[Paper Review] Binary words avoiding a pattern and marked succession rule

Stefano Bilotta, Donatella Merlini+2 more|arXiv (Cornell University)|Mar 29, 2011

Advanced Combinatorial Mathematics15 references3 citations

TL;DR

This paper studies binary words avoiding the pattern $1^{j+1}0^j$ for $j \geq 1$ using Riordan arrays to algebraically enumerate such words by the number of ones. It introduces a novel 'jumping and marked succession rule' to describe their growth, and resolves the non-invertibility of the rule by proposing an algorithm that constructs all binary words with a fixed number of ones and filters out those containing the forbidden pattern.

ABSTRACT

In this paper we study the enumeration and the construction of particular binary words avoiding the pattern $1^{j+1}0^j$. By means of the theory of Riordan arrays, we solve the enumeration problem and we give a particular succession rule, called jumping and marked succession rule, which describes the growth of such words according to their number of ones. Moreover, the problem of associating a word to a path in the generating tree obtained by the succession rule is solved by introducing an algorithm which constructs all binary words and then kills those containing the forbidden pattern.

Motivation & Objective

• To enumerate binary words avoiding the specific pattern $1^{j+1}0^j$ according to the number of ones.
• To develop a combinatorial framework using succession rules that model the growth of such words across levels of a generating tree.
• To resolve the issue of non-unique path association in the generating tree by introducing a constructive algorithm.
• To generalize the approach for forbidden patterns of the form $1^j0^i$ with $0 < i < j$, and explore multi-pattern and multi-alphabet cases.

Proposed method

• Employ algebraic techniques based on proper Riordan arrays to solve the enumeration problem for binary words avoiding $1^{j+1}0^j$.
• Introduce a 'jumping and marked succession rule' that allows sons to be generated at different levels and labels to be marked or unmarked.
• Construct a generating tree where each row of the Riordan array corresponds to a level, and labels represent the number of ones in the words.
• Design an algorithm that generates all binary words with a fixed number of ones and eliminates those containing the forbidden pattern $1^{j+1}0^j$.
• Use path representations in Dyck-like paths to model the forbidden pattern and apply cut-and-paste operations to define killing paths.
• Apply mappings from the succession rule productions to unambiguously associate each word to a unique path in the generating tree via the algorithmic filtering process.

Experimental results

Research questions

• RQ1 How can binary words avoiding the pattern $1^{j+1}0^j$ be enumerated according to the number of ones using algebraic methods?
• RQ2 Can a succession rule be defined that captures the growth of such words across multiple levels and with marked labels?
• RQ3 Why is it not possible to uniquely associate a word to a path in the generating tree derived from the succession rule?
• RQ4 What algorithmic approach enables the construction of all valid binary words with a fixed number of ones while excluding those with the forbidden pattern?
• RQ5 How can the framework be generalized to other forbidden patterns, such as $1^j0^i$ with $0 < i < j$?

Key findings

• The number of binary words avoiding $1^{j+1}0^j$ is algebraically enumerated using proper Riordan arrays, establishing a direct link between the array entries and the count of such words by number of ones.
• A jumping and marked succession rule is constructed that describes the growth of these words across different levels of the generating tree, with marked labels accounting for structural ambiguities.
• The succession rule cannot uniquely associate a word to a path in the generating tree due to the non-uniqueness of label mappings across levels.
• An algorithm is proposed that constructs all binary words with a fixed number of ones and filters out those containing the forbidden pattern $1^{j+1}0^j$, resolving the path association problem.
• For each word $\omega$ in the set of invalid words, there exists a unique corresponding word $\omega'$ with the same number of rise steps and forbidden patterns, differing only in the marking of the last forbidden pattern.
• The killing procedure is univocally determined by the path structure and the rule productions, ensuring a one-to-one correspondence between invalid words and their marked counterparts.