Combinatorial Proof of the Log-Concavity of the Numbers of Permutations with k Runs

Let p= p1 p2 } } } pn be a permutation of the set [1, 2, ..., n] written in the one-line notation. We say that p get changes direction at position i, if either pi&1pi+ j , or p i&1>pi>pi+1 , in other words, when p i is either a peak or a valley. We say that p has k runs if there are k&1 indices i so that p changes direction at these positions. So, for example, p=3561247 has 3 runs as p changes direction when i=3 and when i=4. A geometric way to represent a permutation and its runs by a diagram is shown in Fig. 1. The runs are the line segments (or edges) between two consecutive entries where p changes direction. So a permutation has k runs if it can be represented by k line segments so that the segments go ``up'' and ``down'' doi:10.1006 jcta.1999.3040, available online at http: www.idealibrary.com on