Power sum polynomials

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Definition:The power sum polynomials $p_\la$ are defined by the formulas

$$p_\la = p_{\la_1}p_{\la_2}\dots,$$

$$p_r = m_\la, \qquad {\rm where}\;\; \la=(r,0,\dots,0)$$

Lemma 3   We have

$$p_{\la}=a_\la m_\la+\sum_{\mu>\la}b_{\la\mu}m_mu, \qquad b_{\la\mu}\in{\Bbb Z}_{\geq 0},$$

where $a_\la$ is a natural number. Therefore $\{p_{\la},\; \la=(\la_1\geq\dots\geq\la_n\geq 0)\}$ form a basis of symmetric polynomials.

Exercise:Proof the lemma. $\;\Box$