2 research outputs found
Computing Depths of Patterns
Pattern avoidance is an important research topic in combinatorics on words which dates
back to Thue’s construction of an infinite word over three letters that avoids squares,
i.e., a sequence with no two adjacent identical factors. This result
finds applications in various algebraic contexts where more general patterns than squares
are considered. A more general form of pattern avoidance has recently emerged to allow for
undefined positions in sequences. New concepts on patterns such as depth have been
introduced and a number of questions have been raised, some of them we answer. In the
process, we prove a strict bound on the number of square occurrences in
an unavoidable pattern, and consequently, any pattern with more square occurrences than
distinct variables is avoidable over three letters. We also provide an algorithm that
determines whether a given pattern is of bounded depth, and if so, computes its depth
Computing Depths of Patterns
Pattern avoidance is an important research topic in combinatorics on words which dates
back to Thue’s construction of an infinite word over three letters that avoids squares,
i.e., a sequence with no two adjacent identical factors. This result
finds applications in various algebraic contexts where more general patterns than squares
are considered. A more general form of pattern avoidance has recently emerged to allow for
undefined positions in sequences. New concepts on patterns such as depth have been
introduced and a number of questions have been raised, some of them we answer. In the
process, we prove a strict bound on the number of square occurrences in
an unavoidable pattern, and consequently, any pattern with more square occurrences than
distinct variables is avoidable over three letters. We also provide an algorithm that
determines whether a given pattern is of bounded depth, and if so, computes its depth