Enumeration of words that contain the pattern 123 exactly once

TL;DR

This paper introduces a new enumeration of words containing exactly one occurrence of the pattern 123, providing formulas and generating functions, extending previous permutation pattern results to words.

Contribution

It adapts Zeilberger's combinatorial proof method to words, deriving formulas for words with exactly one 123 pattern, a problem not previously studied.

Findings

Derived a formula for counting words with exactly one 123 pattern.

Established algebraic equations for generating functions of such words.

Extended permutation pattern enumeration techniques to words.

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.