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

Miklós Bóna·Richard Ehrenborg
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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.

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Miklós Bóna, Richard Ehrenborg. 1999-02-03. A combinatorial proof of the log-concavity of the numbers of permutations with $k$ runs. https://arxiv.org/abs/math/9902020

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