On the number of partitions with a fixed largest part (Q2707960)

Let \(P_k(n)\) denote the number of partitions of \(n\) with largest part equal to \(k\). In this article the author derives several formulas for calculating \(P_k(n)\). For example, he shows that if \(1\leq k\leq n\), then NEWLINE\[NEWLINEP_k(n) = P(n-k)-\sum_{i=k+1}^{n-k} P_i(n-k),NEWLINE\]NEWLINE where \(P(m)\) is the number of partitions of \(n\). Similar formulas are derived for \(Q_k(n)\), the number of partitions of \(n\) into distinct parts with largest part being \(k\).