Global structure of integer partitions sequences

Abstract

Integer partitions are deeply related to many phenomena in statistical physics. A question naturally arises which is of interest to physics both on "purely" theoretical and on practical, computational grounds. Is it possible to apprehend the global pattern underlying integer partition sequences and to express the global pattern compactly, in the form of a "matrix" giving all of the partitions of N into exactly M parts? This paper demonstrates that the global structure of integer partitions sequences (IPS) is that of a complex tree. By analyzing the structure of this tree, we derive a closed form expression for a map from (N, M) to the set of all partitions of a positive integer N into exactly M positive integer summands without regard to order. The derivation is based on the use of modular arithmetic to solve an isomorphic combinatoric problem, that of describing the global organization of the sequence of all ordered placements of N indistinguishable balls into M distinguishable non-empty bins or boxes. This work has the potential to facilitate computations of important physics and to offer new insights into number theoretic problems.