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# Tournament Sequences and Meeussen Sequences

Cook, Matthew and Kleber, Michael (2000) Tournament Sequences and Meeussen Sequences. Electronic Journal of Combinatorics, 7 (1). R44. ISSN 1077-8926. http://resolver.caltech.edu/CaltechAUTHORS:COOejc00

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## Abstract

A tournament sequence is an increasing sequence of positive integers (t1; t2; : : :) such that t1 = 1 and ti+1 <= 2ti. A Meeussen sequence is an increasing sequence of positive integers (m1;m2; : : :) such that m1 = 1, every nonnegative integer is the sum of a subset of the {mi}, and each integer mi − 1 is the sum of a unique such subset. We show that these two properties are isomorphic. That is, we present a bijection between tournament and Meeussen sequences which respects the natural tree structure on each set. We also present an efficient technique for counting the number of tournament sequences of length n, and discuss the asymptotic growth of this number. The counting technique we introduce is suitable for application to other well-behaved counting problems of the same sort where a closed form or generating function cannot be found.

Item Type: Article Submitted: March 22, 2000; Accepted: September 5, 2000 Partially supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. The authors would like to acknowledge W. Meeussen for suggesting the question, and N. J. A. Sloane for his Encyclopedia of Integer Sequences [9], which led us to notice the coincidence. Thanks also to J. Polito for useful conversations, J. Shallit for helpful comments on an earlier draft of this work, and D. Knuth for excellent suggestions about the asymptotics questions. CaltechAUTHORS:COOejc00 http://resolver.caltech.edu/CaltechAUTHORS:COOejc00 http://www.combinatorics.org/Volume_7/Abstracts/v7i1r44.html No commercial reproduction, distribution, display or performance rights in this work are provided. 768 CaltechAUTHORS Archive Administrator 29 Sep 2005 26 Dec 2012 08:41

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