The original question was to find the number of ways to split an integer, $n$, into any number of partitions where each of the parts belong to the set $\lbrace 1,3,4,9\rbrace$. Assuming I did this right I found the answer to be simply:
The $x^n$ coefficient of $$(1-x)^{-1}(1-x^3)^{-1}(1-x^4)^{-1}(1-x^9)^{-1}.$$
But now the restriction has been made that the number of parts in the composition that equal $9$ has to be less than or equal to the number of parts that equal $4$. If someone could help me figure out the correct generating function I'd be really thankful!
