Ascent sequences and the binomial convolution of Catalan numbers

TL;DR

This paper explores pattern-avoiding ascent sequences and demonstrates that their counts relate to the binomial convolution of Catalan numbers, completing classifications for certain pattern avoidances.

Contribution

It establishes new enumeration results for ascent sequences avoiding specific patterns, linking them to binomial convolutions of Catalan numbers and completing Wilf classifications.

Findings

Number of 201,210-avoiding ascent sequences equals binomial convolution of Catalan numbers.

Number of 0021-avoiding ascent sequences also equals binomial convolution of Catalan numbers.

Completes Wilf classification for single patterns of length 4 in ascent sequences.

Abstract

In this paper, we consider two sets of pattern-avoiding ascent sequences: those avoiding both 201 and 210 and those avoiding 0021. In each case we show that the number of such ascent sequences is given by the binomial convolution of the Catalan numbers. The result for $\{201, 210\}$-avoiders completes a family of results given by Baxter and the current author in a previous paper. The result for 0021-avoiders, together with previous work of Duncan, Steingr\'{i}msson, Mansour, and Shattuck, completes the Wilf classification of single patterns of length 4 for ascent sequences.