Ascent sequences and 3-nonnesting set partitions

Abstract

Listen

Translate article

A sequence x=x1x2⋯xn is said to be an ascent sequence of length n if it satisfies x1=0 and 0≤xi≤asc(x1x2⋯xi−1)+1 for all 2≤i≤n, where asc(x1x2⋯xi−1) is the number of ascents in x1x2⋯xi−1. Recently, Duncan and Steingrímsson 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.