Counting permutations by alternating descents

We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers. We give two proofs of the formula. The first uses a system of differential equations. The second proof derives the generating function directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function is an "alternating" analogue of David and Barton's generating function for permutations with no increasing runs of length 3 or more. Our general results give further alternating analogues of permutation enumeration formulas, including results of Chebikin and Remmel

Enumeration of permutations by the parity of descent position

Noticing that some recent variations of descent polynomials are special cases of Carlitz and Scoville's four-variable polynomials, which enumerate permutations by the parity of descent and ascent position, we prove a q-analogue of Carlitz-Scoville's generating function by counting the inversion number and a type B analogue by enumerating the signed permutations with respect to the parity of desecnt and ascent position. As a by-product of our formulas, we obtain a q-analogue of Chebikin's formula for alternating descent polynomials, an alternative proof of Sun's gamma-positivity of her bivariate Eulerian polynomials and a type B analogue, the latter refines Petersen's gamma-positivity of the type B Eulerian polynomials.Comment: 26 page

Two-sided permutation statistics via symmetric functions

Given a permutation statistic $\operatorname{st}$, define its inverse statistic $\operatorname{ist}$ by $\operatorname{ist}(\pi):=\operatorname{st}(\pi)$. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of $\operatorname{st}_{1}$ and $\operatorname{ist}_{2}$ whenever $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$ are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs, and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of $\operatorname{st}_{1}$ and $\operatorname{ist}_{2}$ can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$. Our work leads to a rederivation of Stanley's generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and $\gamma$-positivity.Comment: 43 page