arXiv 1206.35912012-06-15math.CO

森林和循环的斯特林数

Stirling numbers of forests and cycles

Do Trong Thanh, David Galvin

备注：: 17 pages

摘要：

对于图$G$和正整数$k$，{ \em图Stirling数} $S(G,k)$是$G$的顶点集划分成$k$非空独立集的个数。等价地，它是$G$的正确着色的数量，这些着色完全使用$k$颜色，如果两种着色仅在颜色名称上不同，则可以识别两种着色。如果$G$是$n$顶点上的空图，则$S(G,k)$简化为$S(n,k)$，即熟悉的第二类Stirling数。在本文中，我们首先考虑森林的Stirling数。我们证明了，如果$(F^{c(n)}_n)_{n\geq 0}$是具有$n$顶点和$c(n)=o(\sqrt{n/\log n})$分量的$F^{c(n)}_n$的森林序列，并且如果$X^{c(n)}_n$是以概率正比于$S(F^{c(n)}_n,k)$的概率取值$k$的随机变量，（即，$X^{c(n)}_n$是$F^{c(n)}_n$的均匀选择的划分为非空独立集合的类的数目），则$X^{c(n)}_n$是渐近正态的，意味着适当地归一化它在分布上趋向于标准正态。这推广了Harper关于普通Stirling数的一个开创性结果。一路上，我们给递归计算序列$(S(F^c_n,k))_{k \geq 0}$的生成函数，这些功能都有真正的零，并表现出三种不同的交错模式之间的零对连续的生成函数。接下来我们考虑循环的Stirling数。我们建立渐近正态性的类的数量在一个统一选择的分区的$C_n$（$n$顶点上的循环）到非空的独立集。我们给出了一个计算序列$(S(C_n,k))_{k \geq 0}$的生成函数的递推公式，并利用这个公式直接证明了以前只以非常间接的方式得出的对数周期结果。

英文摘要：

For a graph $G$ and a positive integer $k$, the {\em graphical Stirling number} $S(G,k)$ is the number of partitions of the vertex set of $G$ into $k$ non-empty independent sets. Equivalently it is the number of proper colorings of $G$ that use exactly $k$ colors, with two colorings identified if they differ only on the names of the colors. If $G$ is the empty graph on $n$ vertices then $S(G,k)$ reduces to $S(n,k)$, the familiar Stirling number of the second kind. In this note we first consider Stirling numbers of forests. We show that if $(F^{c(n)}_n)_{n\geq 0}$ is any sequence of forests with $F^{c(n)}_n$ having $n$ vertices and $c(n)=o(\sqrt{n/\log n})$ components, and if $X^{c(n)}_n$ is a random variable that takes value $k$ with probability proportional to $S(F^{c(n)}_n,k)$ (that is, $X^{c(n)}_n$ is the number of classes in a uniformly chosen partition of $F^{c(n)}_n$ into non-empty independent sets), then $X^{c(n)}_n$ is asymptotically normal, meaning that suitably normalized it tends in distribution to the standard normal. This generalizes a seminal result of Harper on the ordinary Stirling numbers. Along the way we give recurrences for calculating the generating functions of the sequences $(S(F^c_n,k))_{k \geq 0}$, show that these functions have all real zeroes, and exhibit three different interlacing patterns between the zeroes of pairs of consecutive generating functions. We next consider Stirling numbers of cycles. We establish asymptotic normality for the number of classes in a uniformly chosen partition of $C_n$ (the cycle on $n$ vertices) into non-empty independent sets. We give a recurrence for calculating the generating function of the sequence $(S(C_n,k))_{k \geq 0}$, and use this to give a direct proof of a log-concavity result that had previously only been arrived at in a very indirect way.

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