Complete homogeneous symmetric polynomials with repeating variables

We consider polynomials of the form ${\mathrm{h}}_m(y_1^{[{\mathrm{\varkappa }}_1]},\dots ,y_n^{[{\mathrm{\varkappa }}_n]})$, where ${\mathrm{h}}_m$ is the complete homogeneous polynomial of degree $m$ and $y_j^{[{\mathrm{\varkappa }}_j]}$ denotes $y_j$ repeated ${\mathrm{\varkappa }}_j$ times. Using the decomposition of the generating function into partial fractions we represent such polynomials in the form $${\mathrm{h}}_m(y_1^{[{\mathrm{\varkappa }}_1]},\dots ,y_n^{[{\mathrm{\varkappa }}_n]})=\sum _{j=1}^n\sum _{r=1}^{{\mathrm{\varkappa }}_j}\left(\genfrac{}{}{0px}{}{r+m-1}{r-1}\right)A_{y,\mathrm{\varkappa },j,r}{y}_j^m,$$ where $A_{y,\mathrm{\varkappa },j,r}$ are some coefficients that do not depend on $m$. We also provide an alternative proof using the inverse of the confluent Vandermonde matrix.