Problem #PRU-61525

Problems Geometry Solid geometry Visual geometry in space

Problem

We denote by \(P_{k, l}(n)\) the number of partitions of the number \(n\) into at most \(k\) terms, each of which does not exceed \(l\). Prove the equalities:

a) \(P_{k, l}(n) - P_{k, l-1}(n) = P_{k-1, l}(n-l)\);

b) \(P_{k, l}(n) - P_{k-1, l} (n) = P_{k, l-1}(n-k)\);

c) \(P_{k, l}(n) = P_{l, k} (n)\);

d) \(P_{k, l}(n) = P_{k, l} (kl - n)\).