Counting strictly increasing and non-decreasing functions $f$ is non-decreasing if $x \lt y$ implies $f(x) \leq f(y)$ and increasing if $x < y$ implies $f(x) < f(y)$. 

  
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*How many $f: [a]\to [b]$ are nondecreasing?
  
*How many $f: [a] \to [b]$ are strictly increasing?

Where $[a]=\{1,2\ldots a\}$ and $[b]=\{1,2\ldots b\}$

For the first part if you know how many of each of the $i\in [k]$ there are the on image of a non decreasing function, then only that function can have that property. Therefore the number of non deacreasing functions is on $1-1$ correspondance with the number of the solution.
$$x_1+x_2+\ldots x_k=n$$
An example of that correspondance
For instance for $n=5$ and $k=7$, if you know that in the image of $f$ there are two $1$,
two $3$, and one $7$. Then the only function with that property (which is non-decreasing) is:
\begin{align}
f(1)=1\\
f(2)=1\\
f(3)=2\\
f(4)=2\\
f(5)=7
\end{align}
