On naturally labelled posets and permutations avoiding 12–34

Abstract

A partial order $\prec$ on [n] is naturally labelled (NL) if x $\prec$ y implies $x<y$. We establish a bijection between {3, 2+2}-free NL posets and 12–34-avoiding permutations, determine functional equations satisfied by their generating function, and use series analysis to investigate their asymptotic growth, presenting evidence of stretched exponential behaviour. We also exhibit bijections between 3-free NL posets and various other objects, and determine their generating function. The connection between our results and a hierarchy of combinatorial objects related to interval orders is described.