Bijections for rhombic alternative tableaux

TL;DR

This paper develops new bijections connecting rhombic alternative tableaux with permutations and assemblées, leading to refined enumeration formulas and solutions to open questions about combinatorial statistics.

Contribution

It introduces novel bijections between RAT and assemblées, extending known permutation correspondences, and addresses open problems regarding combinatorial statistics and mappings.

Findings

Refined enumeration formula for RAT

Bijection mapping free cells to crossings

An $r!$-to-$1$ map from marked Laguerre histories to assemblées

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.