Enumeration of words that contain the pattern 123 exactly once

Abstract

Enumeration problems related to words avoiding patterns as well as permutations that contain the pattern $123$ exactly once have been studied in great detail. However, the problem of enumerating words that contain the pattern $123$ exactly once is new and will be the focus of this paper. Previously, Doron Zeilberger provided a shortened version of Alexander Burstein's combinatorial proof of John Noonan's theorem that the number of permutations with exactly one $321$ pattern is equal to $\frac{3}{n} \binom{2n}{n+3}$. Surprisingly, a similar method can be directly adapted to words. We are able to use this method to find a formula enumerating the words with exactly one $123$ pattern. Further inspired by Nathaniel Shar and Zeilberger's paper on generating functions enumerating 123-avoiding words with $r$ occurrences of each letter, we examine the algebraic equations for generating functions for words with $r$ occurrences of each letter and with exactly one $123$ pattern.