Counting -avoiding permutations

A poset is ( 3 + 1 ) -free if it contains no induced subposet isomorphic to the disjoint union of a 3-element chain and a 1-element chain. These posets are of interest because of their connection with interval orders and their appearance in the ( 3 + 1 ) -free Conjecture of Stanley and Stembridge. The dimension 2 posets P are exactly the ones which have an associated permutation π where i ≺ j in P if and only if i < j as integers and i comes before j in the one-line notation of π . So we say that a permutation π is ( 3 + 1 ) -free or ( 3 + 1 ) -avoiding if its poset is ( 3 + 1 ) -free. This is equivalent to π avoiding the permutations 2341 and 4123 in the language of pattern avoidance. We give a complete structural characterization of such permutations. This permits us to find their generating function.