Ascent sequences avoiding pairs of patterns

TL;DR

This paper studies ascent sequences avoiding pairs of length-3 patterns, providing exact counts for 16 pattern pairs, and establishes a bound similar to Erdős–Szekeres for these sequences.

Contribution

It offers exact enumeration results for ascent sequences avoiding two patterns of length 3 and introduces an Erdős–Szekeres type theorem for these sequences.

Findings

Exact enumeration for 16 pattern pairs

Bijections to Dyck paths and permutations

An Erdős–Szekeres type bound for ascent sequences

Abstract

Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrimsson. In this paper, we consider ascent sequences of length $n$ avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdos-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound.