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When I've read Cohn et al. paper 2005, I've met a strange object in lemma 1.1 that looks like number of combinations: ${N\choose\mu}$ where $\mu = (\mu_1, ... , \mu_n)$ be the vector of nonnegative integers for which $\sum_{i=1}^n \mu_i = N$. After "reverse engineering" of lemma's proof I've got that ${N\choose\mu}=\prod_i \left({\mu_i \over N}\right)^{-\mu_i}$, but what does it mean?

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That notation usually denotes a multinomial coefficient: $$\binom{N}{\mu}=\binom{N}{\mu_1,\ldots,\mu_n}=\frac{N!}{\mu_1!\cdots\mu_n!}$$

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Note: This is written as an answer as I was unable to get the $\LaTeX$ code working as a comment, but really it should be a comment.

Just to add to @ZevChonoles comment, it is a generalistion of the binomial coefficient. Indeed \begin{align*}\left(\begin{array}{cc}N\\p\end{array}\right)=\left(\begin{array}{cc}N\\p,N-p\end{array}\right)=\left(\begin{array}{cc}N\\N-p\end{array}\right).\end{align*}

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