1
Descents, Eulerian numbers, and refined descent counts
▶
1.1
Descents of a permutation
1.2
Eulerian numbers
1.3
The refined count \(\beta _n(S)\)
2
Min-max trees and the \(\psi _{i}\) operators
▶
2.1
The min-max tree of a sequence
2.2
Windows, internal vertices, and left children
2.3
The operators \(\psi _{i}\)
3
The cd-index and Stanley’s Fact #3
▶
3.1
The cd-letter alphabet
3.2
The cd \(\to \) ab expansion
3.3
Stanley’s identity for \(\Phi _{w}\)
3.4
Stanley Fact #3
4
The toggle action and the \(\omega (S)\) map
▶
4.1
The toggle action
4.2
The \(\omega (D)\) map
4.3
The set \(\leftrightarrow \) seq bridges
4.4
The alternating descent set and equation (1.65)
5
Proposition 1.6.4 and Corollary 1.6.5
▶
5.1
The witness construction
5.2
Stanley Proposition 1.6.4
5.3
Stanley Corollary 1.6.5 — the headline
5.4
What this gives a Stanley reader
6
Cycles, inversions, and the major index
▶
6.1
Cycle representation of permutations
6.2
Stirling numbers of the first kind
6.3
Inversions
6.4
The major index
6.5
Foata’s first fundamental bijection and MacMahon’s equidistribution
6.6
The \(q\)-factorial generating function
6.7
The \(q\)-Eulerian polynomial — joint \((\mathrm{maj},\operatorname {des})\) distribution
6.8
Longest alternating subsequences (Stanley §1.6.2)
6.9
André’s reflection method (Stanley §1.6.4)
Dependency graph
Permutation descents, Eulerian numbers, and the cd-index
A Rocq / MathComp formalization of Stanley EC1 §1.4 + §1.6
Marc Lelarge