Multivariate generating function

Question

I am investigating the perturbation of the Jordan canonical form. In my work I must calculate the number of ways to factor $p^ {n-k} q^k$ where $p$ and $q$ are distinct primes (https://oeis.org/A054225). This sequence is generated by the function: $$ \prod_{i=1}^t \prod_{j=0}^i \frac{1}{1-x^iy^j}=1+x+xy+2x^2+2x^2y+2x^2y^2+3x^3+4x^3y+4x^3y^2+3x^3y^3+\ldots. $$ I've never solved a multivariable generating function before. Could you advise please, how to prove that the function is generated for the sequence, or advise the literature on this subject?

Any help would be greatly appreciated.

Answer 1

The given g.f. directly follows from the sequence definition.

Indeed, the sequence gives the number of partitions of vector $(n,m)$ into the sum of vectors with nonnegative integer components. Each possible vector $(i,j)$ is encoded by the term $$\frac{1}{1-x^iy^j} = 1 + x^iy^j + x^{2i}y^{2j} + x^{3i}y^{3j} + \dots,$$ where the summand $x^{ki}y^{kj}$ encodes the contribution of this vector when it is taken $k$ times (the powers of $x$ and $y$ stand for the first and second component of $k\cdot (i,j)$, respectively). Then, the total contribution of all vectors in a partition results in $x^ny^m$, and the coefficient of this monomial in the product gives the number of different partitions.