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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

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

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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.

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  • $\begingroup$ Thank you! I also think that to get some nice expressions on have to specify something. What about arbitrary $x$ and some fixed $y$? $\endgroup$
    – Sasha
    Commented Dec 30, 2016 at 18:02
  • $\begingroup$ I think, exercises 7.30 and 7.32 in Volume 2 are not directly related to my question. $\endgroup$
    – Sasha
    Commented Dec 30, 2016 at 18:05
  • $\begingroup$ This (with $x$ free), still, might be expecting much. I could be wrong. Yes, the exercises are only quotients and then you sum. $\endgroup$
    – T. Amdeberhan
    Commented Dec 30, 2016 at 18:06

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