Probabilistic Entry Swapping Bijections for Non-Attacking Fillings

Abstract: Non-attacking fillings are combinatorial objects central to the theory of Macdonald polynomials. A probabilistic bijection for partition-shaped non-attacking fillings was introduced by Mandelshtam (2024) to prove a compact formula for symmetric Macdonald polynomials. In this work, we generalize this probabilistic bijection to composition-shaped non-attacking fillings. As an application, we provide a bijective proof to extend a symmetry theorem for permuted-basement Macdonald polynomials established by Alexandersson (2019), proving a version with fewer assumptions.