A connection between alternating sign matrices and totally symmetric self-complementary plane partitions

Abstract

We give a lattice path interpretation for totally symmetric self-complementary plane partitions. This is a first step in solving the long standing problem of enumerating such plane partitions. Another outstanding problem in enumerative combinatories is the search for a bijection between alternating sign matrices and totally symmetric self-complementary plane partitions. From the lattice path interpretation, we discover a new statistic on totally symmetric self-complementary plane partitions which should correspond to the position of the 1 in the top row of an alternating sign matrix under such a bijection.