Non-overlapping descents and ascents in stack-sortable permutations

Abstract: The Eulerian polynomials $A_n(x)$ give the distribution of descents over permutations. It is also known that the distribution of descents over stack-sortable permutations (i.e. permutations sortable by a certain algorithm whose internal storage is limited to a single stack data structure) is given by the Narayana numbers $\frac{1}{n}{n \choose k}{n \choose k+1}$. On the other hand, as a corollary of a much more general result, the distribution of the statistic ``maximum number of non-overlapping descents'', MND, over all permutations is given by $\sum_{n,k \geq 0}D_{n,k}x^k\ frac{t^n}{n!}=\frac{e^t}{1-x(1+(t-1)e^t)}$. In this paper, we show that the distribution of MND over stack-sortable permutations is given by $\frac{1}{n+1}{n+1\choose 2k+1}{n+k \choose k}$. We give two proofs of the result via bijections with rooted plane (binary) trees allowing us to control MND. Moreover, we show combinatorially that MND is equidistributed with the statistic MNA, the maximum number of non-overlapping ascents, over stack-sortable permutations. The last fact is obtained by establishing an involution on stack-sortable permutations that gives equidistribution of 8 statistics.