### Abstract

This entry provides a basic library for number partitions, defines the
two-argument partition function through its recurrence relation and relates
this partition function to the cardinality of number partitions. The main
proof shows that the recursively-defined partition function with arguments
n and k equals the cardinality of number partitions of n with exactly k parts.
The combinatorial proof follows the proof sketch of Theorem 2.4.1 in
Mazur's textbook `Combinatorics: A Guided Tour`. This entry can serve as
starting point for various more intrinsic properties about number partitions,
the partition function and related recurrence relations.