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In 1990 Bender, Canfield and McKay gave an asymptotic formula for the numberof connected graphs on $[n]$ with $m$ edges, whenever $n$ and the nullity$m-n+1$ tend to infinity. Asymptotic formulae for the number of connected$r$-uniform hypergraphs on $[n]$ with $m$ edges and so nullity $t=(r-1)m-n+1$were proved by Karo\'nski and \L uczak for the case $t=o(\log n/\log\log n)$,and Behrisch, Coja-Oghlan and Kang for $t=\Theta(n)$. Here we prove such aformula for any $r\ge 3$ fixed, and any $t=t(n)$ satisfying $t=o(n)$ and$t\to\infty$ as $n\to\infty$. This leaves open only the (much simpler) case$t/n\to\infty$, which we will consider in future work. ( arXiv:1511.04739 )  Our approach is probabilistic. Let $H^r_{n,p}$ denote the random $r$-uniformhypergraph on $[n]$ in which each edge is present independently withprobability $p$. Let $L_1$ and $M_1$ be the numbers of vertices and edges inthe largest component of $H^r_{n,p}$. We prove a local limit theorem giving anasymptotic formula for the probability that $L_1$ and $M_1$ take any given pairof values within the `typical' range, for any $p=p(n)$ in the supercriticalregime, i.e., when $p=p(n)=(1+\epsilon(n))(r-2)!n^{-r+1}$ where$\epsilon^3n\to\infty$ and $\epsilon\to 0$; our enumerative result then followseasily.  Taking as a starting point the recent joint central limit theorem for $L_1$and $M_1$, we use smoothing techniques to show that `nearby' pairs of valuesarise with about the same probability, leading to the local limit theorem.Behrisch et al used similar ideas in a very different way, that does not seemto work in our setting.  Independently, Sato and Wormald have recently proved the special case $r=3$,with an additional restriction on $t$. They use complementary, more enumerativemethods, which seem to have a more limited scope, but to give additionalinformation when they do work.

Counting connected hypergraphs via the probabilistic method