On the number of distinct-decks: Enumeration and bounds

Johan Chrisnata, Han Mao Kiah, Sankeerth Rao Karingula, Alexander Vardy, Eitan Yaakobi, Hanwen Yao

Abstract: The $ k $-deck of a sequence is defined as the multiset of all its subsequences of length $ k $. Let $ D_k(n) $ denote the number of distinct $ k $-decks for binary sequences of length $ n $. For binary alphabet, we determine the exact value of $ D_k(n) $ for small values of $ k $ and $ n $, and provide asymptotic estimates of $ D_k(n) $ when $ k $ is fixed.Specifically, for fixed $ k $, we introduce a trellis-based method to compute $ D_k(n) $ in time polynomial in $ n $. We then compute $ D_k(n) $ for $ k \in \{3,4,5,6\} $ and $ k \leqslant n \leqslant 30 $. We also improve the asymptotic upper bound on $ D_k(n) $, and provide a lower bound thereupon. In particular, for binary alphabet, we show that $ D_k(n) = O\bigl(n^{(k-1)2^{k-1}+1}\bigr) $ and $ D_k(n) = \Omega(n^k) $. For $ k = 3 $, we moreover show that $ D_3(n) = \Omega(n^6) $ while the upper bound on $ D_3(n) $ is $ O(n^9) $. Keywords: k-deck problem, subsequence reconstruction, trellis. Mathematics Subject Classification: Primary: 68R05, 68R15. Citation: \begin{equation} \\ \end{equation}