I am working on a problem involving determining the order polynomial $\Omega_P(k)$ of a partial order $P$, which counts the number of order-preserving transformations/maps from $P$ to the $k$-chain (containing $k$ points and with length $(k-1)$). I'm aware that various sources (e.g., Handbook of Enumerative Combinatorics by Bona, 2015) show that $$\Omega_P(k)=a_nk^n+a_{n-1}k^{n-1}+\ldots+a_1k+a_0,$$
where $n=|P|$ and $a_n=e(P)/|P|!$. Here, $e(P)$ denotes the number of linear extensions of $P$. My question is, what else is known about the remaining coefficients $a_{n-1},\ldots,a_1,a_0$? In particular, I am assuming that $n=k$ (thus thinking of both the poset and the chain length as functions of $n$), and am hoping to prove something of the form
$$\Omega_P(n)=\Theta(n^n),$$
which will be useful.
Besides the main problem, I am also interested in any known closed form expressions of order polynomials for special classes of posets. (E.g., if $P=P_{N,r}$, the graded disjoint union of $N$ chains having $r$ elements each, then $\Omega_P(k)={r+k-1\choose k-1}^N$; see Reiner, Stanton and Welker 2003)
(P.S. I posted this on StackExchange earlier, but realised that there weren't any specific tags that would facilitate the reach of the question to the hands of the appropriate audience.)
