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](https://mathoverflow.net/questions/343005/order-polynomial-of-shifted-double-staircase).



Unified framework for posets with order polynomial product formulas
I wrote a [survey of posets with order polynomial product formulas](https://arxiv.org/abs/2006.01568). 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.


