A combinatorial proof of the log-concavity of the numbers of permutations with $k$ runs

Abstract

We combinatorially prove that the number $R(n,k)$ of permutations of length $n$ having $k$ runs is a log-concave sequence in $k$, for all $n$. We also give a new combinatorial proof for the log-concavity of the Eulerian numbers.

10 pages, 4 figures