Klazar trees and perfect matchings

TL;DR

This paper introduces Klazar trees and a bijection with perfect matchings, providing combinatorial explanations for a recurrence relation related to weighted ordered trees.

Contribution

It defines Klazar violators in increasing ordered trees and establishes a bijection to perfect matchings, explaining a recurrence relation through combinatorial methods.

Findings

Defined Klazar violators and Klazar trees.

Established a bijection between increasing ordered trees and perfect matchings.

Derived an explicit formula for perfect matchings with no even-to-larger-odd matches.

Abstract

Martin Klazar computed the total weight of ordered trees under 12 different notions of weight. The last and perhaps most interesting of these weights, w_{12}, led to a recurrence relation and an identity for which he requested combinatorial explanations. Here we provide such explanations. To do so, we introduce the notion of a "Klazar violator" vertex in an increasing ordered tree and observe that w_{12} counts what we call Klazar trees--increasing ordered trees with no Klazar violators. A highlight of the paper is a bijection from n-edge increasing ordered trees to perfect matchings of [2n]={1,2,...,2n} that sends Klazar violators to even numbers matched to a larger odd number. We find the distribution of the latter matches and, in particular, establish the one-summation explicit formula sum_{k=1}^{lfloor n/2 rfloor}(2k-1)!!^2 StirlingPartition{n+1}{2k+1} for the number of perfect matchings of [2n] with no even-to-larger-odd matches. The proofs are mostly bijective.