| Abstract: | An original algorithm is presented that generates both restricted integer compositions and restricted integer partitions that
can be constrained simultaneously by (a) upper and lower bounds on the number of summands (“parts”) allowed, and (b) upper
and lower bounds on the values of those parts. The algorithm can implement each constraint individually, or no constraints
to generate unrestricted sets of integer compositions or partitions. The algorithm is recursive, based directly on very fundamental
mathematical constructs, and given its generality, reasonably fast with good time complexity. A general, closed form solution
to the open problem of counting the number of integer compositions doubly restricted in this manner also is presented; its
formulaic link to an analogous solution for counting doubly-restricted integer partitions is shown to mirror the algorithmic
link between these two objects. |
