OEIS A800260: Lambda calculus, rooted 3-polytopes, and hidden connections in combinatorics

Today we unpack lambda calculus, a foundational formal language for defining and applying functions, and then trace how its ideas echo through the world of A800260. What is lambda calculus? It’s a minimal syntax for building and applying anonymous functions using three essential pieces: variables, lambda abstractions (functions), and applications (function calls). We’ll walk through a tiny example: the identity function λx. x and how (λx. x) a reduces to a; we’ll also cover alpha-conversion (renaming bound variables) and beta-reduction (the actual computation step). You’ll see why untyped vs simply-typed lambda calculus matter, and how Church encodings let you represent data like booleans and numbers purely with functions. We’ll note that lambda calculus captures exactly what a simple computer can compute (it’s Turing complete), which is why it’s central to theory and to the design of functional programming languages. Then we connect the dots to A800260. This OEIS entry counts rooted simplicial 3-polytopes, and, through elegant bijections, ties to rooted planar maps, Tamari lattice intervals, and certain pattern-avoiding permutations. Although lambda calculus might seem far from “polytopes and maps,” it shares a unifying theme: different objects—geometric shapes, trees, and expressions—can encode the same combinatorial structure and thus the same counting sequence. We’ll sketch how a tree-like representation of lambda terms can reflect the same structural skeleton that underlies Tamari intervals and planar maps, offering a glimpse into why a single number sequence can surface in such diverse domains. If you want, we can add a step-by-step mini-tutorial on lambda calculus with more examples, or dive into a concrete bijection that connects these ideas more tightly.

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