(I'm not sure if this question belongs here, but stack overflow doesn't work with LaTeX so here it is)
An inefficient solution to a scheduling problem similar to the 2018 Google Hashcode problem (unless you already know of this problem I wouldn't recommend reading it for this, it will likely overcomplicated this).
Given $n$ tasks and $m$ agents and that 1 agent can do 1 task at a time, you want to assign agents to tasks such that all tasks are completed in the minimum amount of time.
An extremely expensive algorithm is an application of depth-first-search, constructing a tree of every possible assignment and from that finding the shortest path (minimum time required).
Considering a problem with 2 agents and 3 tasks, for each node in each layer you have so many outgoing edges (considering at layer $i$ there are $n+1-i$ tasks remaining):
- $mn = 2 \cdot 3 = 6$
- $m(n-1)=2 \cdot 2 = 4$
- $m(n-2)=2 \cdot 1 = 2$
As such the number of outgoing edges you have in each layer are:
- $mn=\frac{n!}{(n-1)!}m^1=2 \cdot 3 = 6$
- $mn\cdot m(n-1)=\frac{n!}{(n-2)!}m^2=(2 \cdot 3)(2 \cdot 2) = 24$
- $mn\cdot m(n-1)\cdot m(n-2)=\frac{n!}{(n-3)!}m^3=(2 \cdot 3)(2 \cdot 2)(2 \cdot 1) = 48$
Therefore one could give the total number of edges in the network as $t$ where:
$t=\sum\limits_{i=1}^{n} \frac{n!}{(n-i)!}m^i$
At this point I am stuck and do not know how I could simplify it further.
$$t_{m,n}=\sum\limits_{i=1}^{n} \frac{n!}{(n-i)!}m^i=e^{\frac{1}{m}}\, m^n\, \Gamma \left(n+1,\frac{1}{m}\right)-1$$ where appears the incompleta gamma function.
$$\left( \begin{array}{ccc} m & n & t_{m,n} \\ 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 15 \\ 1 & 4 & 64 \\ 1 & 5 & 325\\ 2 & 1 & 2 \\ 2 & 2 & 12 \\ 2 & 3 & 78 \\ 2 & 4 & 632 \\ 2 & 5 & 6330\\ 3 & 1 & 3 \\ 3 & 2 & 24 \\ 3 & 3 & 225 \\ 3 & 4 & 2712 \\ 3 & 5 & 40695 \\ 4 & 1 & 4 \\ 4 & 2 & 40 \\ 4 & 3 & 492 \\ 4 & 4 & 7888 \\ 4 & 5 & 157780 \\ 5 & 1 & 5 \\ 5 & 2 & 60 \\ 5 & 3 & 915 \\ 5 & 4 & 18320 \\ 5 & 5 & 458025 \end{array} \right)$$