Remarks on direct computation for \(P(n,k)\) (Q1908954)

Let \(P(n, k)\) and \(P(n)\) be the number of partitions of \(n\) into \(k\) parts greater than or equal to 1, and the number of all partitions of \(n\), respectively. All results deduced in this paper are simple consequences of the equality \(P(n, m)= P(n- m)\) holding for every \(m\geq n/2\). For example, \(P(2m, m)= P(2m+ s, m+ s)\) holds for any integer \(s\geq 0\) (Theorem 2.1) since both members are equal to \(P(m)\). Another identity proposed here is \[ P(2m- r- 1, m- r- 1)= P(m)- \sum^r_{t= 0} P(t) \] for every \(m\geq 3\) and \(0\leq r\leq m/2\).