Thinking more about this, I can try to find the number of different systems of distinct representatives. Given $x$ and $n$ as above, let $t = \lfloor nx \rfloor$. Then, for $k=1,\ldots, (n-1)$, let $A_k = \{\lceil kx \rceil, \ldots, t-n+k \}$ and $A_n = \{t\}$. Phrased this way, the problem becomes counting the number of systems of distinct representatives.