Bijections between planar maps and planar linear normal λ-terms with connectivity condition

Abstract

The enumeration of linear λ-terms has attracted quite some attention recently, partly due to their link to combinatorial maps. Zeilberger and Giorgetti (2015) [19] gave a recursive bijection between planar linear normal λ-terms and planar maps, which, when restricted to 2-connected λ-terms (i.e., without closed sub-terms), leads to bridgeless planar maps. Inspired by this restriction, Zeilberger and Reed (2019) [20] conjectured that 3-connected planar linear normal λ-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 λ-terms and planar maps, whose restriction to 2-connected λ-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.