Bijections for rhombic alternative tableaux
Abstract
We generalize well-known bijections between alternative tableaux and permutations to bijections between rhombic alternative tableaux (RAT) and assembl\'ees of permutations. We show how these various bijections are connected. As a consequence, we find a refined enumeration formula for RAT. One of our bijections carries many statistics from RAT to assembl\'ees; notably, it sends the number of free cells to the number of crossings, which answers a question of Mandelshtam and Viennot. We also find an r!-to-1 map from marked Laguerre histories to assembl\'ees, answering a question of Corteel and Nunge.
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