Klazar trees and perfect matchings

Martin Klazar computed the total weight of ordered trees under 12 different notions of weight. The last and perhaps most interesting of these weights, w12, 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 w12 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 ∑⌊n/2⌋ k=1 (2k−1)!!2 { n+1 2k+1 } for the number of perfect matchings of [2n] with no even-to-larger-odd matches. The proofs are mostly bijective.