## Science University Research Symposium (SURS)

#### Title

Bijective Proofs of Word Avoidance Identities

#### Publication Date

2022

#### College

Sciences and Mathematics, College of

#### Department

Math and Computer Science, Department of

#### SURS Faculty Advisor

Dr. Mary Goodloe

#### Presentation Type

Oral Presentation

#### Abstract

Our object of study is the set of length-* n* words on a given alphabet that all avoid a particular contiguous subword

*. We call this set an avoidance set and denote it as*

**p***A*For certain subwords

**(**_{n}**p**).*and*

**p***, the avoidance sets*

**q***A*and

**(**_{n}**p**)*A*are the same size for all natural numbers

**(**_{n}**q**)*. We call such*

**n***and*

**p***avoidant equivalent subwords. It has been proven that if*

**q***and*

**p**

**q***have similar border patterns, then they are avoidant equivalent subwords. A proof of this via generating functions is credited to a 1966 paper of Solov'ev. Generating function approaches to counting word avoidances have since improved, but a bijective proof of this theorem is nowhere to be found. Does there exist some natural correspondence between the avoidance sets of avoidant equivalent subwords? In other words, how are the individual words of these avoidance sets related? Our talk discusses examples of avoidant equivalent subwords, the condition of similar border patterns, and the main result of our research: a natural correspondence between avoidance sets in many cases.*

#### Recommended Citation

Hollars, Isaiah; Carrigan, Julia; and Rowland, Eric, "Bijective Proofs of Word Avoidance Identities" (2022). *Science University Research Symposium (SURS)*. 8.

https://repository.belmont.edu/surs/8