Counting (3+1)-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 posers P are exactly the ones which have an associated permutation pi where i < j in P if and only if i < j as integers and i comes before j in the oneline notation of pi. So we say that a permutation pi is (3 + 1)-free or (3 + 1)-avoiding if its poser is (3 + 1)-free. This is equivalent to pi 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. (C) 2011 Elsevier Ltd. All rights reserved.