On the enumeration of interval graphs

Abstract:

We present upper and lower bounds for the number $i_n$ of interval graphs on $n$ vertices. Answering a question posed by Hanlon, we show that the ordinary generating function $I(x) = \sum _{n\ge 0} i_n x^n$ for the number $i_n$ of $n$-vertex interval graphs has radius of convergence zero. We also show that the exponential generating function $J(x) = \sum _{n\ge 0} i_n x^n/n!$ has radius of convergence at least $1/2$.