  Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1
whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We
put ASMs into a larger context by studying the order ideals of subposets of a
certain poset, proving that they are in bijection with a variety of interesting
combinatorial objects, including ASMs, totally symmetric self--complementary
plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young
tableaux, and totally symmetric plane partitions. We use this perspective to
prove an expansion of the tournament generating function as a sum over TSSCPPs
which is analogous to a known formula involving ASMs.
