Counting (3+1) - Avoiding permutations

A poset is {\it $(\text{3}+\text{1})$-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 $(\text{3}+\text{1})$-free Conjecture of Stanley and Stembridge. The dimension 2 posets $P$ are exactly the ones which have an associated permutation $\pi$ where $i\prec j$ in $P$ if and only if $i<j$ as integers and $i$ comes before $j$ in the one-line notation of $\pi$. So we say that a permutation $\pi$ is {\it $(\text{3}+\text{1})$-free} or {\it $(\text{3}+\text{1})$-avoiding} if its poset is $(\text{3}+\text{1})$-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.