Strings with maximally many distinct subsequences and substrings.(English)Zbl 1034.68067

Electron. J. Comb. 11, No. 1, Research paper R8, 10 p. (2004); printed version J. Comb.11, No. 1 (2004).
Summary: A natural problem in extremal combinatorics is to maximize the number of distinct subsequences for any length-$$n$$ string over a finite alphabet $$\Sigma$$; this value grows exponentially, but slower than $$2^n$$. We use the probabilistic method to determine the maximizing string, which is a cyclically repeating string. The number of distinct subsequences is exactly enumerated by a generating function, from which we also derive asymptotic estimates. For the alphabet $$\Sigma={1,2}, (1,2,1,2,...)$$ has the maximum number of distinct subsequences, namely Fib $$(n+3)-1\sim ((1+\sqrt5)/2)^{n+3} / \sqrt5$$.
We also consider the same problem with substrings in lieu of subsequences. Here, we show that an appropriately truncated de Bruijn word attains the maximum. For both problems, we compare the performance of random strings with that of the optimal ones.

MSC:

 68R15 Combinatorics on words 05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.) 05A15 Exact enumeration problems, generating functions 05A16 Asymptotic enumeration

Keywords:

extremal combinatorics
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