The [hook length formula](https://en.wikipedia.org/wiki/Hook_length_formula) give a simple product expression for the number of standard Young tableaux of a given shape $\lambda$, where $\lambda$ is an integer partition, or equivalently, the number of ways to build the Ferrers diagram of $\lambda$ from the empty partition by adding boxes one at a time.


I'm aware that the hook length formula has analogues for certain ('$d$-complete') posets, I have a rather more pedestrian question: is there a hook length formula for plane partitions?


Here what I mean by 'hook-length formula for plane partitions' is the following. Consider the plane partition $\Lambda$ as a union of $1 \times 1 \times 1$ cubes, the $3D$ analogue of the Ferrers diagram of a partition, and let $SYT(\Lambda)$ be the number of sequences $\emptyset \subset \Lambda^{(1)} \subset \ldots \subset \Lambda^{(N)} = \Lambda$, where each $\Lambda^{(i)}$ is a valid plane partition and $\Lambda^{(i)},\Lambda^{(i+1)}$ differ by a single $1 \times 1 \times 1$ cube (such sequences are just the $3D$ analogues of standard Young tableaux). My question is, is there a simple product formula for the number $|SYT(\Lambda)|$? Or is there good reason to think such a formula should or should not exist?



hook length formula for plane partitions
I'll convert my comments to an answer.


You are asking about the number of linear extensions of a poset $P$ which is a finite order ideal (downwards closed set) in $\mathbb{N}^3$. With [SageMath](https://www.sagemath.org/) I was easily able to compute that the number of linear extensions of $P=[3]\times[3]\times[3]$ is $6405442434150 = 2 \times 3 \times 5^2 \times 607 \times 70350823$. This number has a huge prime factor in it, which rules out the possibility of any product formula like you had in mind.


In general, there are many two-dimensional arrays of numbers which have remarkable enumerative properties. Depending on your viewpoint, these remarkable properties are either the result of connections to representation theory or to planar statistical mechanical models. But at any rate, their naive 3D analogs fail to behave nicely from an enumerative point of view. For instance, you were asking about 3D Standard Young Tableaux. On the other hand, 100 years ago, MacMahon considered "3D plane partitions" (also called "solid partitions"), i.e., 3D arrays of numbers that are weakly decreasing in all directions. He conjectured a simple product formula for their generating function, but his conjecture was wrong already in the $x^6$ coefficient. See for instance the [Wikipedia page on solid partitions](https://en.wikipedia.org/wiki/Solid_partition) and see also this [prior MathOverflow question about 3D analogs of tableaux, et cetera](https://mathoverflow.net/questions/125846).


[ Nevertheless, higher-dimensional partitions have been investigated to some degree. For example, there are some nontrivial asymptotic results, as mentioned on the Wikipedia page linked above. Also, very recently there have been some interesting enumerative results related to higher-dimensional partitions when you "change the question you are asking": see, e.g., the preprint ["MacMahon's statistics on higher-dimensional partitions"](https://arxiv.org/abs/2009.00592) by Amanov and Yeliussizov, or the preprint ["Fully complementary higher dimensional partitions"](https://arxiv.org/abs/2301.12272) by Aigner that was just posted to the arXiv today! ]


