This is a classic problem in combinatorics on words. A known technique is to use **recursive counting with states**, but we can estimate via careful case analysis.

Sun, 03 May 2026 09:22:56 +0000

Classic Combinatorics on Words: Recursive Counting with States and Case Analysis for Problem Solving In the rich and elegant domain of combinatorics on words, one recurring challenge captivates mathematicians and computer scientists alike: counting distinct strings under specific constraints—such as avoiding repeated substrings—ordered sequences, or recurrent patterns. This classic problem often resists brute-force enumeration due to exponential growth in possibilities, making recursive counting with well-defined states a powerful go-to technique. Yet, even with recursive methods, a careful case analysis is essential to efficiently estimate solutions without exhaustive computation. ### Understanding the Core Problem
A classic instance involves counting the number of valid strings of length n over a finite alphabet that avoid any repetition of contiguous substrings—commonly applied in areas like coding theory, learned models, and automata design. For example, a simple constraint might be: no substring of length k repeats within the string. The challenge lies in ensuring that each recursive step captures not only valid extensions but also avoids overcounting or dead-end branches. ### Recursive Strategy with State Modeling
The standard approach pioneers recursive algorithms using state-based enumeration. Here’s how it works in principle: