MathOverflow

The Eilenberg-Watts theorem gives an equivalence of categories between the category of cocontinuous additive functors between module categories $\mathrm{Mod}_R\xrightarrow{} \mathrm{Mod}_S$ for two unital rings $R$ and $S$ and the category ${}_{R}\mathrm...

In a brand new village of $N$ people, villagers work together daily in shifts of $S \geq 2$ people, selected uniformly at random from all possible $K$ subsets of villagers. During the shifts, all villagers involved get to know each other.
What is the expected time taken until at least $M$ villagers mutually know each other?...

If $E$ is a $k$-dimensional complex vector bundle over an $n$-dimensional CW-complex $X$, such that $n\leq 2k-1$, then $E$ has a one-dimensional trivial subbundle (or equivalently, a global non-vanishing section). This is Proposition 1.1 in Chapter ...

I hope this is a suitable MO question. As a project I've been trying to simplify down an inclusion-exclusion style combinatorial expression:
\begin{equation}f(n) = \sum\limits_{k=1}^{\Omega(n)} \frac{(-1)^{(k+1)} H(n, k)}{k+1}\end{equation}
where $H(n, k)$ is the number of ordered ways to express $n$ as a product of $k$ integer multipliers greater than 1. Let $n$ have the following factorization: $n=p_1^{\alpha_1}p_2^{\alpha_2}...p_T^{\alpha_T}$ and $\Omega(n)=\alpha_1 + \alpha_2 + ......