Generating Functions for Alternating Descents and Alternating Major Index

摘要

In 2008, Chebikin introduced the alternating descent set, AltDes(σ), of a permutation σ = σ ₁ ··· σ _n in the symmetric group S _n as the set of all i such that either i is odd and σ _i > σ _i+1 or i is even and σ _i < σ _i+1. We can then define altdes(σ) = |AltDes(σ)| and altmaj(s) = å_{i Î AltDes(s)}i{{rm altmaj}(sigma) = sum_{i in AltDes(sigma)}i}. In this paper, we compute a generating function for the joint distribution of altdes(σ) and altmaj(σ) over S _n . Our formula is similar to the formula for the joint distribution of des and maj over the symmetric group that was first proved by Gessel. We also compute similar generating functions for the groups B _n and D _n and for r-tuples of permutations in S _n . Finally we prove a general extension of these formulas in cases where we keep track of descents only at positions r, 2r, . . ..