Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters

Data stored in synthetic DNA is retrieved by shotgun sequencing, which returns short subsequences rather than the stored word itself. A natural abstraction of this readout is the $k$-deck of a word: the vector recording how often each word of length $k$ occurs as a subsequence. Two stored words are distinguishable from their readouts exactly when their $k$-decks differ, so the number $D_{q,k}(n)$ of distinct $k$-decks of words of length $n$ over an alphabet of size $q$ measures what a length-$k$ readout retains. We analyse the degrees of freedom remaining in a $k$-deck once all shorter decks are fixed. Within each class of words having prescribed letter multiplicities, the length-$k$ entries are confined to an affine subspace whose dimension is exactly the number of Lyndon words with the same multiplicities, which we give in closed form as a Möbius sum. Writing $L_q(j)$ for the number of Lyndon words of length $j$ over an alphabet of size $q$, we deduce the improved upper bound $$D_{q,k}(n)=O\text{\hspace{0.17em}⁣}\left(n^{E_q(k)}\right),E_q(k)=\sum _{j=1}^{k}j{L}_q(j)-1.$$ In the case of a binary alphabet this bound satisfies $D_{2,k}(n)=O\text{\hspace{0.17em}⁣}\left(n^{4\cdot 2^{k-1}}\right)$. We then prove matching lower bounds in the first two nontrivial cases: $D_{q,2}(n)=\mathrm{\Theta }\text{\hspace{0.17em}⁣}\left(n^{q^2-1}\right)$ for every alphabet size $q$, and $D_{2,3}(n)=\mathrm{\Theta }(n^9)$ for the binary alphabet. The latter confirms, for $q=2$ and $k=3$, our conjecture that the upper bound has the correct degree for every $q$ and $k$.