An explicit formula for the number of permutations with a given number of alternating runs

TL;DR

This paper derives an explicit formula for counting permutations of n elements with exactly k alternating runs, providing a precise mathematical tool for analyzing permutation structures.

Contribution

The paper introduces a new explicit formula for R(n,k), the number of permutations with k alternating runs, advancing combinatorial enumeration methods.

Findings

Provides an explicit formula for R(n,k)

Enables direct computation of permutations with given alternating runs

Enhances understanding of permutation structure

Abstract

Let $R(n,k)$ denote the number of permutations of ${1,2,...,n}$ with $k$ alternating runs. In this note we present an explicit formula for the numbers $R(n,k)$.