Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤ
Values
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,0,1,1,0,1,0,0,1,0] => [1,3,5,2,6,4] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,0,1,1,1,0,0,1,0,0] => [1,3,2,6,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,4,2,6,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,5,2,6,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,2,5,6,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,0,0,1,1,0,0] => [1,4,2,5,3,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,5,2,6,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,1,0,0,0,1,0] => [1,4,2,3,6,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,6,3,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,5,2,6,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,1,0,0,0,1,0,0,1,0] => [1,2,5,3,6,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,1,0,0,1,0,0,0,1,0] => [1,5,2,3,6,4] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,6,3,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,7,6] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [2,3,1,6,4,7,5] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3,7,6] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,6,7,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,6,3,7,5] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,4,3,5,7,6] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,4,3,7,5,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [2,4,1,5,3,7,6] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,1,6,3,7,5] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [2,1,3,5,4,7,6] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,0,0,1,0,0,1,0,1,0] => [2,1,5,3,6,7,4] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0] => [2,1,5,3,6,4,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,0,0,1,0,1,0,0,1,0] => [2,1,5,6,3,7,4] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [2,1,5,3,4,7,6] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [2,1,5,3,7,4,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [2,5,1,6,3,7,4] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0] => [2,1,3,6,4,7,5] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,1,0,0,1,0,0,0,1,0] => [2,1,6,3,4,7,5] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,1,1,0,0,1,0,0,1,0,0] => [2,1,6,3,7,4,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,3,4,5,2,7,6] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [1,3,4,2,6,7,5] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [1,3,4,2,6,5,7] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [1,3,4,6,2,7,5] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,3,4,2,5,7,6] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [1,3,4,2,7,5,6] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,5,6,7,4] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,5,6,4,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,7,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,7,4,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,5,4,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [1,3,5,2,6,7,4] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [1,3,5,2,6,4,7] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [1,3,5,6,2,7,4] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,7,6] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,7,4,6] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,3,2,4,6,7,5] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2,4,6,5,7] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,0] => [1,3,2,6,4,7,5] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [1,3,2,6,7,4,5] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [1,3,2,6,4,5,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,0,1,0,0,0,1,0] => [1,3,6,2,4,7,5] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,0,1,0,0,1,0,0] => [1,3,6,2,7,4,5] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,3,2,4,5,7,6] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,1,0,0,0,1,0,0] => [1,3,2,4,7,5,6] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,0,1,1,1,1,0,0,1,0,0,0] => [1,3,2,7,4,5,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [3,1,4,5,2,7,6] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [3,1,4,2,6,7,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [3,1,4,2,6,5,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [3,1,4,6,2,7,5] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,0,0,1,1,1,0,0,0,1,0] => [3,1,4,2,5,7,6] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,0,0,1,1,1,0,0,1,0,0] => [3,1,4,2,7,5,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [3,4,1,5,2,7,6] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,0,1,1,0,0,1,0,0,1,0] => [3,4,1,6,2,7,5] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,0,0,0,1,1,0,0,1,0] => [3,1,2,5,4,7,6] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,0,0,1,0,0,1,0,1,0] => [3,1,5,2,6,7,4] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,0,0,1,0,0,1,1,0,0] => [3,1,5,2,6,4,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,0,0,1,0,1,0,0,1,0] => [3,1,5,6,2,7,4] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,0,0,1,1,0,0,0,1,0] => [3,1,5,2,4,7,6] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [3,1,5,2,7,4,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [3,5,1,6,2,7,4] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,1,0,0,0,1,0,0,1,0] => [3,1,2,6,4,7,5] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [3,1,6,2,4,7,5] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [3,1,6,2,7,4,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,2,4,5,3,7,6] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,2,4,3,6,7,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [1,2,4,6,3,7,5] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,2,4,3,5,7,6] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [1,2,4,3,7,5,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [1,4,2,5,6,7,3] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,1,0,0,1,0,1,1,0,0] => [1,4,2,5,6,3,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,1,0,0,1,1,0,0,1,0] => [1,4,2,5,3,7,6] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,1,0,0,1,1,0,1,0,0] => [1,4,2,5,7,3,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [1,4,2,5,3,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [1,4,5,2,6,7,3] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [1,4,5,2,6,3,7] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [1,4,5,6,2,7,3] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,1,0,0,1,0,1,1,0,0,0,1,0] => [1,4,5,2,3,7,6] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3
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Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
This is the length of the shortest cycle in the graph.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
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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