Asymptotic bounds for the number of closed and privileged words
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A word w has a border u if u is a non-empty proper prefix and suffix of u. A word w is said to be closed if |w| ≤ 1 or if w has a border that occurs exactly twice in w. A word w is said to be privileged if |w| ≤ 1 or if w has a privileged border that occurs exactly twice in w. Let C_k(n) (resp. P_k(n)) be the number of length-n closed (resp. privileged) words over a k-letter alphabet. In this paper we improve existing upper and lower bounds on C_k(n) and P_k(n). We prove that C_k(n) ∈Θ(k^n/n). Let log_k^∘ 0(n) = n and log_k^∘ j(n) = log_k(log_k^∘ j-1(n)) for j≥ 1. We also prove that for all j≥ 0 there exist constants N_j, c_j, and c_j' such that c_jk^n/nlog_k^∘ j(n)∏_i=1^jlog_k^∘ i(n)≤ P_k(n) ≤ c_j'k^n/n∏_i=1^jlog_k^∘ i(n) for all n>N_j.
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