Non-overlapping descents and ascents in stack-sortable permutations

The Eulerian polynomials An(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 1 (n )( n ). On the other n k k+1 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 n-ary sumation n,k >= 0Dn,kxk tn et n! = 1-x(1+(t-1)et ) . In this paper, we show that the distribution of MND over stack-sortable permutations is given by 1 ( n+1 )(n+k). We give two proofs of the result via bijections with rooted n+1 2k+1 k 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.(c) 2023 Elsevier B.V. All rights reserved.