Enumeration of partitions via socle reduction

This paper is concerned with the long-standing problem of enumerating higher dimensional partitions. We introduce a socle reduction technique, through which the sought after count becomes equivalent to the count of simpler classes of partitions, satisfying suitable constraints on their embedding dimension and socle type. We provide exact formulas for the generating functions of several infinite families of such partitions, and design a procedure enumerating them in the general case. As a biproduct of our technique, we are able to determine the number of partitions of size up to 32 in any dimension,  extending the previously known results, which were limited to size $\leqslant 26$.