On naturally labelled posets and permutations avoiding 12-34

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.