Pattern-avoiding ascent sequences of length 3

TL;DR

This paper investigates pattern-avoiding ascent sequences of length 3, providing new algorithms, asymptotic conjectures, and growth constants for sequences avoiding specific patterns, expanding understanding of their combinatorial properties.

Contribution

It introduces polynomial and exponential algorithms for certain pattern-avoiding ascent sequences and conjectures their asymptotic behaviors, including growth constants and generating function properties.

Findings

Polynomial time algorithms for 000 and 110-avoiding sequences

Exponential time algorithms for 100 and 120-avoiding sequences

Proved zero radius of convergence for three cases and identified stretched-exponential growth for 120-avoiding sequences

Abstract

Pattern-avoiding ascent sequences have recently been related to set-partition problems and stack-sorting problems. While the generating functions for several length-3 pattern-avoiding ascent sequences are known, those avoiding 000, 100, 110, 120 are not known. We have generated extensive series expansions for these four cases, and analysed them in order to conjecture the asymptotic behaviour. We provide polynomial time algorithms for the 000 and 110 cases, and exponential time algorithms for the 100 and 120 cases. We also describe how the 000 polynomial time algorithm was detected somewhat mechanically given an exponential time algorithm. For 120-avoiding ascent sequences we find that the generating function has stretched-exponential behaviour and prove that the growth constant is the same as that for 201-avoiding ascent sequences, which is known. The other three generating functions have zero radius of convergence, which we also prove. For 000-avoiding ascent sequences we give what we believe to be the exact growth constant. We give the conjectured asymptotic behaviour for all four cases.