Ascent sequences avoiding pairs of patterns
Andrew M. Baxter, and Lara K. Pudwell

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 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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · semigroups and automata theory · Algorithms and Data Compression
