生成模奇数平衡Steinhaus图的泛整数序列
A universal sequence of integers generating balanced Steinhaus figures modulo an odd number
摘要：
在本文中，我们部分地解决了一个公开的问题，由于J. C. Molluzzo在1976年提出了一个正整数$n$模的平衡Steinhaus三角形的存在性，即包含$\mathbb{Z}/n\mathbb{Z}$的所有元素的Steinhaus三角形具有相同的重数。对于每一个奇数$n$，通过线性元胞自动机生成杨辉三角形模$n$，在$\mathbb{Z}/n\mathbb{Z}$中建立一个轨道，其中包含无穷多个平衡Steinhaus三角形.在$\mathbb{Z}/n\mathbb{Z}$中，这个轨道是从一个称为通用序列的整数序列中得到的。证明了在$n$是奇素数幂的情况下，对于至少$2/3$的容许大小，存在平衡的Steinhaus三角形.其他平衡的斯坦豪斯图形，如斯坦豪斯三角形、广义杨辉三角形、帕斯卡三角形或菱形，也出现在通用序列模$n$奇数的轨道上。我们证明了平衡广义杨辉三角形的存在性至少$2/3$的容许大小，在$n$是一个奇素数的权力的情况下，和平衡菱形的存在性的所有容许的大小，在$n$是一个平方免费的奇数。
英文摘要：
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 called the universal sequence. 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, such 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 admissible sizes, in the case where $n$ is a square-free odd number.
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