Tournament Sequences and Meeussen Sequences

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. MSC: 11B99 (Primary), 05A15, 05A16 (Secondary). Partially supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship 1 

• Type: Journal
• Year: 2000
• Where: COMBINATORICS
• Authors: Matthew Cook, Michael Kleber
• Keywords: COMBINATORICS 2000, Increasing Sequence, Positive Integers, Tournament Sequences