Automatic Generation of Theorems and Proofs on Enumerating Consecutive-Wilf classes

TL;DR

This paper presents automated methods for enumerating consecutive-Wilf classes in permutations, enabling the derivation of functional equations and explicit formulas that advance combinatorial enumeration research.

Contribution

It introduces two automated approaches, implemented in Maple packages, for deriving functional equations and explicit formulas in pattern-avoidance enumeration, surpassing manual ad-hoc methods.

Findings

Automated derivation of functional equations for generating functions.

Automatic simplification leading to explicit formulas.

Discovery of new explicit results beyond human capability.

Abstract

This article, dedicated to Herbert Saul Wilf on the occaison of his forthcoming 80-th birthday, describes two complementary approaches to enumeration, the "positive" and the "negative", each with its advantages and disadvantages. Both approaches are amenable to automation, and when applied to the currently active subarea, initiated in 2003 by Sergi Elizalde and Marc Noy, of enumerating consecutive-Wilf classes (i.e. consecutive pattern-avoidance) in permutations, were successfully pursued by DZ's two current PhD students, Andrew Baxter and Brian Nakamura. The Maple packages SERGI and ELIZALDE, implementing the algorithms enable the computer to "do research" by deriving, "all by itself", functional equations for the generating functions that enable polynomial-time enumeration for any set of patterns. In the case of ELIZALDE (the "negative" approach), these functional equations can be sometimes (automatically!) simplified, and imply "explicit" formulas, that previously were derived by humans using ad-hoc methods. We also get lots of new "explicit" results, beyond the scope of humans, but we have to admit, that we still need humans to handle "infinite families" of patterns, but this too, no doubt, will soon be automatable, and we leave it as a challenge to the (human and/or computer) reader. The Maple packages, and lots of sample output, is available from the webpage of this article: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/auto.html