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arXiv · 2603.12700

Bijections for rhombic alternative tableaux

Sylvie Corteel·Jang Soo Kim·Olya Mandelshtam·Philippe Nadeau
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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\'{e}es; 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\'{e}es, answering a question of Corteel and Nunge.

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Sylvie Corteel, Jang Soo Kim, Olya Mandelshtam, Philippe Nadeau. 2026-03-13. Bijections for rhombic alternative tableaux. https://arxiv.org/abs/2603.12700

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