Ascent sequences and 3-nonnesting set partitions

TL;DR

This paper proves a conjecture linking 210-avoiding ascent sequences to 3-nonnesting set partitions by establishing a bijection through growth diagrams for Ferrers shape fillings.

Contribution

It confirms the conjecture by Duncan and Steingrímsson, introducing a novel bijection via growth diagrams for 01-fillings of Ferrers shapes.

Findings

210-avoiding ascent sequences are in bijection with 3-nonnesting set partitions

The proof uses growth diagrams for Ferrers shape fillings

The conjecture by Duncan and Steingrímsson is validated

Abstract

A sequence x=x_1 x_2...x_n $ is said to be an ascent sequence of length $n$ if it satisfies x_1=0 and $0\leq x_i\leq asc(x_1x_2...x_{i-1})+1$ for all $2\leq i\leq n$, where $asc(x_1x_2... x_{i-1})$ is the number of ascents in the sequence $x_1x_2... x_{i-1}$. Recently, Duncan and Steingr\'{\i}msson proposed the conjecture that 210-avoiding ascent sequences of length $n$ are equinumerous with 3-nonnesting set partitions of $\{1,2,..., n\}$. In this paper, we confirm this conjecture by showing that 210-avoiding ascent sequences of length $n$ are in bijection with 3-nonnesting set partitions of $\{1,2,..., n\}$ via an intermediate structure of growth diagrams for 01-fillings of Ferrers shapes.