The asymptotic number of acyclic digraphs. I

We obtain an asymptotic formula forAn,q, the number of digraphs withn labeled vertices,q edges and no cycles. The derivation consists of two separate parts. In the first we analyze the generating function forAn,q so as to obtain a central limit theorem for an associated probability distribution. In the second part we show combinatorially thatAn,q is a smooth function ofq. By combining these results, we obtain the desired asymptotic formula.