In Pittel (1989)'s solution to a problem of Knuth (1976) on the expected number of stable matchings between $n$ men and $n$ women under uniform random preferences, it was shown that, as $n \to \infty$, $$ \int_{[0,1]^{2n}} \prod_{\substack{1 \le i, j \le n \\ i \neq j}}(1 - x_iy_j) \, dx_1 \cdots dx_n dy_1 \cdots dy_n = (e^{-1}+o(1))\frac{n \log n}{n!}. $$ The LHS can be rewritten, using the language of graphons, as $t(H_n, W)$, where $H_n$ is $K_{n,n}$ minus a perfect matching and $W(x,y) = 1 - xy$.

Is there a general technique for determining the asymptotic expression of $t(H_n, W)$ as $n \to \infty$, for a fixed graphon $W$ and a nice sequence of graphs $H_n$ (e.g., $K_n$, $K_{n,n}$)?

Are there other interesting cases where the asymptotics have been determined (perhaps with applications)?