Enumeration of trees by inversions

Mallows and Riordan “The Inversion Enumerator for Labeled Trees,” Bulletin of the American Mathematics Society, vol. 74 119681 pp. 92-94) first defined the inversion polynomial, JJ9) 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 9 = 0, t l . 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 w e do not have a closed form.