关于有限循环群中Steinhaus三角形的Molluzzo问题
On a problem of Molluzzo concerning Steinhaus triangles in finite cyclic groups
摘要：
设$X$是一个长度为$m\geq 1$的有限序列，其长度为$\mathbb{Z}/n\mathbb{Z}$ . $X$的\textit{derived sequence} $\partial X$是通过将$X$的连续项成对相加而获得的长度为$m-1$的序列。$X$的迭代导出序列的集合（直到长度达到1）确定三角形，即\textit{Steinhaus triangle $ΔX$ generated by the sequence $X$}。我们说$X$是\textit{balanced}，如果它的Steinhaus三角形$ΔX$包含$\mathbb{Z}/n\mathbb{Z}$的每个元素具有相同的重数。$m$是$\mathbb{Z}/n\mathbb{Z}$中平衡序列的长度的一个明显的必要条件是$n$整除二项式系数$\binom{m+1}{2}$。这是一个开放的问题，以确定是否在$m$上这个条件也是充分的。这个问题由Hugo Steinhaus在1963年为$n=2$提出，并由John C. Molluzzo在1976年为$n\geq3$。到目前为止，只有1972年海科·哈伯斯（Heiko Harborth）解决了$n=2$一案。本文对$n=3^k$对所有$k\geq1$都成立的情况下的Molluzzo问题给出了肯定的回答.此外，对每一个奇数$n\geq3$，我们构造了$\mathbb{Z}/n\mathbb{Z}$中的无穷多个平衡序列.这是通过分析由算术级数生成的斯坦豪斯三角形来实现的。相反，对于任何$n$，即使具有$n\geq4$，也不知道在$\mathbb{Z}/n\mathbb{Z}$中是否存在无穷多个平衡序列。至于算术级数，甚至对于$n$，我们表明它们从来都不是平衡的，除了恰好8例发生在$n=2$和$n=6$。
英文摘要：
Let $X$ be a finite sequence of length $m\geq 1$ in $\mathbb{Z}/n\mathbb{Z}$. The \textit{derived sequence} $\partial X$ of $X$ is the sequence of length $m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of iterated derived sequences of $X$, until length 1 is reached, determines a triangle, the \textit{Steinhaus triangle $ΔX$ generated by the sequence $X$}. We say that $X$ is \textit{balanced} if its Steinhaus triangle $ΔX$ contains each element of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. An obvious necessary condition for $m$ to be the length of a balanced sequence in $\mathbb{Z}/n\mathbb{Z}$ is that $n$ divides the binomial coefficient $\binom{m+1}{2}$. It is an open problem to determine whether this condition on $m$ is also sufficient. This problem was posed by Hugo Steinhaus in 1963 for $n=2$ and generalized by John C. Molluzzo in 1976 for $n\geq3$. So far, only the case $n=2$ has been solved, by Heiko Harborth in 1972. In this paper, we answer positively Molluzzo's problem in the case $n=3^k$ for all $k\geq1$. Moreover, for every odd integer $n\geq3$, we construct infinitely many balanced sequences in $\mathbb{Z}/n\mathbb{Z}$. This is achieved by analysing the Steinhaus triangles generated by arithmetic progressions. In contrast, for any $n$ even with $n\geq4$, it is not known whether there exist infinitely many balanced sequences in $\mathbb{Z}/n\mathbb{Z}$. As for arithmetic progressions, still for $n$ even, we show that they are never balanced, except for exactly 8 cases occurring at $n=2$ and $n=6$.
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