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, 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,\ldots,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.
