Most frequent subsequences in a word

We prove that every $n$-letter word over $k$-letter alphabet contains some word as a subsequence in at least $k^{n\mathrm{/}4k(1+o(1))}$ many ways, and that this is sharp as $k\to \mathrm{\infty }$. For fixed $k$, we show that the analogous number deviates from ${\mu }_k^n$, for some constant ${\mu }_k$, by a factor of at most $n$.