This paper finds precise formulas and large-scale approximations for two related graph quantities: the number of acyclic orientations and certain negative values of the chromatic polynomial for complete multipartite graphs. An acyclic orientation is a way to put arrows on the edges so there is no directed cycle. The chromatic polynomial counts how many proper colorings a graph has for a given number of colors. For these particular graphs the authors give exact integral representations and then use saddle‑point methods to read off asymptotic behavior as sizes grow. The work settles conjectures and describes several growth regimes that appear in OEIS sequences the authors study.

The main technical tool is an exact “Gamma‑type’’ integral. For a complete multipartite graph with part sizes λ1,…,λr they write the acyclic orientation count as an integral of a product of explicit polynomials Pm(t), where Pm(t) is built from Stirling numbers of the second kind. They extend this to a Gamma‑weighted integral Hs(Kλ) = (1/Γ(s)) ∫0^∞ e−t t^{s−1} ∏i P_{λi}(t) dt, which recovers the acyclic count when s=1 and connects to the Tutte polynomial when the graph is connected. The paper also gives a simple combinatorial interpretation: AO(Kλ)/N! equals the expected product of factorials of run lengths in a random permutation of the vertices, after merging consecutive vertices of the same part (this helps with probabilistic estimates).

To turn those integrals into usable asymptotics the authors apply saddle‑point and analytic‑combinatorics‑in‑several‑variables (ACSV) techniques. Roughly speaking, they rewrite sums as integrals, find the point or points where the integrand contributes most (the saddle points), and expand the integrand around those points to obtain leading terms and corrections. In the multivariate cases they analyze singularities near strictly minimal critical points, while in product regimes they use one‑dimensional Laplace expansions for products of the Pm(t) polynomials.