On the number of distinct k-decks: Enumeration and bounds

Abstract

<p style='text-indent:20px;'>The <i><inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-deck</i> of a sequence is defined as the multiset of all its subsequences of length <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id="M4">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> denote the number of distinct <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula>-decks for binary sequences of length <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>. For binary alphabet, we determine the exact value of <inline-formula><tex-math id="M7">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> for small values of <inline-formula><tex-math id="M8">\begin{document}$ k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ n $\end{document}</tex-math></inline-formula>, and provide asymptotic estimates of <inline-formula><tex-math id="M10">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M11">\begin{document}$ k $\end{document}</tex-math></inline-formula> is fixed.</p><p style='text-indent:20px;'>Specifically, for fixed <inline-formula><tex-math id="M12">\begin{document}$ k $\end{document}</tex-math></inline-formula>, we introduce a trellis-based method to compute <inline-formula><tex-math id="M13">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> in time polynomial in <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math></inline-formula>. We then compute <inline-formula><tex-math id="M15">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M16">\begin{document}$ k \in \{3,4,5,6\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ k \leqslant n \leqslant 30 $\end{document}</tex-math></inline-formula>. We also improve the asymptotic upper bound on <inline-formula><tex-math id="M18">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula>, and provide a lower bound thereupon. In particular, for binary alphabet, we show that <inline-formula><tex-math id="M19">\begin{document}$ D_k(n) = O\bigl(n^{(k-1)2^{k-1}+1}\bigr) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}$ D_k(n) = \Omega(n^k) $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M21">\begin{document}$ k = 3 $\end{document}</tex-math></inline-formula>, we moreover show that <inline-formula><tex-math id="M22">\begin{document}$ D_3(n) = \Omega(n^6) $\end{document}</tex-math></inline-formula> while the upper bound on <inline-formula><tex-math id="M23">\begin{document}$ D_3(n) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M24">\begin{document}$ O(n^9) $\end{document}</tex-math></inline-formula>.</p>