A universal sequence of integers generating balanced Steinhaus figures modulo an odd number - Mathematics > Combinatorics

A universal sequence of integers generating balanced Steinhaus figures modulo an odd number - Mathematics > Combinatorics - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

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.

Autor: Jonathan Chappelon LMPA

Fuente: https://arxiv.org/