Compositions With \(m\) Distinct Parts

We study \(F(n,m)\), the number of compositions of \(n\) in which repetition of parts is allowed, but exactly \(m\) distinct parts are used. We obtain explicit formulas, recurrence relations, and generating functions for \(F(n,m)\) and for auxiliary functions related to \(F\). We also consider the analogous functions for partitions.