Digital Convexity and Combinatorics on Words
Alessandro De Luca, Gabriele Fici, Andrea Frosini
TL;DR
The paper studies digital convexity in binary words as a combinatorial proxy for convex planar curves, establishing a precise link to Christoffel and central words and to Sturmian (balanced) words. It develops inflation/deflation operations and uses Lyndon factorizations to characterize digitally convex words: a word with Parikh vector $(a,b)$ can be generated from the lower Christoffel word $w_{a,b}$ via inflation and from $1^b0^a$ via deflation. It also analyzes structural and enumerative aspects, showing the set of digitally convex words forms a factorial language closed under reversal and complement, with a meet-semilattice structure under the dominance order and a minimal forbidden words characterization, along with Euler-transform-based counting. These results bridge digital geometry with classical word combinatorics, enabling constructive synthesis and analysis of digital convex approximations.
Abstract
An upward (resp. downward) digitally convex word is a binary word that best approximates from below (resp. from above) an upward (resp. downward) convex curve in the plane. We study these words from the combinatorial point of view, formalizing their geometric properties and highlighting connections with Christoffel words and finite Sturmian words. In particular, we study from the combinatorial perspective the operations of inflation and deflation on digitally convex words.