Crucial words for abelian powers

TL;DR

This paper investigates the minimal length of crucial words over n-letter alphabets that avoid abelian powers, providing exact lengths for cubes and bounds for higher powers, advancing understanding of combinatorial word properties.

Contribution

It determines the minimal lengths of crucial words avoiding abelian cubes and offers a construction for words avoiding higher abelian powers, extending prior work on abelian squares.

Findings

Minimal crucial words avoiding abelian cubes have length 9n-13 for n >= 5.

A construction for crucial words avoiding abelian k-th powers with length k^2(n-1)-k-1.

Provided lower bounds for lengths of crucial words avoiding higher abelian powers for n >= 4 and k >= 4.

Abstract

A word is "crucial" with respect to a given set of "prohibited words" (or simply "prohibitions") if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A "minimal crucial word" is a crucial word of the shortest length. A word W contains an "abelian k-th power" if W has a factor of the form X_1X_2...X_k where X_i is a permutation of X_1 for 2<= i <= k. When k=2 or 3, one deals with "abelian squares" and "abelian cubes", respectively. In 2004 (arXiv:math/0205217), Evdokimov and Kitaev showed that a minimal crucial word over an n-letter alphabet A_n = {1,2,..., n} avoiding abelian squares has length 4n-7 for n >= 3. In this paper we show that a minimal crucial word over A_n avoiding abelian cubes has length 9n-13 for n >= 5, and it has length 2, 5, 11, and 20 for n=1, 2, 3, and 4, respectively. Moreover, for n >= 4 and k >= 2, we give a construction of length k^2(n-1)-k-1 of a crucial word over A_n avoiding abelian k-th powers. This construction gives the minimal length for k=2 and k=3. For k >= 4 and n >= 5, we provide a lower bound for the length of crucial words over A_n avoiding abelian k-th powers.