Double cosets are worth studying due to their interesting statistical interpretations, but the double cosets LS_nL of the symmetric group with respect to a Sylow p-subgroup L are not yet well-understood. Even determining the cardinality of LS_nL is an open problem. I will share a novel approach to indexing these double cosets, using families of antisymmetric matrices with certain properties. This approach is currently restricted to the case n = p^a, and is still conjectural. But the main conjectures have some empirical validation in the form of computerized tests. I will describe the conjectured index set, then outline how one would use it in order to determine the cardinality of LS_nL.