I'm trying to solve this problem : Let S be the multiset {$∞ * e_1$, $∞ * e_2$, $∞ * e_3$, $∞ * e_4$}. Determine the generating function for the sequence $h_0, h_1, h_2, …., h_n, …,$ where $h_n$ is the number of n-combinations of S with the following added restrictions:
1) The element $e_1$ does not occur, and $e_2$ occurs at most once.
2) The element $e_1$ occurs 1, 3, or 11 times, and the element $e_2$ occurs 2, 4, or 5 times.
My attempt :
For the 1), I thought that the generating series
$g(x) = (1+x+x^2+…)(1+x+x^2+…)(1+x)$ ; Namely, the first term was because of $e_3$ and the second one was because of $e_4$, and the last term was because of $e_2$
For the 2), I thought that the generating series
$g(x) = (1+x+x^2+…)(1+x+x^2+…)(x+x^3+x^11)(x^2+x^4+x^5)$ ; Namely, the first term was because of $e_3$ and the second one was because of $e_4$, and the third term was because of $e_1$ the last term was because of $e_2$
Is this correct attempt? Thanks
Best Answer
Those two expressions are correct, but we can write them neater. It is well-known that $$1+x+x^2+x^3+\cdots=\frac1{1-x}$$ so the first series can be shortened to $$\frac{1+x}{(1-x)^2}$$ while the second series can be shortened to $$\frac{(x+x^3+x^{11})(x^2+x^4+x^5)}{(1-x)^2}$$ Incidentally, the first series gives the odd numbers in their entirety.