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\title{MCCME}
\author{mrkvmrmrkvmr }
\date{March 2020}

\begin{document}
\textbf{7.1}

\textbf{а} $\prod\limits_{i = 1}^{\infty} (1 - t^i) = \prod\limits_{i = 1}^{\infty}(1 + t^i + t^{2i} + \dots )$,
рассмотрим множитель при $t^{n}$ --- это окажется $p(n)$,
потому что степень $n$ может получится как раз таки $p(n)$ способами (просто раскрываем скобки).
\textbf{б} $\prod\limits_{i = 1}^{m} (1 - t^i) = \prod\limits_{i = 1}^{m}(1 + t^i + t^{2i} + \dots )$, рассмотрим множитель при $t^n$ --- это окажется $p(n, m)$, потому что степень $n$ может получится как раз таки $p(n, m)$ способами (просто раскрываем скобки).

\textbf{7.2}

\textbf{а} Из лекции известно
$$\sum\limits_{k = 1}^{\infty} p_k t^{k - 1}= p(t) = \frac{H(t)^\prime}{H(t)} = \frac{E(-t)^\prime}{E(-t)}$$
$\sum\limits_{k = 1}^{\infty} p_k t^{k - 1} \sum\limits_{k = 0}^{\infty} (-1)^k e_k(x_1, \dots , x_k) t^k = \sum\limits_{k = 0}^{\infty} (-1)^k k e_k(x_1, \dots , x_k) t^k$
\end{document}