Identifier
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Images
=>
Cc0005;cc-rep-0Cc0012;cc-rep-1
[1,0]=>[(1,2)]=>[2,1]=>[2] [1,0,1,0]=>[(1,2),(3,4)]=>[2,1,4,3]=>[2,2] [1,1,0,0]=>[(1,4),(2,3)]=>[4,3,2,1]=>[1,2,1] [1,0,1,0,1,0]=>[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,2,2] [1,0,1,1,0,0]=>[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,2,1] [1,1,0,0,1,0]=>[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[1,2,1,2] [1,1,0,1,0,0]=>[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[1,2,2,1] [1,1,1,0,0,0]=>[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[1,1,2,1,1] [1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,2,2,2] [1,0,1,0,1,1,0,0]=>[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,2,1,2,1] [1,0,1,1,0,0,1,0]=>[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,2,1,2] [1,0,1,1,0,1,0,0]=>[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,2,2,1] [1,0,1,1,1,0,0,0]=>[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,1,2,1,1] [1,1,0,0,1,0,1,0]=>[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[1,2,1,2,2] [1,1,0,0,1,1,0,0]=>[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[1,2,1,1,2,1] [1,1,0,1,0,0,1,0]=>[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[1,2,2,1,2] [1,1,0,1,0,1,0,0]=>[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[1,2,2,2,1] [1,1,0,1,1,0,0,0]=>[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[1,2,1,2,1,1] [1,1,1,0,0,0,1,0]=>[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[1,1,2,1,1,2] [1,1,1,0,0,1,0,0]=>[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[1,1,2,1,2,1] [1,1,1,0,1,0,0,0]=>[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[1,1,2,2,1,1] [1,1,1,1,0,0,0,0]=>[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[1,1,1,2,1,1,1] [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]=>[2,2,2,2,2] [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]=>[2,2,2,2,2,2]
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
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.