Ascent sequences and 3-nonnesting set partitions

A sequence x=x_1 x_2…x_n $issaidtobeanascentsequenceoflength$n$ifitsatisfies{x}_1=0and$0\leq x_i\leq asc(x_1x_2…x_{i-1})+1$forall$2\leq i\leq n$,where$asc(x_1x_2… x_{i-1})$ is the number of ascents in the sequence $x_1{x}_2\mathrm{.}\mathrm{.}\mathrm{.}{x}_{i-1}$. 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 $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$. 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.