Bijections for rhombic alternative tableaux
Combinatorics
2026-03-16 v1
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 -to- map from marked Laguerre histories to assembl\'{e}es, answering a question of Corteel and Nunge.
Keywords
Cite
@article{arxiv.2603.12700,
title = {Bijections for rhombic alternative tableaux},
author = {Sylvie Corteel and Jang Soo Kim and Olya Mandelshtam and Philippe Nadeau},
journal= {arXiv preprint arXiv:2603.12700},
year = {2026}
}
Comments
38 pages, 18 figures