Using Noonan-Zeilberger Functional Equations to enumerate (in Polynomial Time!) Generalized Wilf classes

TL;DR

This paper introduces a simplified polynomial-time method using Noonan-Zeilberger functional equations to efficiently enumerate Wilf classes for permutations with any pattern length and number of occurrences.

Contribution

It presents a new, simpler approach to generate enumeration schemes for Wilf classes, overcoming complexity issues of previous methods for larger pattern occurrences.

Findings

Polynomial-time enumeration for increasing pattern lengths

Efficient counting for any number of pattern occurrences

Simplification of previous complex functional equation methods

Abstract

One of the most challenging problems in enumerative combinatorics is to count Wilf classes, where you are given a pattern, or set of patterns, and you are asked to find a "formula", or at least an efficient algorithm, that inputs a positive integer n and outputs the number of permutations avoiding that pattern. In 1996, John Noonan and Doron Zeilberger initiated the counting of permutations that have a prescribed, r, say, occurrences of a given pattern. They gave an ingenious method to generate Functional Equations, alas, with an unbounded number of "catalytic variables", but then described a clever way, using multivariable calculus, how to get enumeration schemes. Alas, their method becomes very complicated for r larger than 1. In the present article we describe a far simpler way to squeeze the necessary information, in polynomial time, for increasing patterns of any length, and for any number of occurrences, r.