= Solution
The <Young permutation module> $M^\lambda$ has a <basis> of <tabloids>, equivalently ordered row sets of sizes $\lambda_1,\ldots,\lambda_r$. A tabloid is fixed by $\sigma$ precisely when every cycle of $\sigma$ lies entirely in one row. Assigning a length-$q$ cycle to row $j$ contributes $x_j^q$. Distinct cycles can be assigned independently, so the <character> is
$$
\boxed{\chi_{M^\lambda}(\sigma)=[x_1^{\lambda_1}\cdots x_r^{\lambda_r}]\prod_{q=1}^n(x_1^q+\cdots+x_r^q)^{m_q}.}
$$
Padding to $n$ variables with zero row sizes gives exactly the same coefficient. Explicitly, the <character of a Young permutation module> is
$$
\sum_{\substack{a_{qj}\ge0\colon\ \sum_j a_{qj}=m_q\\ \sum_q q a_{qj}=\lambda_j}}
\prod_q\frac{m_q!}{\prod_j a_{qj}!}.
$$
Here $a_{qj}$ counts the length-$q$ cycles assigned to row $j$. Rows remain distinguished even when their sizes are equal, so there is no further division by permutations of equal rows.
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