Lenart's bijection via bumpless pipe dreams (Q2684896)

Summary: Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to \textit{C. Lenart} [J. Algebr. Comb. 20, No. 3, 263--299 (2004; Zbl 1056.05146)] and \textit{A. Weigandt} [J. Comb. Theory, Ser. A 182, Article ID 105470, 52 p. (2021; Zbl 1475.05172)], respectively. Recently, \textit{Y. Gao} and \textit{D. Huang} [``The canonical bijection between pipe dreams and bumpless pipe dreams'', Preprint, \url{arXiv:2108.11438}] have defined a bijection between the former two sets. In this note we show for vexillary permutations that the Gao-Huang bijection preserves the associated tableaux, giving a new proof of Lenart's result [loc. cit.]. Our methods extend to give a recording tableau for any bumpless pipe dream.