Accepted manuscript ISSN 2590-9770 The Art of Discrete and Applied Mathematics https://doi.org/10.26493/2590-9770.1377.8e9 (Also available at http://adam-journal.eu) On avoiding 1233 Toufik Mansour Department of Mathematics, University of Haifa, 3498838 Haifa, Israel Mark Shattuck Department of Mathematics, University of Tennessee, 37996 Knoxville, TN Received 12 August 2020, accepted 16 June 2021 Abstract In this paper, we establish a recurrence relation for finding the generating function for the number of k-ary words of length n that avoid 1233 for arbitrary k. Comparable generating function formulas may also be found counting words where a single permutation pattern of length three is avoided in addition to 1233. Keywords: k-ary words, Kernel method, Avoiding 1233. Math. Subj. Class.: 05A15, 05A05 1 Introduction We denote the set of all words of length n over the alphabet [k]= {1,...,k} by [k] n and refer to members of [k] n as k-ary words. Let π = π 1 ··· π n ∈ [k] n and τ = τ 1 ··· τ m ∈ [ℓ] m such that each letter from [ℓ] appears at least once in τ (possibly with repetitions). We say that π contains τ if there exist indices 1 ≤ i 1 < ··· <i m ≤ n such that π ia Φπ i b if and only if τ a Φτ b , for any relation Φ ∈{<, =,>} and a, b ∈ [m]. In this context, the word τ is called a pattern, and it is said that π avoids τ if π fails to contain τ per the preceding definition. The area of permutation pattern avoidance has received considerable attention in recent decades; see, e.g., [13] and references therein. Alon and Friedgut [2] extended this study to avoidance on k-ary words in obtaining an upper bound on the number of permutations of length n that avoid a given pattern. The question of pattern avoidance on permutations was initiated by Knuth [6], who found that the number of permutations of length n that avoid the pattern τ for any τ ∈ S 3 is given by the n-th Catalan number 1 n+1 ( 2n n ) . Later, Simion and Schmidt [12] extended this result by determining the number of permutations E-mail addresses: tmansour@univ.haifa.ac.il (Toufik Mansour), shattuck@math.utk.edu (Mark Shattuck) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/