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

TL;DR

The paper establishes two direct bijections between planar linear normal λ-terms and planar maps under connectivity constraints. It first proves a 3-connected correspondence: 3-connected planar linear normal λ-terms with $n$ variables correspond to bipartite planar maps with $n-2$ edges via degree trees, enabling refined enumerations and asymptotic statistics to be transferred from maps to λ-terms. A second direct bijection uses v-trees obtained from skeletons and a one-corner decomposition of planar maps, relating planar linear normal λ-terms to planar maps with a 1-to-1 mapping that preserves 2-connectivity to loopless maps. The results provide new enumerative tools, show the distinctness of the new bijections from prior work (even after dualization), and connect β(0,1)-tree frameworks to the λ-term/map correspondence, with generating function identities linking the two domains.

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) 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) 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.

Figures (6)

• Figure 1: Example of a normal planar $\lambda$-term, its skeleton and syntactic diagram.
• Figure 2: Example of the bijection $\varphi$ from reduced skeletons in $\mathcal{RS}$ to degree trees (zeros in $\ell_\Lambda$ are omitted).
• Figure 3: Example of the bijection $\psi$ from skeletons in $\mathcal{S}^{(1)}$ to v-trees.
• Figure 4: Illustration of one-corner decomposition of planar maps
• Figure 5: Example of construction of one-corner components $U_i$ with $\Pi(U_i) = M$ for a given planar map $M$.

Theorems & Definitions (42)

• Conjecture 1: 3-conn-conj
• Theorem 2
• Theorem 3
• Remark 1
• Proposition 2.1
• proof
• Proposition 2.2: See grygiel-yu
• proof
• Proposition 2.3: See grygiel-yu
• proof