A *plane partition* of $n$ is an table of integers $A=(a\_{ij})$ which add up to $n$ and non-increase in rows and columns. For example,
$$A= \begin{matrix} 331 \\
32 \ \ \\
11 \ \ 
\end{matrix}
$$
is a plane partition of $(3+3+1)+(3+2)+(1+1)=14$. 


One can view plane partitions as an arrangement of cubes stacked in the corner, see more on [Wikipedia](https://en.wikipedia.org/wiki/Plane_partition). 
Define *margins* to be 1-dim projections of cubes on all coordinate axis. These margins are triples $(\lambda,\mu,\nu)$ of partitions of $n$. For example, for $A$ as above we have $\lambda = (7,5,2)$, $\mu=(7,6,1)$ and $\nu=(7,4,3)$. 


**Question:** What is the smallest $n$ for which there exist two different plane partitions of $n$ with the same margins $(\lambda,\mu,\nu)$? 


**Note:** I know there are two different plane partitions of $2100$ with equal margins. This is a consequence of $p\_2(n)<p(n)^3$ for $n\ge 2100$, where $p\_2(n)$ is the number of plane partitions. Unfortunately, 2100 is way too large, and $p\_2(2100)\approx 1.47\cdot 10^{141}$. So it would be nice to find a small explicit example or an argument why e.g. there is none, say, for $n\le 200$.


UPDATE (Jan 29): Thank you, John Machacek, Gjergji Zaimi and Brian Hopkins, and apologies for being unclear. No, I don't worry for the symmetries, so John's first answer is the correct one. It's so nice and clean and Gjergji's argument is so simple and convincing, there is not much to add other than the motivation.


This is related to the study of [Kronecker coefficients](https://en.wikipedia.org/wiki/Kronecker_coefficient) and a bound by Ernesto Vallejo ([here](https://link.springer.com/article/10.1023/A:1008795704190)) which should be better known: 
$$g(\lambda,\mu,\nu)\ge p(\lambda,\mu,\nu)$$
where $g(\lambda,\mu,\nu)$ is the Kronecker coefficient and $p(\lambda,\mu,\nu)$ is the number of (labeled) plane partitions with margins $\lambda,\mu,\nu$. So I became interested if this bound is ever effective. From asymptotic considerations I can see that $g(\lambda,\mu,\nu)\ge p\_2(n)/p(n)^3 = \exp\Theta(n^{2/3})$, but what are those partitions? I then found an easy construction of such $\lambda,\mu,\nu$ and this many plane partitions as long as there is *one* example with $p(\lambda,\mu,\nu)\ge 2$. So now my pathetic 2100 can be replaces with 13. Nice. I will link my paper here discussing these and other bounds once we put it on the arXiv (joint with Greta Panova). 



Plane partitions with equal margins
Hopefully I understood the definitions and computed correctly. How about $n = 13$ with
$$A = \begin{matrix} 3 3 1 \\ 2 1 1 \\ 2 \ \ \end{matrix}$$
and
$$B = \begin{matrix} 3 2 2 \\ 3 1 \ \\ 1 1 \ \end{matrix}$$
where all margins are $(7,4,2)$.


