Problem #PRU-61525

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