patitions of the number n [closed]

Question

I'm having difficult with the following question :

A. Show that the number of partitions of n where in each one of them the even numbers appears at most once equals to the number of partitions of n in which every number appears at most three times .

Answer 1

For A consider the generating function;

$$\dfrac{(1+x^2)(1+x^4)(1+x^6)\dots}{(1-x)(1-x^3)(1-x^5)\dots}$$

This reads as '$0$ or $1$ of any even number, and any number of any odd numbers'.

We make progress by multiplying top and bottom by $(1+x)$ to give:

$$\dfrac{(1+x)(1+x^2)(1+x^4)(1+x^6)\dots}{(1+x)(1-x)(1-x^3)(1-x^5)\dots}$$ $$=\dfrac{(1+x+x^2+x^3)(1+x^4)(1+x^6)\dots}{(1-x^2)(1-x^3)(1-x^5)\dots}$$

We do the same with $(1+x^2)$:

$$\dfrac{(1+x+x^2+x^3)(1+x^2)(1+x^4)(1+x^6)\dots}{(1+x^2)(1-x^2)(1-x^3)(1-x^5)\dots}$$ $$=\dfrac{(1+x+x^2+x^3)(1+x^2+x^4+x^6)(1+x^6)\dots}{(1-x^4)(1-x^3)(1-x^5)\dots}$$

The numerators $(1+x^k+x^{2k}+x^{3k})$ allow for $0,1,2$ or $3$ $k$'s.

We can do this trick for all the numerators using the denominators with exponent $d$ to remove all numerations with exponent $2^kd$. The denominator is left as:

$$(1-x^\infty)^\infty$$

which plays no part in the value of the coefficients of $x^k$ for finite $k$.