On the number of distinct <i>k</i>-decks: Enumeration and bounds

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