= Solution
\b[There is exactly one solution: $X_i=i$ for $i=1,\ldots,n$.] Strict increase makes all $n$ values distinct. Since all lie in the $n$-element domain $\{1,\ldots,n\}$, they must use every value, and their increasing order fixes their positions.
Equivalently, $X_i\ge i$ by propagation from the lower endpoint, and $X_i\le i$ by propagating backwards from $X_n\le n$. For general $m\ge n$, choosing any $n$-element subset determines one increasing solution, giving the <binomial coefficient> $\binom mn$; the case $m=n$ gives $1$.
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