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The set of 123-avoiding permutations (alias words in \({\{1, \ldots, n\}}\) with exactly 1 occurrence of each letter) is famously enumerated by the ubiquitous Catalan numbers, whose generating function C(x) famously satisfies the algebraic equation \({C(x) = 1 + xC(x)^{2}}\). Recently, Bill Chen, Alvin Dai, and Robin Zhou found (and very elegantly proved) an algebraic equation satisfied by the generating function enumerating 123-avoiding words with two occurrences of each of \({\{1, \ldots, n\}}\). Inspired by the Chen-Dai-Zhou result, we present an algorithm for finding such an algebraic equation for the ordinary generating function enumerating 123- avoiding words with exactly r occurrences of each of \({\{1, \ldots, n\}}\) for any positive integer r, thereby proving that they are algebraic, and not merely D-finite (a fact that is promised by WZ theory). Our algorithm consists of presenting an algebraic enumeration scheme, combined with the Buchberger algorithm.

The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with r Occurrences of Each of 1, 2, . . . , n Are Always Algebraic