ENUMERATION OF TREES BY INVERSIONS

Mallows and Riordan ''The Inversion Enumerator for Labeled Trees,'' Bulletin of the American Mathematics Society, vo1. 74 [1968] pp. 92-94) first defined the inversion polynomial, J(n)(q) for trees with n vertices and found its generating function. In the present work, we define inversion polynomials for ordered, plane, and cyclic trees, and find their values at q = 0, +/- 1. Our techniques involve the use of generating functions (including Lagrange inversion), hypergeometric series, and binomial coefficient identities; induction, and bijections. We also derive asymptotic formulae for those results for which we do not have a closed form. (C) 1995 John Wiley & Sons, Inc.