The \(r\)-complete partitions (Q1382837)

A partition of \(n\) into \(k\) parts is called a complete partition if for each \(1 \leq m \leq n\), \(m\) can be written as the sum of parts of the partition. (For example \((1,1,2)\) is a complete partition of 4, but \((1,3)\) is not.) This paper studies the enumeration of \(r\)-complete partitions -- a generalization of complete partitions, where the parts of the partition may appear in the sum with multiplicity up to \(r\). Hence \((1,3)\) is a 2-complete partition of 4.