Identifier
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Images
=>
Cc0005;cc-rep-0Cc0012;cc-rep-1Cc0012;cc-rep-2
[1,0]=>[(1,2)]=>[(1,2)]=>[2,1]=>[1,1] [1,0,1,0]=>[(1,2),(3,4)]=>[(1,2),(3,4)]=>[2,1,4,3]=>[1,2,1] [1,1,0,0]=>[(1,4),(2,3)]=>[(1,3),(2,4)]=>[3,4,1,2]=>[2,2] [1,0,1,0,1,0]=>[(1,2),(3,4),(5,6)]=>[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[1,2,2,1] [1,0,1,1,0,0]=>[(1,2),(3,6),(4,5)]=>[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[1,3,2] [1,1,0,0,1,0]=>[(1,4),(2,3),(5,6)]=>[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[2,3,1] [1,1,0,1,0,0]=>[(1,6),(2,3),(4,5)]=>[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[2,2,2] [1,1,1,0,0,0]=>[(1,6),(2,5),(3,4)]=>[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[3,3] [1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8)]=>[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[1,2,2,2,1] [1,0,1,0,1,1,0,0]=>[(1,2),(3,4),(5,8),(6,7)]=>[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[1,2,3,2] [1,0,1,1,0,0,1,0]=>[(1,2),(3,6),(4,5),(7,8)]=>[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[1,3,3,1] [1,0,1,1,0,1,0,0]=>[(1,2),(3,8),(4,5),(6,7)]=>[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[1,3,2,2] [1,0,1,1,1,0,0,0]=>[(1,2),(3,8),(4,7),(5,6)]=>[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[1,4,3] [1,1,0,0,1,0,1,0]=>[(1,4),(2,3),(5,6),(7,8)]=>[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[2,3,2,1] [1,1,0,0,1,1,0,0]=>[(1,4),(2,3),(5,8),(6,7)]=>[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[2,4,2] [1,1,0,1,0,0,1,0]=>[(1,6),(2,3),(4,5),(7,8)]=>[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[2,2,3,1] [1,1,0,1,0,1,0,0]=>[(1,8),(2,3),(4,5),(6,7)]=>[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[2,2,2,2] [1,1,0,1,1,0,0,0]=>[(1,8),(2,3),(4,7),(5,6)]=>[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[2,3,3] [1,1,1,0,0,0,1,0]=>[(1,6),(2,5),(3,4),(7,8)]=>[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[3,4,1] [1,1,1,0,0,1,0,0]=>[(1,8),(2,5),(3,4),(6,7)]=>[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[3,3,2] [1,1,1,0,1,0,0,0]=>[(1,8),(2,7),(3,4),(5,6)]=>[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[3,2,3] [1,1,1,1,0,0,0,0]=>[(1,8),(2,7),(3,6),(4,5)]=>[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[4,4] [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)]=>[2,1,4,3,6,5,8,7,10,9]=>[1,2,2,2,2,1] [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)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[1,2,2,2,2,2,1]
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
Kasraoui-Zeng
Description
The Kasraoui-Zeng involution for perfect matchings.
This yields the perfect matching with the number of nestings and crossings exchanged.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.