What does ${N\choose\mu}$ mean where $\mu = (\mu_1, ... , \mu_n)$ be the vector of nonnegative integers for which $\sum_{i=1}^n \mu_i = N$? 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? 

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