When studying representation theory, special functions or various other topics one is very likely to encounter the following identity at some point:
$$\det \left(\frac{1}{x \_i+y \_j}\right) \_{1\le i,j \le n}=\frac{\prod \_{1\le i < j\le n} (x \_j-x \_i)(y \_j-y \_i)}{\prod \_{i,j=1}^n (x \_i+y \_j)}$$
This goes under the name of Cauchy's determinant identity and has various generalizations and analogous statements. There is also a lot of different proofs using either analysis or algebra. In my case I have always seen it introduced (or motivated) as an identity that plays an important role in combinatorics, but I realized that I haven't really seen this identity in a combinatorial context before. In this question I'm asking for a combinatorial interpretation of the above identity. A bonus to someone who can give such an interpretation to Borchardt's variation:
$$\det \left(\frac{1}{(x \_i+y \_j)^2}\right) \_{1\le i,j \le n}=\frac{\prod \_{1\le i < j\le n} (x \_j-x \_i)(y \_j-y \_i)}{\prod \_{i,j=1}^n (x \_i+y \_j)} \cdot \text{per}\left(\frac{1}{x \_i +y \_j}\right) \_{1 \le i,j \le n}$$
(This seems a little too ambitious though, and I would be happy to accept an answer of just the first question)



What role does Cauchy's determinant identity play in combinatorics?
See also pp. 397--398 of *Enumerative Combinatorics*, vol. 2. Cauchy's determinant is given in a slightly different but equivalent form. It is explained there that the evaluation of the determinant is equivalent to the fundamental identity $\prod(1-x\_iy\_j)^{-1} =\sum\_\lambda s\_\lambda(x)s\_\lambda(y)$ in the theory of symmetric functions.


