Enumeration of graded -avoiding posets

The notion of ( 3 + 1 ) -avoidance has shown up in many places in enumerative combinatorics, but the natural goal of enumerating all ( 3 + 1 ) -avoiding posets remains open. In this paper, we enumerate graded ( 3 + 1 ) -avoiding posets for both reasonable definitions of the word “graded.” Our proof consists of a number of structural theorems followed by some generating function computations. We also provide asymptotics for the growth rate of the number of graded ( 3 + 1 ) -avoiding posets.