论文标题
在将球放入有序的垃圾箱的组合上
On the Combinatorics of Placing Balls into Ordered Bins
论文作者
论文摘要
在本文中，我们使用枚举组合技术的技术来研究以下问题：我们计算将$ n $ balls分为非发行的方法的数量，订购的垃圾箱，以使最拥挤的垃圾箱完全具有$ K $ balls。我们找到了可能出现的三种不同情况的封闭表格：$ k> \ frac {n} {2} $，$ k = \ frac {n} {2} $，当存在$ j <k $时，$ n = 2k + j $。作为我们的证据的直接结果，我们找到了一个封闭形式，用于$ x_1 + x_1 + x_2 + \ dots + x _ {\ ell} = n $，最大可获得的$ \ {x_1，x_2，x_2，x_2，\ dots，x _ {\ ell}代数关系彼此。该问题被概括为找到一个公式，该公式列举了$ n，\ ell，k $的没有特定条件的方式的总数。随后，证明和解释了与此枚举有关的各种其他身份和估计。
In this paper, we use techniques of enumerative combinatorics to study the following problem: we count the number of ways to split $n$ balls into nonempty, ordered bins so that the most crowded bin has exactly $k$ balls. We find closed forms for three of the different cases that can arise: $k > \frac{n}{2}$, $k = \frac{n}{2}$, and when there exists $j < k$ such that $n = 2k + j$. As an immediate result of our proofs, we find a closed form for the number of positive integer solutions to $x_1 + x_2 + \dots + x_{\ell} = n$ with the attained maximum of $\{x_1, x_2, \dots, x_{\ell}\}$ being equal to $k$, when $n$ and $k$ have one of the aforementioned algebraic relationships to each other. The problem is generalized to find a formula that enumerates the total number of ways without specific conditions on $n, \ell, k$. Subsequently, various additional identities and estimates related to this enumeration are proven and interpreted.
