Bijections between planar maps and planar linear normal $\lambda$-terms with connectivity condition

TL;DR

This paper establishes new direct bijections between classes of planar linear normal $mbda$-terms and various classes of planar maps, confirming conjectures and revealing combinatorial connections.

Contribution

It provides the first direct bijection between 3-connected planar linear normal $mbda$-terms and bipartite planar maps, and introduces a novel bijection linking planar $mbda$-terms to loopless planar maps.

Findings

Confirmed the conjecture relating 3-connected $mbda$-terms and bipartite planar maps.

Established a new bijection between planar $mbda$-terms and loopless planar maps.

Explored enumerative consequences of these bijections.

Abstract

The enumeration of linear $\lambda$-terms has attracted quite some attention recently, partly due to their link to combinatorial maps. Zeilberger and Giorgetti (2015) gave a recursive bijection between planar linear normal $\lambda$-terms and planar maps, which, when restricted to 2-connected $\lambda$-terms (i.e., without closed sub-terms), leads to bridgeless planar maps. Inspired by this restriction, Zeilberger and Reed (2019) conjectured that 3-connected planar linear normal $\lambda$-terms have the same counting formula as bipartite planar maps. In this article, we settle this conjecture by giving a direct bijection between these two families. Furthermore, using a similar approach, we give a direct bijection between planar linear normal $\lambda$-terms and planar maps, whose restriction to 2-connected $\lambda$-terms leads to loopless planar maps. This bijection seems different from that of Zeilberger and Giorgetti, even after taking the map dual. We also explore enumerative consequences of our bijections.