Some Results on Digital Segments and Balanced Words

Abstract

We exhibit combinatorial results on Christoffel words and binary balanced words that are motivated by their geometric interpretation as approximations of digital segments. We give a closed formula for counting the exact number of balanced words with $a$ zeroes and $b$ ones. We also study minimal non-balanced words.

Figures (5)

• Figure 1: The lower Christoffel word $w_{7,4}=00100100101$. The upper Christoffel word $W_{7,4}=10100100100$ is the reversal of $w_{7,4}$.
• Figure 2: The region $R(7,4)$ of the discrete plane whose contour is delimited by the lower Christoffel word $w_{7,4}$ and the upper Christoffel word $W_{7,4}$ is precisely the region of points in the grid that have Euclidean distance smaller than $\sqrt{2}$ from the Euclidean segment joining $(0,0)$ to $(7,4)$.
• Figure 3: The central word $C=010010010$ is the central factor of the primitive lower Christoffel word $w_{7,4}=0C1$. It encodes the intersections of the Euclidean segment joining $(0,0)$ and $(7,4)$ with the grid ($0$ for a vertical intersection and $1$ for a horizontal intersection).
• Figure 4: The standard factorization $0Q1\cdot 0P1 = 001 \cdot 00100101$ (left) and the palindromic factorization $0P0\cdot 1Q1 = 00100100\cdot 101$ (right) of the lower Christoffel word $w_{7,4}$. The point $S$ determined by the standard factorization is the closest to the Euclidean segment, while the point $S'$ determined by the palindromic factorization is the farthest.
• Figure 5: A three-dimensional plot showing the number of binary balanced words with Parikh vector $(a,b)$ for $a$ and $b$ ranging from $1$ to $36$.

Theorems & Definitions (47)

• Proposition 1
• Definition 2
• Remark 3
• Definition 4
• Theorem 5: Combinatorial Structure of Central Words
• Example 6
• Proposition 7: Pirillo2001
• Proposition 8: DBLP:journals/tcs/BerstelL97
• Example 9
• Proposition 10: DBLP:journals/ejc/BertheLR08