Laboratoire de Physique Théorique
Toulouse - UMR 5152

Several combinatorial problems in physics, mathematics and computer science lead to a natural generalization of the partitions of integers
— these are called higher-dimensional partitions and were first introduced by MacMahon. Two-dimensional or plane partitions have a nice generating function like the one due to Euler for usual partitions. It was shown in 1967 by Atkin et. al. that a similar generating function guessed by MacMahon for dimensions >2 was wrong. One needs to take recourse to exact enumeration. It was also shown by Atkin et. al. that in order to enumerate partitions of a positive integer N in any dimension, one needs to compute (N-1) independent numbers.This could be, say, the partitions of N in dimensions 1,..., (N-1). We improve on the 1967 result by showing that one only need [(N-1)/2] independent numbers to obtain the result. This reduction is summarized by a simple transform which leads to a new combinatorial problem that we hope can be used for direct enumerations. As a by-product, we are able to compute partitions of all integers <=25 in any dimension which would have been impossible without the reduction.