{{1,2},{3,5},{4,6},{7,8}}
[1,0,1,1,0,1,0,0]=>[(1,2),(3,8),(4,5),(6,7)]=>{{1,2},{3,8},{4,5},{6,7}}=>{{1,2},{3,5},{4,7},{6,8}}
[1,0,1,1,1,0,0,0]=>[(1,2),(3,8),(4,7),(5,6)]=>{{1,2},{3,8},{4,7},{5,6}}=>{{1,2},{3,6},{4,7},{5,8}}
[1,1,0,0,1,0,1,0]=>[(1,4),(2,3),(5,6),(7,8)]=>{{1,4},{2,3},{5,6},{7,8}}=>{{1,3},{2,4},{5,6},{7,8}}
[1,1,0,0,1,1,0,0]=>[(1,4),(2,3),(5,8),(6,7)]=>{{1,4},{2,3},{5,8},{6,7}}=>{{1,3},{2,4},{5,7},{6,8}}
[1,1,0,1,0,0,1,0]=>[(1,6),(2,3),(4,5),(7,8)]=>{{1,6},{2,3},{4,5},{7,8}}=>{{1,3},{2,5},{4,6},{7,8}}
[1,1,0,1,0,1,0,0]=>[(1,8),(2,3),(4,5),(6,7)]=>{{1,8},{2,3},{4,5},{6,7}}=>{{1,3},{2,5},{4,7},{6,8}}
[1,1,0,1,1,0,0,0]=>[(1,8),(2,3),(4,7),(5,6)]=>{{1,8},{2,3},{4,7},{5,6}}=>{{1,3},{2,6},{4,7},{5,8}}
[1,1,1,0,0,0,1,0]=>[(1,6),(2,5),(3,4),(7,8)]=>{{1,6},{2,5},{3,4},{7,8}}=>{{1,4},{2,5},{3,6},{7,8}}
[1,1,1,0,0,1,0,0]=>[(1,8),(2,5),(3,4),(6,7)]=>{{1,8},{2,5},{3,4},{6,7}}=>{{1,4},{2,5},{3,7},{6,8}}
[1,1,1,0,1,0,0,0]=>[(1,8),(2,7),(3,4),(5,6)]=>{{1,8},{2,7},{3,4},{5,6}}=>{{1,4},{2,6},{3,7},{5,8}}
[1,1,1,1,0,0,0,0]=>[(1,8),(2,7),(3,6),(4,5)]=>{{1,8},{2,7},{3,6},{4,5}}=>{{1,5},{2,6},{3,7},{4,8}}
[1,0,1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8),(9,10)]=>{{1,2},{3,4},{5,6},{7,8},{9,10}}=>{{1,2},{3,4},{5,6},{7,8},{9,10}}
[1,0,1,0,1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>{{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}=>{{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}
          

Map

to tunnel matching

Description

Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.

Map

to set partition

Description

Return the set partition corresponding to the perfect matching.

Map

Chen Deng Du Stanley Yan

Description

A map that swaps the crossing number and the nesting number of a set partition.