Truncating a generating function

Question

It is true that the generating function for the number of ways to partition an integer $n$ into a sum of ones is \begin{align} f(x) = 1 + x + x^2 + x^3 + \cdots \end{align} But I don't see how the $1$ corresponds to no ones in your partition, $x$ corresponds to one one in your partition, $x^2$ corresponds to two ones in your partition, etc. So in order to have the size of your partition less than $k$, you would truncate this generating function and consider the generating function \begin{align} f(x) = 1+x+x^2+\cdots+x^k. \end{align} Could anyone clarify this?

Answer 1

But I don't see how the $1$ corresponds to no ones in your partition, $x$ corresponds to one one in your partition, $x^2$ corresponds to two ones in your partition, etc.

What is a generating function? In the simplest case of one variable, it's a function $$A(x) = a_0 + a_1 x + a_2 x^2 + \ldots$$ whose coefficients $a_n$ correspond to a sequence of interest. When the sequence of interest is

the number of ways to partition an integer $n$ into a sum of ones

we want $a_n$ to be the number of partitions of $n$ into ones. Therefore $a_0$ is the number of partitions of $0$ into ones, $a_1$ is the number of partitions of $1$ into ones, etc. If you have two ones in your partition, their sum is $2$ and so they count towards $a_2$; since this is the only way to partition $2$ into ones, $a_2 = 1$ and so the coefficient of $x^2$ is $1$.

Answer 2

The number of ways to partition and integer $\, n \,$ into a sum of ones is one because there is only one way to do it. That is, for all $\, n \ge 0 \,$ the sequence $\, a_n = 1. \,$ Thus, the generating function of this sequence is, as you asked, $\, f(x) = 1 + x + x^2 + x^3 + \dots. \,$ If you place more restrictions on the partition, then the generating function will change to reflect those restrictions. Agian, as you asked, if you restrict to size of partition always less than $\, k, \,$ then the the sequence $\, a_n = 1 \,$ if $\, n < k \,$ and $\, a_n = 0 \,$ if $ n\ge k \,$ and thus its generating function is now $\, f(x) = 1 + x + x^2 + \dots + x^{k-1}. \,$