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Abstract: In this paper, we partially solve an open problem, due to J.C. Molluzzo in1976, on the existence of balanced Steinhaus triangles modulo a positiveinteger $n$, 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 automatongenerating the Pascal triangle modulo $n$, which contains infinitely manybalanced Steinhaus triangles. This orbit, in $\mathbb{Z}-n\mathbb{Z}$, isobtained from an integer sequence called the universal sequence. We show thatthere exist balanced Steinhaus triangles for at least $2-3$ of the admissiblesizes, in the case where $n$ is an odd prime power. Other balanced Steinhausfigures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascaltrapezoids or lozenges, also appear in the orbit of the universal sequencemodulo $n$ odd. We prove the existence of balanced generalized Pascal trianglesfor at least $2-3$ of the admissible sizes, in the case where $n$ is an oddprime power, and the existence of balanced lozenges for all admissible sizes,in the case where $n$ is a square-free odd number.



Author: Jonathan Chappelon LMPA

Source: https://arxiv.org/



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