  Recently, Lai and Rohatgi discovered a shuffling theorem for lozenge tilings
of doubly-dented hexagons, which generalized the earlier work of Ciucu. Later,
Lai proved an analogous theorem for centrally symmetric tilings, which
generalized some other previous work of Ciucu. In this paper, we give a unified
proof of these two shuffling theorems, which also covers the weighted case.
Unlike the original proofs, our arguments do not use the graphical condensation
method but instead rely on a well-known tiling enumeration formula due to Cohn,
Larsen, and Propp. Fulmek independently found a similar proof of Lai and
Rohatgi's original shuffling theorem. Our proof also gives a combinatorial
explanation for Ciucu's recent conjecture relating the total number and the
number of centrally symmetric lozenge tilings.
