A Binary Tree Decomposition Space of Permutation Statistics

Based on the binary tree decomposition of a permutation, a natural vector space of permutation statistics is defined. Besides containing many well known permuta­ tion statistics, this space provides the general context for an archetypal recurrence relationship that contains most of the classic combinatorial sequences and some of their known generalizations. Permutation statIstIcs have arisen in connection with various applica­ tions of combinatorics to a wide spectrum of mathematics (including differential operators, orthogonal polynomials, hypergeometric functions, probability, and sorting problems). However, in spite of an increased interest in permutation statistics, no systematic study of them has been done. The basic purpose of this paper is to propose a possible framework for such a study. The approach being advocated for unifying this area consists of the introduction and examination of natural vector spaces of statistics which (Ire defined in terms of permutation decompositions. The central focus here will be on a vector space which arises in connection with the binary tree decomposition of a permutation. Besides containing many well-known per­ mutation statistics (such as the descent number, the inversion number, the trough number, the minimum component number, and others), this binary tree decomposition statistic space provides the setting for an archetypal recurrence relationship that contains many classic combinatorial sequences (including the Catalan numbers, the Fibonacci numbers, the Eulerian poly­ nomials, the Stirling polynomials of both kinds, the Hermite polynomials, and the Bell polynomials).