How to find the weight function and corresponding set given the generating series?
Is there a general method for this kind of problems, I am preparing for an olympiad. Consider the below example:
$\prod\limits_{k=0}^{99} (1+x^{1\cdot10k}+x^{2\cdot10k}+x^{3\cdot10k}+x^{4\cdot10k}+x^{5\cdot10k}+x^{6\cdot10k}+x^{7\cdot10k}+x^{8\cdot10k}+x^{9\cdot10k})$
How should I solve this? For simpler generating series it might be easy but for questions like this one, I am unable to come up with a solution.
Please help, thanks!
Before like terms are collected, a typical term of the product has an exponent of the form
$$a_0\cdot 10\cdot0+a_1\cdot10\cdot 1+\ldots+a_{99}\cdot10\cdot99=\sum_{k=0}^{99}10ka_k\;,$$
where each $a_k\in\{0,1,2,3,4,5,6,7,8,9\}$; we can take this to be the weight of one element of the set in question. There are many sets that could be made to work with a suitable definition of the weight function, but the simplest is $D^{100}$, where $D=\{0,1,2,3,4,5,6,7,8,9\}$. Each element of $D^{100}$ is a $100$-tuple $a=\langle a_0,a_1,\ldots,a_{99}\rangle$, and we define
$$w(a)=\sum_{k=0}^{99}10ka_k\;.$$
Then
$$\sum_{a\in D^{100}}x^{w(a)}=\prod_{k=0}^{99}\sum_{i=0}^9x^{10ik}\;,$$
as desired. The element $a=\langle a_0,a_1,\ldots,a_{99}\rangle\in D^{100}$ corresponds to the term $\prod_{k=0}^{100}x^{10a_kk}=x^{w(a)}$ in the expanded product on the righthand side.