TITLE: Sum of the ratios of Schur functions
QUESTION [7 upvotes]: 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
$$
\sum_\lambda \frac{s_\lambda(x)}{ s_\lambda(y)}=\sum_{k=0}^\infty \frac{x^k}{y^k}=\frac{1}{1-x/y}.
 $$

REPLY [3 votes]: 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.