Enumeration of trees by inversions

Abstract

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, w e 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. 0 1995 John Wiley & Sons, Inc.

On the Analysis of Linear Probing Hashing

“…An inversion in a rooted labeled tree is a pair (i, j) such that i is on the path from the root to j and i > j. Exact generating functions have been first found by Mallows and Riordan [31] in the case of "Cayley" trees and other families of trees are considered in [11]. (Q) Connectivity in graphs.…”

“…In this way, moment generating functions are obtained immediately from the corresponding computation for full tables. The analysis still relies on the functions f r = U ∂ q r F introduced in (11). We have…”

A Unifying Approach for Proving Hook-Length Formulas for Weighted Tree Families

“…As another example we consider the family C of so-called labelled cyclic trees (also called labelled mobile trees) considered in [2,4,5]. Each node v in such a labelled tree is either an end-node or there is attached a cycle of children, i.e., one might assume that the children of each node are arranged via circular shifts such that the child with smallest label is always the leftmost child.…”

“…Trees with a special treatment of the root. Some combinatorial tree families (as, e.g., plane trees as considered in [5], and so-called non-crossing trees [14]) are not equivalent to weighted ordered trees as introduced in Section 2, since the root of any such tree has to be treated in a separate way. However, one can easily extend the concept of weighted ordered trees to cover also such situations.…”

Local extrema in random trees

“…A functional equation satisfied by the generating function for the number of labelled trees according to the number of inversions is given in [7]. Related results are contained in [6]. A formula for the expected number of inversions of a uniformly random labelled tree is given in [9].…”