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Let's say we have the expression $$\sum_{k_1=1}^\infty\sum_{k_2=1}^\infty\sum_{k_3=1}^\infty\sum_{k_4=1}^\infty\sum_{k_5=1}^\infty x^{k_1+k_2+k_3+k_4+k_5} f(k_1,k_2+k_3,k_4+k_5)$$ where $f(a,b,c)$ is a symmetric function in its three variables.

It is easy to see that it is a power expansion and $x^5$ is the lowest power, this terms is $$x^5f(1,2,2)$$ In the same way we can get the coefficients for $x^6$ and $x^7$ in the power expansion $$=x^5f(1,2,2)+x^6[f(2,2,2)+4f(1,2,3)]+x^7[6f(1,2,4)+4f(1,3,3)+5f(2,2,3)]\dots$$

I want a closed form expresion for the coefficients 1, 1, 4, 6, 4, 5, ...

We can see the numerical arguments of the function as a partition (of the exponent). Furthermore, we can see the argument of the function in the left-hand-side as a kind of partitions with $k$'s. In this case it will be (with my notation) $$(k_1,k_2+k_3,k_4+k_5)=k_{(1,2,2)}$$ So, the expression with its expansion can be written as $$\sum x^{\sum_{j=1}^5 k_j} f(k_{(1,2,2)})=\sum_{m=5}^\infty x^m\sum_{\mu_m^{(3)}} C^{\mu^{(3)}_m}_{(1,2,2)} f(\mu^{(3)}_m)$$

I think this has to do with the "stars and bars" construction. But I cannot find an expression for the $C$.

This example was for a partition $\mu_5^{(3)}=(1,2,2)$ of $n=5$ with cycle number $c=3$. (note that $c$ is always the number of variables).

For $n$ $k$'s and $c$ variables $$\sum_{all \ k's} x^{\sum_{j=1}^n k_j} f(k_{\mu^{(c)}_n})=\sum_{m=n}^\infty x^m\sum_{\mu_m^{(c)}} C^{\mu^{(c)}_m}_{\mu^{(c)}_n} f(\mu^{(c)}_m)$$

What is the closed form expression for $C^{\mu^{(c)}_m}_{\mu^{(c)}_n}$?

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