Unified framework for posets with order polynomial product formulas One of the most celebrated results in algebraic combinatorics is the Hook Length Formula of Frame-Robinson-Thrall which counts the number of standard Young tableaux of given partition shape. Such SYTs can be viewed as linear extensions of a poset (namely, the poset of the shape). There are also product formulas enumerating SYTs of shifted shape, and linear extensions of tree posets. In fact, Proctor has defined a family of finite posets, called the "d-complete posets", which are defined axiomatically and which all enjoy a hook length-style product formula enumerating their linear extensions. See e.g. https://arxiv.org/abs/1704.05792. The d-complete posets include "all major examples" of posets with product formulas enumerating their linear extensions, as far as I am aware. (EDIT: This might be overselling the unifying power of the d-complete class; see the comments.)
If $P$ is a finite poset then its order polynomial $\Omega_P(m)$ is defined by 
$$\Omega_P(m) := \# \textrm{ weakly order preserving maps $P\to \{1,2,\ldots,m\}$}.$$
It is known that $\Omega_P(m)$ is a polynomial of degree $\#P$ and its leading coefficient is $1/\#P!$ times the number of linear extensions of $P$. ($\Omega_P(m)$ is basically the Ehrhart polynomial of the order polytope of $P$.) Certain posets have product formulas for their order polynomials. For instance, this is true of the rectangle poset $P = [a] \times [b]$ for which we have the celebrated formula of MacMahon:
$$ \Omega_P(m) = \prod_{i=1}^{a} \prod_{j=1}^{b} \frac{i+j+m-2}{i+j-1}.$$ And there are similar product formulas for the order polynomials of all the minuscule posets (see https://www.sciencedirect.com/science/article/pii/S0195669884800372; in fact, the minuscule posets have a product formula for a $q$-analog of their order polynomials). But this is true also for instance of the root poset $P=\Phi^+(A_n)$ of the Type A root system which has
$$\Omega_{P}(m) = \prod_{1\leq i \leq j \leq n} \frac{i+j+2m-2}{i+j}$$
(see https://www.sciencedirect.com/science/article/pii/S019566981380128X). More generally, the root posets of coincidental type have product formulas for their order polynomials (see S4.6.1 of https://conservancy.umn.edu/bitstream/handle/11299/159973/Williams_umn_0130E_14358.pdf).
Question: Is there a framework analogous to the framework of d-complete posets which explains when posets have product formulas for their order polynomials (at least for the "major examples" discussed above)?
EDIT:
I am adding a very closely related question which I am also quite interested in: how many posets have product formulas for their order polynomials, anyways?
Simpler Question: Are there any families of posets which have product formulas for their order polynomials, beyond the following?:


*

*minuscule posets,

*root posets of coincidental type ($\Phi^+(A_n)$, $\Phi^+(B_n)$, $\Phi^+(H_3)$, and $\Phi^+(I_2(\ell))$),

*the ``trapezoid poset'', which has the same order polynomial as the rectangle.


For instance, tree/forest posets have a very simple structure and have a known hook-length style formula enumerating their linear extensions, but I wasn't able to figure out a product formula for their order polynomials.
EDIT 2:
I experimentally found a conjectural additional family of posets having order polynomial product formulas (the ``shifted double staircases'') and I asked for a proof in this follow-up question.
 A: There is a product formula for the order polynomial of an unusual poset in Solution of an Enumerative Problem Connected with Lattice Paths by Kreweras and Niederhausen,  The poset is a product of a chain with a 3-element "V-shaped" poset.
A: Bob Proctor sent me an email explaining that the poset $P$ corresponding to the (unshifted, straight) shape $\lambda = (p+(r-1)b, p+(r-2)b, ..., p+b, p)$ has a product formula for its order polynomial, which can be seen via manipulations on the appropriate determinant. Note that this class includes both rectangles ($b=0$), as well as staircases ($p=1$, $b=1$). A reference for this result (with attribution to Proctor) is Stanley's EC2 Exercise 7.101. It is also mentioned in Proctor's paper "Odd symplectic groups."
A: I wrote a survey of posets with order polynomial product formulas. It does not provide a "unified framework" for these posets, but does put forward a heuristic that they are the posets with good dynamical behavior.
A: There are product formulas for the linear extensions of forests also, and some q-analogs of these. Perhaps the order-polynomials are nice as well.
