2017/12/22 by Mingjia Yang, Yang, Mingjia
Computer Science · Mathematics · #05A05 #05A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1712.08662
openalex publication_date 2017/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (3)/(n) \binom2nn+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.