论文标题
渐近枚举和限制多运动的定律：次指数案例
Asymptotic Enumeration and Limit Laws for Multisets: the Subexponential Case
论文作者
论文摘要
对于给定的组合类$ \ MATHCAL {C} $，我们研究$ \ Mathcal {g} = \ Mathrm {mset}（\ Mathcal {c}）$满足多层结构的满足，即$ \ Mathcal {g} $的任何对象，都是$ \ nter的$ \ nse $ \ nse $ \ nsit $ \ nsit ogriped c。多重性。例如，$ \ mathrm {mset}（\ mathbb {n}）$ is（isomorphic to）正整数的数字分区类别，一个突出且经过充分研究的情况。多材构造自然出现在未标记对象的研究中，例如图形或与数字分区相关的各种结构。我们的主要结果确定了集合$ \ MATHCAL的渐近大小{g} _ {n，n，n} $，其中包含$ \ Mathcal {g} $中的所有多组，具有尺寸$ n $，并且由$ n $ objects组成，并由$ \ n $ n $ n $ n $ \ emph $ n Inf ginity and tort Inting Inting Inting Inting Inting Inting an $ \ MATHCAL {C} $受次指数增长的约束；这是组合应用中特别重要的环境。此外，我们研究了从$ \ Mathcal {g} _ {n，n} $中随机对象的组件分布，我们发现了一种现象，我们施洗了，我们将\ emph {emerth Condemation}：取走最大的组件，以及最小的对象的所有组件，我们都将与$ n，$ n，$ n，$ n，$ n，$ n cefty cefty n efty fefty felty fefty fcement $ n，$ n，$ n，n f。还检索了限制对象的分布。而且，令人惊讶的是，与标记对象的类似结果形成鲜明对比的是，这里的结果均匀地保持在$ n $中。
For a given combinatorial class $\mathcal{C}$ we study the class $\mathcal{G} = \mathrm{MSET}(\mathcal{C})$ satisfying the multiset construction, that is, any object in $\mathcal{G}$ is uniquely determined by a set of $\mathcal{C}$-objects paired with their multiplicities. For example, $\mathrm{MSET}(\mathbb{N})$ is (isomorphic to) the class of number partitions of positive integers, a prominent and well-studied case. The multiset construction appears naturally in the study of unlabelled objects, for example graphs or various structures related to number partitions. Our main result establishes the asymptotic size of the set $\mathcal{G}_{n,N}$ that contains all multisets in $\mathcal{G}$ having size $n$ and being comprised of $N$ objects from $\mathcal{C}$, as $n$ \emph{and} $N$ tend to infinity and when the counting sequence of $\mathcal{C}$ is governed by subexponential growth; this is a particularly important setting in combinatorial applications. Moreover, we study the component distribution of random objects from $\mathcal{G}_{n,N}$ and we discover a phenomenon that we baptise \emph{extreme condensation}: taking away the largest component as well as all the components of the smallest possible size, we are left with an object which converges in distribution as $n,N\to\infty$. The distribution of the limiting object is also retrieved. Moreover and rather surprisingly, in stark contrast to analogous results for labelled objects, the results here hold uniformly in $N$.
