Involution $h$ on Catalan structures

Abstract

We define an involution $h$ on Catalan structures through an abstract framework, prove an equidistribution theorem for four canonical statistics and present a generating function carrying these. This framework encompasses all combinatorial structures with a decomposition mirroring the first-return decomposition of Dyck paths. The fixed points of~$h$ are counted by Catalan numbers. Canonical bijections transport the equidistribution to eight well known concrete families, identifying the canonical statistics with native ones on each. In addition to its primary structure, each Catalan structure has a derived \emph{secondary structure}, and $h$~interchanges primary and secondary structure. The involution factors as $h = \rev \circ \corev \circ \rev$, where $\rev$ and $\corev$ are two simpler involutions, and the composition $M = h \circ \rev$ coincides with Donaghey's automorphism on plane trees. This yields $M^{-1} = \rev \circ M \circ \rev$ and a period theorem: Iterating the secondary structure construction produces a sequence that repeats with period equal to the order of~$M$. It is an open problem to describe $h$ and the canonical statistics explicitly on most of the more than two hundred known families of Catalan structures.