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.

翻译：线性λ-项的枚举近期引起了广泛关注，部分原因在于它们与组合地图的关联。Zeilberger 与 Giorgetti（2015）给出了平面线性正规λ-项与平面地图之间的递归双射，当其限制于2-连通λ-项（即不含闭合子项）时，导出无桥平面地图。受此限制启发，Zeilberger 与 Reed（2019）推测3-连通平面线性正规λ-项具有与二部平面地图相同的计数公式。本文通过给出这两个族之间的直接双射，证实了这一猜想。此外，利用类似方法，我们给出了平面线性正规λ-项与平面地图之间的直接双射，其限制于2-连通λ-项将导出无环平面地图。该双射在取地图对偶后仍与 Zeilberger-Giorgetti 双射不同。我们还探讨了双射的枚举学推论。