Let $x_{i - j}\in\{1,2,\ldots, N\}$, where $1\leq i\leq N$ and $0\leq j\leq i$.

Based on such context, I am interested in an explicit formula for the numbers of configurations of $(x_{i},x_{i-1},\ldots,x_{0})$ which satisfy the following relation:
\begin{align*}
\max\{x_{i},x_{i-1} - 1, x_{i-2} - 2, \ldots, x_{0} - j\} = i
\end{align*}

I do not need a full answer.

I would like to know if anyone could give me some reference so that I could solve it by myself (or at least try).

Any suggestion such as books and articles are more than welcome.