Permutation totally symmetric self-complementary plane partitions

Alternating sign matrices and totally symmetric self-complementary plane partitions are equinumerous sets of objects for which no explicit bijection is known. In this paper, we identify the subset of totally symmetric self-complementary plane partitions corresponding to permutations by giving a statistic-preserving bijection to permutation matrices, which are a subset of alternating sign matrices. This bijection maps the inversion number of the permutation, the position of the one in the last column, and the position of the one in the last row to natural statistics on these plane partitions. We use this bijection to define a new partial order on permutations, and prove this new poset contains both the Tamari lattice and the Catalan distributive lattice as subposets. We also study a new partial order on totally symmetric self-complementary plane partitions arising from this perspective and show is a distributive lattice related to Bruhat order when restricted to permutations.