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On symmetric and antisymmetric balanced binary sequences. (English) Zbl 1083.94006

Summary: Let \(X=(x_1,\dots,x_n)\) be a finite binary sequence of length \(n\), i.e., \(x_i=\pm 1\) for all \(i\). The derived sequence of \( X\) is the binary sequence \(\partial X=(x_1x_2,\dots,x_{n-1}x_n)\) of length \(n-1\), and the derived triangle of \(X\) is the collection \(\Delta X\) of all derived sequences \(\partial^iX\) for \(0\leq i\leq n-1\). We say that \(X\) is balanced if its derived triangle \(\Delta X\) contains as many \(+1\)’s as \(-1\)’s. This concept was introduced by H. Steinhaus [One hundred problems in elementary mathematics. Popular Lectures in Mathematics 7. Oxford etc.: Pergamon Press (1963; Zbl 0116.24102)]. It is known that balanced binary sequences occur in every length \(n\equiv 0\) or \(3 \bmod 4\), and in none other. In this paper, we solve the problem of determining all possible lengths of symmetric and of antisymmetric balanced binary sequences. We prove that (1) there exists a symmetric balanced binary sequence of length \(n\) if and only if \(n\equiv 0,3\) or \(7 \bmod 8\), and (2) there exists an antisymmetric balanced binary sequence of length \(n\) if and only if \(n \equiv 4\bmod 8\).

MSC:

11B75 Other combinatorial number theory
05A05 Permutations, words, matrices

Citations:

Zbl 0116.24102
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