arXiv 2102.009442021-02math.COmath.NT

关于中心二项式系数的divisor

On a divisor of the central binomial coefficient

Matthew Just, Maxwell Schneider

备注：: 16 pages, 11 figures, 1 table

摘要：

众所周知，对于所有$n\geq1$，数字$n+ 1$是中心二项式系数${2n\choose n}$的约数。由于$n$第n个中心二项式系数等于从$(0,0)$到$(n,n)$向北或向东单位步长的格路径的数量，因此自然的问题是是否存在将这些路径划分成$n+ 1$路径集合或$n+1$等数量路径集合的方法。Chung-Feller定理对这个问题给出了一个优雅的答案。我们提出并提供了一个类似的问题的答案$2n-1$，另一个因子的${2n\choose n}$。然后，我们表明我们的主要结果如下从一个更一般的观察二项式系数${n\choose k}$与$n$和$k$相对素数。讨论的情况下，$n$和$k$不是相对的素数也给出了，突出我们的方法的局限性。最后，我们来完整的循环，并给出一个新的解释加泰罗尼亚数字。

英文摘要：

It is well known that for all $n\geq1$ the number $n+ 1$ is a divisor of the central binomial coefficient ${2n\choose n}$. Since the $n$th central binomial coefficient equals the number of lattice paths from $(0,0)$ to $(n,n)$ by unit steps north or east, a natural question is whether there is a way to partition these paths into sets of $n+ 1$ paths or $n+1$ equinumerous sets of paths. The Chung-Feller theorem gives an elegant answer to this question. We pose and deliver an answer to the analogous question for $2n-1$, another divisor of ${2n\choose n}$. We then show our main result follows from a more general observation regarding binomial coefficients ${n\choose k}$ with $n$ and $k$ relatively prime. A discussion of the case where $n$ and $k$ are not relatively prime is also given, highlighting the limitations of our methods. Finally, we come full circle and give a novel interpretation of the Catalan numbers.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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