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


$$
\sum\_\lambda \frac{s\_\lambda(x)}{ s\_\lambda(y)}=\sum\_{k=0}^\infty \frac{x^k}{y^k}=\frac{1}{1-x/y}.
 $$



Sum of the ratios of Schur functions
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.


