2000/02/29 by Matthew Cook, Michael Kleber
Mathematics · #math.CO #msc:11B99 #msc:05A15 #msc:05A16
published as Electronic Journal of Combinatorics 7(1) (2000) #R44. http://www.combinatorics.org/Volume_7/Abstracts/v7i1r44.html · 16 pages, 1 figure. Minor changes only; final version as published in EJC
arxiv created 2000/12/01 · arxiv updated 2009/11/30
A "tournament sequence" is an increasing sequence of positive integers (t1,t2,...) such that t1=1 and ti+1 <= 2 ti. 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.