Partitions without 2

Question

How do I find the generating function for partitions of $n$ that have no part with size $2$? In general, how would I find this for partitions that have no part of size $k$?

Answer 1

Are objects identical or distinct?

If they are distinct, it may help to think of recursion. Suppose we let S(n, t, r) be the number of partitions of n objects into t partitions out of which r partitions have only 1 object. Then

S(n, t, r) = (t-r)S(n-1, t, r) + S(n-1, t-1, r-1)

This is a similar recursion to Stirling numbers of the second kind, and hope this gets you started.

Answer 2

The generating function for $p(n)$ is

$$p(n)=\prod_{k\ge 1}\frac1{1-x^k}=\prod_{k\ge 1}\sum_{j\ge 0}x^{jk}\;:$$

the coefficient of $x^n$ is the number of ways to write $n$ in the form

$$n=j_1\cdot1+j_2\cdot2+j_3\cdot3+\ldots\;.$$

To omit $\ell$ as a summand in the partitions, drop out the factor of $\dfrac1{1-x^\ell}$. Thus, if $p_\ell(n)$ is the desired generating function, you have

$$p_\ell(n)=(1-x^\ell)p(n)\;.$$