# Cardinality of Number Partitions

January 14, 2016

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

### 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.