Identifier
Mp00165:
Perfect matchings
—Chen Deng Du Stanley Yan⟶
Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Images
=>
Cc0012;cc-rep-0Cc0012;cc-rep-1
[(1,2)]=>[(1,2)]=>[2,1]=>[2]
[(1,2),(3,4)]=>[(1,2),(3,4)]=>[2,1,4,3]=>[2,2]
[(1,3),(2,4)]=>[(1,4),(2,3)]=>[4,3,2,1]=>[1,2,1]
[(1,4),(2,3)]=>[(1,3),(2,4)]=>[3,4,1,2]=>[4]
[(1,2),(3,4),(5,6)]=>[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,2,2]
[(1,3),(2,4),(5,6)]=>[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[1,2,1,2]
[(1,4),(2,3),(5,6)]=>[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,2]
[(1,5),(2,3),(4,6)]=>[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,2,1]
[(1,6),(2,3),(4,5)]=>[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,3]
[(1,6),(2,4),(3,5)]=>[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[2,2,2]
[(1,5),(2,4),(3,6)]=>[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[2,3,1]
[(1,4),(2,5),(3,6)]=>[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[1,1,2,1,1]
[(1,3),(2,5),(4,6)]=>[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[1,2,2,1]
[(1,2),(3,5),(4,6)]=>[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,2,1]
[(1,2),(3,6),(4,5)]=>[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,4]
[(1,3),(2,6),(4,5)]=>[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[1,2,3]
[(1,4),(2,6),(3,5)]=>[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[1,3,2]
[(1,5),(2,6),(3,4)]=>[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[1,4,1]
[(1,6),(2,5),(3,4)]=>[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6]
[(1,2),(3,4),(5,6),(7,8)]=>[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,2,2,2]
[(1,3),(2,4),(5,6),(7,8)]=>[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[1,2,1,2,2]
[(1,4),(2,3),(5,6),(7,8)]=>[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,2,2]
[(1,5),(2,3),(4,6),(7,8)]=>[(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[3,2,1,2]
[(1,6),(2,3),(4,5),(7,8)]=>[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,3,2]
[(1,7),(2,3),(4,5),(6,8)]=>[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[3,2,2,1]
[(1,8),(2,3),(4,5),(6,7)]=>[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,2,3]
[(1,8),(2,4),(3,5),(6,7)]=>[(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[2,2,1,3]
[(1,7),(2,4),(3,5),(6,8)]=>[(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[2,2,1,2,1]
[(1,6),(2,4),(3,5),(7,8)]=>[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[2,2,2,2]
[(1,5),(2,4),(3,6),(7,8)]=>[(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[2,3,1,2]
[(1,4),(2,5),(3,6),(7,8)]=>[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[1,1,2,1,1,2]
[(1,3),(2,5),(4,6),(7,8)]=>[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[1,2,2,1,2]
[(1,2),(3,5),(4,6),(7,8)]=>[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,2,1,2]
[(1,2),(3,6),(4,5),(7,8)]=>[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,4,2]
[(1,3),(2,6),(4,5),(7,8)]=>[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[1,2,3,2]
[(1,4),(2,6),(3,5),(7,8)]=>[(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[1,3,2,2]
[(1,5),(2,6),(3,4),(7,8)]=>[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[1,4,1,2]
[(1,6),(2,5),(3,4),(7,8)]=>[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,2]
[(1,7),(2,5),(3,4),(6,8)]=>[(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[5,2,1]
[(1,8),(2,5),(3,4),(6,7)]=>[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[5,3]
[(1,8),(2,6),(3,4),(5,7)]=>[(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[4,2,2]
[(1,7),(2,6),(3,4),(5,8)]=>[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[4,3,1]
[(1,6),(2,7),(3,4),(5,8)]=>[(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[1,3,2,1,1]
[(1,5),(2,7),(3,4),(6,8)]=>[(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[1,4,2,1]
[(1,4),(2,7),(3,5),(6,8)]=>[(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[1,3,1,2,1]
[(1,3),(2,7),(4,5),(6,8)]=>[(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[1,2,2,2,1]
[(1,2),(3,7),(4,5),(6,8)]=>[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,3,2,1]
[(1,2),(3,8),(4,5),(6,7)]=>[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,3,3]
[(1,3),(2,8),(4,5),(6,7)]=>[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[1,2,2,3]
[(1,4),(2,8),(3,5),(6,7)]=>[(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[1,3,1,3]
[(1,5),(2,8),(3,4),(6,7)]=>[(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[1,4,3]
[(1,6),(2,8),(3,4),(5,7)]=>[(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[1,3,2,2]
[(1,7),(2,8),(3,4),(5,6)]=>[(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[1,3,3,1]
[(1,8),(2,7),(3,4),(5,6)]=>[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,4]
[(1,8),(2,7),(3,5),(4,6)]=>[(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[3,2,3]
[(1,7),(2,8),(3,5),(4,6)]=>[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[1,2,2,2,1]
[(1,6),(2,8),(3,5),(4,7)]=>[(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[1,2,3,2]
[(1,5),(2,8),(3,6),(4,7)]=>[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[1,2,2,1,2]
[(1,4),(2,8),(3,6),(5,7)]=>[(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[1,3,2,2]
[(1,3),(2,8),(4,6),(5,7)]=>[(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[1,2,1,2,2]
[(1,2),(3,8),(4,6),(5,7)]=>[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,2,2,2]
[(1,2),(3,7),(4,6),(5,8)]=>[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,2,3,1]
[(1,3),(2,7),(4,6),(5,8)]=>[(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[1,2,1,3,1]
[(1,4),(2,7),(3,6),(5,8)]=>[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[1,3,3,1]
[(1,5),(2,7),(3,6),(4,8)]=>[(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[1,2,2,2,1]
[(1,6),(2,7),(3,5),(4,8)]=>[(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[1,2,3,1,1]
[(1,7),(2,6),(3,5),(4,8)]=>[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[3,4,1]
[(1,8),(2,6),(3,5),(4,7)]=>[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[3,3,2]
[(1,8),(2,5),(3,6),(4,7)]=>[(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[2,1,2,1,2]
[(1,7),(2,5),(3,6),(4,8)]=>[(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[2,1,2,2,1]
[(1,6),(2,5),(3,7),(4,8)]=>[(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[2,1,3,1,1]
[(1,5),(2,6),(3,7),(4,8)]=>[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[1,1,1,2,1,1,1]
[(1,4),(2,6),(3,7),(5,8)]=>[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[1,1,2,2,1,1]
[(1,3),(2,6),(4,7),(5,8)]=>[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[1,2,1,2,1,1]
[(1,2),(3,6),(4,7),(5,8)]=>[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,1,2,1,1]
[(1,2),(3,5),(4,7),(6,8)]=>[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,2,2,1]
[(1,3),(2,5),(4,7),(6,8)]=>[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[1,2,2,2,1]
[(1,4),(2,5),(3,7),(6,8)]=>[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[1,1,2,1,2,1]
[(1,5),(2,4),(3,7),(6,8)]=>[(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[2,3,2,1]
[(1,6),(2,4),(3,7),(5,8)]=>[(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[2,2,2,1,1]
[(1,7),(2,4),(3,6),(5,8)]=>[(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[2,2,3,1]
[(1,8),(2,4),(3,6),(5,7)]=>[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[2,2,2,2]
[(1,8),(2,3),(4,6),(5,7)]=>[(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[3,1,2,2]
[(1,7),(2,3),(4,6),(5,8)]=>[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[3,1,3,1]
[(1,6),(2,3),(4,7),(5,8)]=>[(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[3,1,2,1,1]
[(1,5),(2,3),(4,7),(6,8)]=>[(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[3,2,2,1]
[(1,4),(2,3),(5,7),(6,8)]=>[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,1,2,1]
[(1,3),(2,4),(5,7),(6,8)]=>[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[1,2,1,1,2,1]
[(1,2),(3,4),(5,7),(6,8)]=>[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,2,1,2,1]
[(1,2),(3,4),(5,8),(6,7)]=>[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,2,4]
[(1,3),(2,4),(5,8),(6,7)]=>[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[1,2,1,4]
[(1,4),(2,3),(5,8),(6,7)]=>[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,4]
[(1,5),(2,3),(4,8),(6,7)]=>[(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,2,3]
[(1,6),(2,3),(4,8),(5,7)]=>[(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[3,3,2]
[(1,7),(2,3),(4,8),(5,6)]=>[(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[3,4,1]
[(1,8),(2,3),(4,7),(5,6)]=>[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,5]
[(1,8),(2,4),(3,7),(5,6)]=>[(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[2,2,4]
[(1,7),(2,4),(3,8),(5,6)]=>[(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[2,2,3,1]
[(1,6),(2,4),(3,8),(5,7)]=>[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[2,2,2,2]
[(1,5),(2,4),(3,8),(6,7)]=>[(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[2,3,3]
[(1,4),(2,5),(3,8),(6,7)]=>[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[1,1,2,1,3]
[(1,3),(2,5),(4,8),(6,7)]=>[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[1,2,2,3]
[(1,2),(3,5),(4,8),(6,7)]=>[(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,2,3]
[(1,2),(3,6),(4,8),(5,7)]=>[(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,3,2]
[(1,3),(2,6),(4,8),(5,7)]=>[(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[1,2,3,2]
[(1,4),(2,6),(3,8),(5,7)]=>[(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[1,1,2,2,2]
[(1,5),(2,6),(3,8),(4,7)]=>[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[1,1,3,1,2]
[(1,6),(2,5),(3,8),(4,7)]=>[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[2,4,2]
[(1,7),(2,5),(3,8),(4,6)]=>[(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[2,3,2,1]
[(1,8),(2,5),(3,7),(4,6)]=>[(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[2,3,3]
[(1,8),(2,6),(3,7),(4,5)]=>[(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[2,4,2]
[(1,7),(2,6),(3,8),(4,5)]=>[(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[2,5,1]
[(1,6),(2,7),(3,8),(4,5)]=>[(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[1,1,4,1,1]
[(1,5),(2,7),(3,8),(4,6)]=>[(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[1,1,3,2,1]
[(1,4),(2,7),(3,8),(5,6)]=>[(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[1,1,2,3,1]
[(1,3),(2,7),(4,8),(5,6)]=>[(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[1,2,4,1]
[(1,2),(3,7),(4,8),(5,6)]=>[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,4,1]
[(1,2),(3,8),(4,7),(5,6)]=>[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,6]
[(1,3),(2,8),(4,7),(5,6)]=>[(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[1,2,5]
[(1,4),(2,8),(3,7),(5,6)]=>[(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[1,3,4]
[(1,5),(2,8),(3,7),(4,6)]=>[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[1,4,3]
[(1,6),(2,8),(3,7),(4,5)]=>[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[1,5,2]
[(1,7),(2,8),(3,6),(4,5)]=>[(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[1,6,1]
[(1,8),(2,7),(3,6),(4,5)]=>[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8]
[(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]=>[2,2,2,2,2]
[(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]=>[2,2,2,2,2,2]
Map
Chen Deng Du Stanley Yan
Description
A map that swaps the crossing number and the nesting number of a perfect matching.
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.
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.
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