The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with r occurrences of each of 1,2, ..., n are Always Algebraic

TL;DR

This paper introduces an algorithm to prove that generating functions counting 123-avoiding words with fixed occurrences are algebraic, extending previous results and confirming their algebraic nature for any fixed number of occurrences.

Contribution

The paper develops an algorithm that constructs algebraic equations for generating functions counting 123-avoiding words with fixed occurrences, generalizing prior specific cases.

Findings

Generated algebraic equations for various fixed occurrences

Confirmed algebraic nature of these generating functions for all fixed r

Extended previous results from specific cases to general r

Abstract

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