  For a simple finite graph G denote by {G \brace k} the number of ways of
partitioning the vertex set of G into k non-empty independent sets (that is,
into classes that span no edges of G). If E_n is the graph on n vertices with
no edges then {E_n \brace k} coincides with {n \brace k}, the ordinary Stirling
number of the second kind, and so we refer to {G \brace k} as a graph Stirling
number.
  Harper showed that the sequence of Stirling numbers of the second kind, and
thus the graph Stirling sequence of E_n, is asymptotically normal ---
essentially, as n grows, the histogram of ({E_n \brace k})_{k \geq 0}, suitably
normalized, approaches the density function of the standard normal
distribution.
  In light of Harper's result, it is natural to ask for which sequences
(G_n)_{n \geq 0} of graphs is there asymptotic normality of ({G_n \brace k})_{k
\geq 0}. Do and Galvin conjectured that if for each n, G_n is acylic and has n
vertices, then asymptotic normality occurs, and they gave a proof under the
added condition that G_n has no more than o(\sqrt{n/\log n}) components.
  Here we settle Do and Galvin's conjecture in the affirmative, and
significantly extend it, replacing "acyclic" in their conjecture with
"co-chromatic with a quasi-threshold graph, and with negligible chromatic
number". Our proof combines old work of Navon and recent work of Engbers,
Galvin and Hilyard on the normal order problem in a Weyl algebra, and work of
Kahn on the matching polynomial of a graph.
