Enumeration of higher-dimensional partitions

Determine a closed formula for the numbers of higher-dimensional partitions p_d^n in dimensions n≥4, thereby resolving the major unsolved enumeration problem in higher-dimensional partition theory.

Background

The paper studies the enumeration of (n−1)-dimensional partitions, denoted p_dn, and develops socle-reduction methods for computing these numbers. Although MacMahon proposed a product formula, that formula fails for n≥4 beyond low-order terms. The authors emphasize that exact enumeration in higher dimensions remains largely inaccessible and motivate their refined generating functions and algorithms as a way to compute substantial finite data rather than solve the general problem.

References

Computing the numbers $p_dn$ for $n\geqslant 4$ is a major unsolved problem in Combinatorics; quoting the words of Stanley, ``almost nothing significant is known'' Sec.~7.20, p.~365.

— Enumeration of partitions via socle reduction  (2501.10267 - Graffeo et al., 17 Jan 2025) in Section 1, Introduction, Overview