The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with   r occurrences of each of 1,2, ..., n are Always Algebraic
  Recently, Bill Chen, together with his disciples Alvin Dai and Robin Zhou, discovered, and very elegantly proved, an algebraic equation satisfied by the generating function enumerating 123-avoiding words with two occurrences of each of 1, ..., n. Inspired by this 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, ... 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 
Nathaniel Shar and Doron Zeilberger
math.CO