Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Multi-statistic enumeration of two-stack sortable permutations. | Multi-statistic enumeration of two-stack sortable permutations Mireille Bousquet-Melou LaBRI Université Bordeaux 1 351 cours de la Libération 33405 Talence Cedex FRANCE bousquet@ Submitted October 1 1997. Accepted April 1 1998 Abstract Using Zeilberger s factorization of two-stack-sortable permutations we write a functional equation of a strange sort that defines their generating function according to five statistics length number of descents number of right-to-left and left-to-right maxima and a fifth statistic that is closely linked to the factorization. Then we show how one can translate this functional equation into a polynomial one. We thus prove that our five-variable generating function for two-stack-sortable permutations is algebraic of degree 20. 1 Introduction The aim of this paper is twofold. Our first intention is to draw attention on the factorization of two-stack-sortable permutations described by Zeilberger in 22 . Among other advantages this factorization preserves many standard statistics defined on permutations and hence constitutes an efficient first step for their enumeration. We thus obtain a functional equation of a strange kind that defines a five-variable generating function for these permutations see Eq. 1 . Our second point is to show how one can solve such an equation or more precisely convert it into a polynomial one in particular we prove that the generating function for two-stack-sortable permutations counted according to the length number of descents number of right-to-left and left-to-right maxima is algebraic of degree 10. In passing we translate some of our results in terms of non-separable maps since two sophisticated bijections have been described between these maps and two-stack-sortable permutations 9 8 14 . We believe that our method can be applied to or at least inspire the solution of other functional equations having similar features. Let us mention that our first point celebrating Zeilberger s .
