  A recent paper examined the global structure of integer partitions sequences
and, via combinatorial analysis using modular arithmetic, derived 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. The output of the IPS map
was a "matrix" having M columns and a number of rows equal to p[N, M], the
number of partitions of N into M parts. The global structure of integer
partition sequences (IPS) is that of a complex tree. In this paper, we examine
the structure of the IPS tree and, by counting the number of directed paths
through the tree, obtain a simple formula which gives, in closed form, the
total number of partitions of N into exactly M parts. By summing over M, we
obtain a transparent alternative to the Hardy-Ramanujan-Rademacher formula for
p(N).
