Avoidance of binary patterns by Catalan words

A certain subset of the multiset permutations of length n satisfying two restrictions has been recently shown to be enumerated by the Catalan number Cn−1. These sequences have been termed Catalan words and are closely related to the 321-avoiding permutations. Here, we consider the problem of avoidance of patterns of type (1,2) wherein the second and third letters within an occurrence of a pattern are required to be adjacent. We derive in several cases functional equations satisfied by the generating functions enumerating members of the avoidance class which we solve by various methods. In one case, the generating function can be expressed in terms of a sum of reciprocals of Chebyshev polynomials, while in another, in terms of a previously studied q-Bell number. Among the sequences arising as enumerators of avoidance classes are the Motzkin and Fibonacci numbers. In several cases, it is more convenient to consider first the problem of avoidance on the subset of Catalan words whose members have no adjacent letters the same before moving to the larger problem on all Catalan words.

Pattern avoidance is an important topic in combinatorics on words which dates back to the beginning of the twentieth century when Thue constructed an infinite word over a ternary alphabet that avoids squares, i.e. , a word with no two adjacent identical factors. This result finds applications in various algebraic contexts where more general patterns than squares are considered. On the other hand, Erdős raised the question as to whether there exists an infinite word that avoids abelian squares, i.e. , a word with no two adjacent factors being permutations of one another. Although this question was answered affirmately years later, knowledge of abelian pattern avoidance is rather limited. Recently, (abelian) pattern avoidance was initiated in the more general framework of partial words, which allow for undefined positions called holes. In this paper, we show that any pattern p with n > 3 distinct variables of length at least 2 n is abelian avoidable by a partial word with infinitely many holes, the bound on the length of p being tight. We complete the classification of all the binary and ternary patterns with respect to non-trivial abelian avoidability, in which no variable can be substituted by only one hole. We also investigate the abelian avoidability indices of the binary and ternary patterns.

A \textit{partial word} is a sequence of symbols over a finite alphabet that may have some undefined positions, called \textit{holes}, that match every letter of the alphabet. Previous work completed the classification of all binary patterns with respect to partial word avoidability. In this paper, we pose the problem of avoiding patterns in partial words very dense with holes. We define the concept of hole sparsity, a measure of the frequency of holes in a partial word, and determine the minimum hole sparsity for all unary patterns in the context of trivial and non-trivial avoidability. Results for more general patterns are also given. Furthermore, we discuss hole spacing and hole density for abelian powers.

A certain subset of the words of length n over the alphabet of non-negative integers satisfying two restrictions has recently been shown to be enumerated by the Catalan number Cn-1. Members of this subset, which we will denote by W(n), have been termed Catalan words or sequences and are closely associated with the 321-avoiding permutations. Here, we consider the problem of enumerating the members of W(n) satisfying various restrictions concerning the containment of certain prescribed subsequences or patterns. Among our results, we show that the generating function counting the members Of W(n) that avoid certain patterns is always rational for four general classes of patterns. Our proofs also provide a general method of computing the generating function for all the patterns in each of the four classes. Closed form expressions in the case of three-letter patterns follow from our general results in several cases. The remaining cases for patterns of length three, which we consider in the final section, may be done by various algebraic and combinatorial methods.

Pattern avoidance in binary trees

Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures

In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare these results to analogous work for contiguous tree patterns. Next, we give an explicit generating function that counts binary trees avoiding a single non-contiguous tree pattern according to number of leaves and show that there is exactly one Wilf class of k-leaf tree patterns for any positive integer k. In addition, we give a bijection between between certain sets of pattern-avoiding trees and sets of pattern-avoiding permutations. Finally, we enumerate binary trees that simultaneously avoid more than one tree pattern.

We show that every binary pattern of length greater than 14 is abelian-2-avoidable. The best known upper bound on the length of abelian-2-unavoidable binary pattern was 118, and the best known lower bound is 7. We designed an algorithm to decide, under some reasonable assumptions, if a morphic word avoids a pattern in the abelian sense. This algorithm is then used to show that some binary patterns are abelian-2-avoidable. We finally use this list of abelian-2-avoidable pattern to show our result. We also discuss the avoidability of binary patterns on 3 and 4 letters.

The undirected repetition threshold and undirected pattern avoidance

Algorithmic combinatorics on partial words, or sequences of symbols over a finite alphabet that may have some do-not-know symbols or holes, has been developing in the past few years. Applications can be found, for instance, in molecular biology for the sequencing and analysis of DNA, in bio-inspired computing where partial words have been considered for identifying good encodings for DNA computations, and in data compression. In this paper, we focus on two areas of algorithmic combinatorics on partial words, namely, pattern avoidance and subword complexity. We discuss recent contributions as well as a number of open problems. In relation to pattern avoidance, we classify all binary patterns with respect to partial word avoidability, we classify all unary patterns with respect to hole sparsity, and we discuss avoiding abelian powers in partial words. In relation to subword complexity, we generate and count minimal Sturmian partial words, we construct de Bruijn partial words, and we construct partial words with subword complexities not achievable by full words (those without holes).

On the aperiodic avoidability of binary patterns with variables and reversals

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.

A partial word is a sequence of symbols over a finite alphabet that may have some undefined positions, called holes, that match every letter of the alphabet. Previous work completed the classification of all unary patterns with respect to partial word avoidability, as well as the classification of all binary patterns with respect to non-trivial partial word avoidability. In this paper, we pose the problem of avoiding patterns in partial words very dense with holes. We define the concept of hole sparsity, a measure of the frequency of holes in a partial word, and determine the minimum hole sparsity for all unary patterns in the context of trivial and non-trivial avoidability.

Computing the partial word avoidability indices of ternary patterns