= Character of a Young permutation module
{title2=$\chi_{M^\lambda}(\mu)=[x^\lambda]\prod_q(\sum_j x_j^q)^{m_q}$}
A <tabloid> is fixed by a permutation exactly when each of its cycles lies wholly within one row. Assigning the distinct length-$q$ cycles to rows gives the displayed coefficient. More explicitly, sum $\prod_q m_q!/\prod_j a_{qj}!$ over nonnegative integers $a_{qj}$ with $\sum_j a_{qj}=m_q$ and $\sum_q q a_{qj}=\lambda_j$. Equal-length rows remain distinguished. Thus the <character> counts fixed ordered row sets, rather than unordered set partitions.
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