On consecutive pattern-avoiding permutations of length 4, 5 and beyond

TL;DR

This paper reviews and extends the understanding of generating functions for consecutive pattern-avoiding permutations of lengths 4 and 5, providing solutions to differential equations, new algorithms, and conjectures about their D-finiteness.

Contribution

It introduces a polynomial-time algorithm for generating series coefficients and provides new solutions and conjectures regarding the nature of these generating functions.

Findings

Solutions to some differential equations for pattern classes

Extended asymptotic calculations for classes of length 4 and 5

Conjecture that solutions are not D-finite or differentially algebraic

Abstract

We review and extend what is known about the generating functions for consecutive pattern-avoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length-4 and twenty-five length-5 consecutive-Wilf classes. D-finite differential equations are known for the reciprocal of the exponential generating functions for four of the length-4 and eight of the length-5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length-4, length-5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not D-finite. For three further length-5 c-Wilf classes we give recurrences for two and a differential-functional equation for a third. For a fourth class we find a new algebraic solution. We give a polynomial-time algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for D-finite solutions for the unsolved classes, leading us to conjecture that the solutions are not D-finite. We have also searched, unsuccessfully, for differentially algebraic solutions.