Ascent sequences and 3-nonnesting set partitions

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 ≤ x i ≤ asc ( x 1 x 2 ⋯ x i − 1 ) + 1 for all 2 ≤ i ≤ n , where asc ( x 1 x 2 ⋯ x i − 1 ) is the number of ascents in x 1 x 2 ⋯ x i − 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.