Number of distinct higher dimensional integer partitions

Question

By a distinct partition, I mean a partition into distinct parts, i.e., $10 = 5+4+1$ is one, but $10=6+2+2$ is not. The number of distinct partitions of $k$ all whose parts are at most $n$ is given by the sequence https://oeis.org/A053632. The link already provides the generating function as well as a recursive formula.

Now, I am interested in higher dimensional distinct partition. By an $q$-dimensional distinct partition of $k$, I mean a sum

$$k = \sum_{i_1,\cdots,i_q} \lambda_{i_1,\cdots,i_q}$$

for positive integers $\lambda_{i_1,\cdots,i_j,\cdots,i_q}$ such that $\lambda_{i_1,\cdots,i_q} < \lambda_{i_1,\cdots,i_j+1,\cdots,i_q}$ for every $j$.

What is known about the number of distinct $q$-dimensional partitions of $k$ into parts of size at most $n$?

Edit: Because that was maybe not clear: I'm aware that 1-, 2- and 3-dimensional partitions are ordinary, plane and solid partitions, respectively, and that there is an abundance of literature about them. However, I am interested in partitions into distinct parts as defined above. For example, the plane partition example on Wikipedia is not a partition into distinct parts. This case seems to be less covered, and I'm asking for literature about that.

Answer 1

For distinct part plane partitions (called strict plane partitions $\mathcal{SP}$ in the references below), MacMahon's formula

$$ \sum_{\pi \in \mathcal{P}} q^{|\pi|} = \prod_{n \ge 1} \left( \frac{1}{1-q^n} \right)^n$$

becomes

$$ \sum_{\pi \in \mathcal{SP}} 2^{k(\pi)}q^{|\pi|} = \prod_{n \ge 1} \left( \frac{1+q^n}{1-q^n} \right)^n$$

where $k(\pi)$ is "the number of connected components" of the plane partition $\pi$ as defined by Vuletić.

M. Vuletić, The Shifted Schur Process and Asymptotics of Large Random Strict Plane Partitions; Int. Math. Res. Notices (2007), Vol. 2007, article ID rnm043, 53 pages.

M. Vuletić, A Generalization of MacMahon's Formula; Trans. Am. Math. Soc. 361(5) (2009) 2789-2804.

These have both been cited several times, so looking through what referenced them might help with your higher dimensional questions.

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As per Richard Stanley's and Sam Hopkins's comments, there's some tension between the question-writer's use of the term "distinct" and their definition (which allows for repeated parts across the entire partition). For plane partitions, I believe that's the difference between the OEIS sequences A117433 and A114736, respectively.