# Sum of the ratios of Schur functions

There are simple expressions for the sums of linear and quadratic combinations of Schur functions over all partitions (including the empty one) $$\sum_\lambda s_\lambda(x)=\prod_{i}\frac{1}{1-x_i}\prod_{i<j}\frac{1}{1-x_ix_j}$$ and $$\sum_\lambda s_\lambda(x) s_\lambda(y)=\prod_{i,j}\frac{1}{1-x_iy_j}.$$

Let us consider the sum of the ratio of two Schur functions, namely $$\sum_\lambda \frac{s_\lambda(x)}{ s_\lambda(y)}$$

Is there any similar expression for this sum?

For example, if $\#x_i=\#y_i=1$, we have

$$\sum_\lambda \frac{s_\lambda(x)}{ s_\lambda(y)}=\sum_{k=0}^\infty \frac{x^k}{y^k}=\frac{1}{1-x/y}.$$

I am skeptical of any "nice" structure while $x$ and $y$ remain so free. It might be somewhat reasonable to check things out under certain specializations.
You may find some interesting quotients, after specializations on $x$ and $y$, in Richard Stanley's Enumerative Combinatorics, Vol. 2, Exercises 7.30 and 7.32.
• Thank you! I also think that to get some nice expressions on have to specify something. What about arbitrary $x$ and some fixed $y$? – Sasha Dec 30 '16 at 18:02
• This (with $x$ free), still, might be expecting much. I could be wrong. Yes, the exercises are only quotients and then you sum. – T. Amdeberhan Dec 30 '16 at 18:06