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Tournament sequences and Meeussen sequences. (English) Zbl 0973.11028
Electron. J. Comb. 7, No. 1, Research paper R44, 16 p. (2000); printed version J. Comb. 7, No. 2 (2000).
A tournament sequence [see P. Capell and T. V. Narayana, Can. Math. Bull. 13, 105-109 (1970; Zbl 0225.60006)] is an increasing sequence of positive integers \((t_1, t_2,\dots)\) such that \(t_1+1\) and \(t_{i+1}\leq 2t_i\). The authors define a Meeussen sequence to be an increasing sequence of positive integers \((m_1, m_2,\dots)\) such that \(m_1=1\), every non-negative integer is the sum of a subset of the \(\{m_i\}\) and each integer \(m_i-1\) is the sum of a unique such subset. They then show that Meeussen sequences are precisely the tournament sequences, by exhibiting a bijection between the two sets of sequences which respects the natural tree structure on each set. They also present an efficient way of counting these sequences, and discuss the asymptotic growth of the number of sequences.

MSC:
11B83 Special sequences and polynomials
05A15 Exact enumeration problems, generating functions
05A16 Asymptotic enumeration
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