Bijective Proofs of Word Avoidance Identities
Sciences and Mathematics, College of
Math and Computer Science, Department of
SURS Faculty Advisor
Dr. Mary Goodloe
Our object of study is the set of length-n words on a given alphabet that all avoid a particular contiguous subword p. We call this set an avoidance set and denote it as An(p). For certain subwords p and q, the avoidance sets An(p) and An(q) are the same size for all natural numbers n. We call such p and q avoidant equivalent subwords. It has been proven that if p and 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.
Hollars, Isaiah; Carrigan, Julia; and Rowland, Eric, "Bijective Proofs of Word Avoidance Identities" (2022). Science University Research Symposium (SURS). 8.