how many numbers in (not-strictly) descending order of string length k Given K numbers $n_1, \, \ldots \,, n_k$ s.t.  $0 \leq n_i \leq b$
How many ways are there to write the numbers $n_1$ to $n_k$ such that they are descending. (Descending here is used in the analysis sense)
Given $k=4$ and $b=4$ we could say that:
$0000$ is descending
$3321$ is descending
yet $2321$ is not.
I know this is a combinatorics problem and I have encountered solutions for strictly descending, but not descending.
Thank you in advance for any help.

For $n_1$, we can choose any number between $0$ and $b$ (inclusive).
For $n_2$, we can choose any number between $0$ and $n_1$ (inclusive), and so forth.
This structure suggests dynamic programming.
Let $f(a,b,k)$ be the number of ways to choose a descending a sequence of length $k$ with numbers between $a$ and $b$ (inclusive).
By the above argument,
$$
f(a,b,k)=\begin{cases}
1 & \text{if }k=0\\
f(a,a,k-1)+f(a,a+1,k-1)\cdots+f(a,b,k-1) & \text{if }k\geq1.
\end{cases}
$$
Next, note that $f(a,b,k) = f(0,b-a,k)$ and hence it is sufficient to check the case of $a=0$.
Lastly, you can verify that $f(0,b,k) = \binom{b+k}{b}$ by induction.
