On some 2-binomial coefficients of binary words: geometrical interpretation, partitions of integers, and fair words

TL;DR

The paper develops a geometric framework for the counts of subwords $ab$ and $ba$ in words, focusing on binary words and the $2$-binomial equivalence. It shows that the equivalence class under $inom{w}{ab}$ and related counts forms a lattice, with canonical representatives init$(w)$ and final$(w)$, and it links these classes to partitions of the integer $inom{w}{ab}$ via two natural partition constructions. It also studies fair words, defined by equal counts of $ab$ and $ba$, connecting fair words to least-squares fitting, showing palindromes are fair and proving a conjecture in the process. The results tie binomial equivalence to Parikh/precedence matrices, provide rewriting-system characterizations, and establish a rich interplay between combinatorics on words and integer partitions, with implications for structure theory and language properties of fair words.

Abstract

The binomial notation (w u) represents the number of occurrences of the word u as a (scattered) subword in w. We first introduce and study possible uses of a geometrical interpretation of (w ab) and (w ba) when a and b are distinct letters. We then study the structure of the 2-binomial equivalence class of a binary word w (two words are 2-binomially equivalent if they have the same binomial coefficients, that is, the same numbers of occurrences, for each word of length at most 2). Especially we prove the existence of an isomorphism between the graph of the 2-binomial equivalence class of w with respect to a particular rewriting rule and the lattice of partitions of the integer (w ab) with (w a) parts and greatest part bounded by (w b). Finally we study binary fair words, the words over {a, b} having the same numbers of occurrences of ab and ba as subwords ((w ab) = (w ba)). In particular, we prove a recent conjecture related to a special case of the least square approximation.

Figures (17)

• Figure 1: word aabaabbababba
• Figure 2: $\binom{w}{ab}$ and $\binom{w}{ba}$ for word $w =$aabaabbababba
• Figure 3: Geometrical interpretations of $S_b(w)$ and $S_b'(w)$
• Figure 4: Number of occurrences of $ab$ in the concatenation of three words
• Figure 5: From $uabv$ to $ubav$

Theorems & Definitions (72)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: Fosse_Richomme2004IPL
• Lemma 2.4
• Lemma 3.1: Cerny2009JALC
• Lemma 3.2: Mateescu_Salomaa_Salomaa_Yu2001TIA and, for the binary case, Prodinger1979DM
• Theorem 3.3: Cerny2009JALC
• Theorem 3.4: Fosse_Richomme2004IPL
• Theorem 3.5
• proof : Proof of the only if part of Theorem \ref{['T_equivalence_reecrite_sim2']}