Let $P_{m}(n)$ gives the number of ways of writing the integer $n$ as a sum of positive integers not lesser than $m$. For example,
$$6=4+2=3+3=2+2+2$$ therefore $P_{2}(6)=4$ $$6=3+3$$ therefore $P_{3}(6)=2$
How can we determine $P_{m}(n)$, by knowing $P(n)$(classic partition function) and $m$ ?
One possibility is to consider the generating function of the unrestricted partition function $p(n)$. It is given by $$ \sum_{n=0}^\infty p(n)x^n = \prod_{k=1}^\infty \left(\frac {1}{1-x^k} \right). $$ Then we obtain a generating function for restricted partitions, taking $m$ fixed and $n$ variable, by $$ \sum_{n \geq 0} p_m(n) x^n = x^m \cdot \prod_{i = 1}^m \frac{1}{1 - x^i}. $$