Identifier
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Images
=>
Cc0005;cc-rep-0Cc0012;cc-rep-1
[1,0]=>[(1,2)]=>[2,1]=>[2,1] [1,0,1,0]=>[(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3] [1,1,0,0]=>[(1,4),(2,3)]=>[4,3,2,1]=>[3,2,4,1] [1,0,1,0,1,0]=>[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5] [1,0,1,1,0,0]=>[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3] [1,1,0,0,1,0]=>[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5] [1,1,0,1,0,0]=>[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1] [1,1,1,0,0,0]=>[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1] [1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7] [1,0,1,0,1,1,0,0]=>[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,8,5] [1,0,1,1,0,0,1,0]=>[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,5,4,6,3,8,7] [1,0,1,1,0,1,0,0]=>[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,5,4,7,6,8,3] [1,0,1,1,1,0,0,0]=>[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3] [1,1,0,0,1,0,1,0]=>[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[3,2,4,1,6,5,8,7] [1,1,0,0,1,1,0,0]=>[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5] [1,1,0,1,0,0,1,0]=>[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[3,2,5,4,6,1,8,7] [1,1,0,1,0,1,0,0]=>[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[3,2,5,4,7,6,8,1] [1,1,0,1,1,0,0,0]=>[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1] [1,1,1,0,0,0,1,0]=>[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[4,3,5,2,6,1,8,7] [1,1,1,0,0,1,0,0]=>[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[4,3,5,2,7,6,8,1] [1,1,1,0,1,0,0,0]=>[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1] [1,1,1,1,0,0,0,0]=>[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1] [1,0,1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[2,1,4,3,6,5,8,7,10,9] [1,0,1,0,1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,1,4,3,6,5,8,7,10,9,12,11]
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 permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].