Pattern Avoidance in Poset Permutations

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on P that avoid the pattern p is denoted A v _P (p). We extend a proof of Simion and Schmidt to show that A v _P (132)=A v _P (123) for any poset P, and we exactly classify the posets for which equality holds.