  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
present a unifying perspective on ASMs and other combinatorial objects by
studying a certain tetrahedral poset and its subposets. We prove the order
ideals of these subposets are in bijection with a variety of interesting
combinatorial objects, including ASMs, totally symmetric self-complementary
plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux,
Catalan objects, tournaments, and totally symmetric plane partitions. We prove
product formulas counting these order ideals and give the rank generating
function of some of the corresponding lattices of order ideals. We also prove
an expansion of the tournament generating function as a sum over TSSCPPs. This
result is analogous to a result of Robbins and Rumsey expanding the tournament
generating function as a sum over alternating sign matrices.
