  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.
