A Symmetry Property of Christoffel Words

TL;DR

The paper introduces strong factor-symmetry as a two-variable analogue of trapezoidal symmetry for word factors, using Parikh images $(p,q)$. It proves that Christoffel words are strongly factor-symmetric and characterizes Christoffel words within primitive Sturmian words, while showing that nonprimitive cases $w=u^k$ (with $u$ primitive) satisfy strong factor-symmetry iff $u$ is a Christoffel word. As a byproduct, factor-symmetry of $w$ corresponds to $u$ being a conjugate of a Christoffel word, and the authors provide a bijection between factors of complementary Parikh images and a geometric path interpretation via attractors. The work connects algebraic reciprocity of the factor-counting polynomial with geometric and combinatorial constructions, yielding a structural framework for Christoffel, Sturmian, and trapezoidal words with implications for factor arrays and their supports.

Abstract

Motivated by the theory of trapezoidal words, whose sequences of cardinality of factors by length are symmetric, we introduce a bivariate variant of this symmetry. We show that this symmetry characterizes Christoffel words, and establish other related results.

Figures (1)

• Figure 1: Paths of lower and upper Christoffel words $w=aabaabaabab$ and $\tilde{w}$ and the function $\delta_w$

Theorems & Definitions (6)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3