I am trying to grasp this example from the book A Walk Through Combinatorics:
Show that $\sum_{n \ge 0} p_d(n)x^n = \prod_{i \ge 1}(1+x^i)$ where $p_d$ stands for partitions of $n$ into all distinct parts.
I have seen this post but I don't understand the solution. I don't understand what the 3rd and 4th sentences mean, and how we concluded the generating function holds.
Could someone please share any hints/solutions with me about this problem?
Let me explain it my own words. We want to partition an integer such that its partitions must be all distinct. Then, for example, we cannot have two partitions such that both of them cannot have three elements.
To see what it is wanted to mean , lets use an example:
For $n=4$, lets write its partitions:
$4$
$3,1$
$2,2$
$2,1 ,1$
$1,1,1,1$
If we want all partitions are distinct, then $\{2,2\}$,$\{2,1 ,1\}$,$\{1,1,1,1\}$ must be discarded. So, when we want to find the partitions such that each partition block is disitnct, then each type of block size can be counted at most once.
Then, for example, there can be a partition block of size $4$ at most once, there can be a partition block of size $3$ at most once etc.
lets write its G.F:
If we want to find the number of partitions of distinct size of integer $n$, then $$[x^n](1+x^1)(1+x^2)(1+x^3)(1+x^4)...(1+x^n)$$
For example, $(1+x^2)$ means either have none block of size $2$ (by $x^0=1$) or have only once block of size $2$.
