Enumeration of permutations by number of alternating runs

TL;DR

This paper explores the enumeration of permutations based on their alternating runs, providing a grammatical framework, convolution formulas, and linking to Andre permutations.

Contribution

It introduces a grammatical approach to count permutations by alternating runs and connects these counts to Andre permutations, offering new combinatorial insights.

Findings

Developed a grammatical description of R(n,k)

Derived convolution formulas for generating functions

Established a link between alternating runs and Andre permutations

Abstract

Let R(n,k) denote the number of permutations of {1,2,...,n} with k alternating runs. We find a grammatical description of the numbers R(n,k) and then present several convolution formulas involving the generating function for the numbers R(n,k). Moreover, we establish a connection between alternating runs and Andre permutations.