ON THE NUMBER OF DISTINCT k-DECKS: ENUMERATION AND BOUNDS

The k-deck of a sequence is defined as the multiset of all its sub-sequences 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 ∈ {3, 4, 5, 6} and k ⩽ n ⩽ 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⁽n^{(k−1)2k−1} +1⁾ and D_k (n) = Ω(n^k). For k = 3, we moreover show that D₃ (n) = Ω(n⁶) while the upper bound on D₃ (n) is O(n⁹).