Mathematics – Combinatorics
Scientific paper
2009-08-18
Mathematics
Combinatorics
31 pages ; 10 figures
Scientific paper
In this paper, we partially solve an open problem, due to J. C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer $n$, that are Steinhaus triangles containing all the elements of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. For every odd number $n$, we build an orbit in $\mathbb{Z}/n\mathbb{Z}$, by the linear cellular automaton generating the Pascal triangle modulo $n$, which contains infinitely many balanced Steinhaus triangles. This orbit, in $\mathbb{Z}/n\mathbb{Z}$, is obtained from an integer sequence said to be universal. We show that there exist balanced Steinhaus triangles for at least 2/3 of the admissible sizes, in the case where $n$ is an odd prime power. Other balanced Steinhaus figures, as Steinhaus trapezoids, generalized Pascal triangles, Pascal trapezoids or lozenges, also appear in the orbit of the universal sequence modulo $n$ odd. We prove the existence of balanced generalized Pascal triangles for at least 2/3 of the admissible sizes, in the case where $n$ is an odd prime power, and the existence of balanced lozenges for all the admissible sizes, in the case where $n$ is a square-free odd number.
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