Generating functions and relations between different bases

We have the generating functions

$$E(t):=\sum_{i=0}^n e_it^i=\prod_{i=1}^n(1+x_it),$$

$$H(t):=\sum_{i=0}^\infty h_it^i=\prod_{i=1}^n\frac1{1-x_it},$$

$$P(t):=\sum_{i=1}^\infty p_it^{i-1}=\sum_{i=1}^n \frac{x_i}{1-x_it}.$$

Note that the first equality is a version of Vieta theorem. We have the relations

$$H(t)E(-t)=1, \qquad H'(t)=P(t)H(T),$$

therefore

$$\sum_{i=0}^r(-1)^se_ih^{r-i}=0,$$ (1)

$$rh_r=\sum_{i=1}^rp_ih_{r-i}.\notag$$ (2)

Exercise:Use the relation $P(t)=(\log H(t))'$ to show that

$$h_r=\sum_{\vert\la\vert=r} \frac{p_\la}{z_\la},\qquad z_\la=\prod_{i=1}^n(i^{m_i}m_i!),$$

where $m_i$ is the number of parts of $\la$ equal $i$. $\;\Box$