Experimental Mathematics

On a Problem of Steinhaus Concerning Binary Sequences

Shalom Eliahou and Delphine Hachez

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Abstract

A finite {\small $\pm 1$} sequence X yields a binary triangle {\small $\Delta X$} whose first row is X, and whose {\small $(k+1)$}th row is the sequence of pairwise products of consecutive entries of its kth row, for all {\small $k \geq 1$}. We say that X is balanced if its derived triangle {\small $\Delta X$} contains as many +1s as {\small $-$}1s. In 1963, Steinhaus asked whether there exist balanced binary sequences of every length {\small $n \equiv$} 0 or 3 mod 4. While this problem has been solved in the affirmative by Harborth in 1972, we present here a different solution. We do so by constructing strongly balanced binary sequences, i.e., binary sequences of length n all of whose initial segments of length {\small $n-4t$} are balanced, for {\small $0 \leq t \leq n/4$}. Our strongly balanced sequences do occur in every length {\small $n \equiv$} 0 or 3 mod 4. Moreover, we provide a complete classification of sufficiently long strongly balanced binary sequences.

Article information

Source
Experiment. Math. Volume 13, Issue 2 (2004), 215-230.

Dates
First available in Project Euclid: 20 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.em/1090350936

Mathematical Reviews number (MathSciNet)
MR2068895

Zentralblatt MATH identifier
1070.11008

Subjects
Primary: 05A05: Permutations, words, matrices 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
Secondary: 11B75: Other combinatorial number theory

Keywords
Steinhaus balanced binary sequence derived sequence derived triangle

Citation

Eliahou, Shalom; Hachez, Delphine. On a Problem of Steinhaus Concerning Binary Sequences. Experiment. Math. 13 (2004), no. 2, 215--230. https://projecteuclid.org/euclid.em/1090350936.


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