论文标题
宽广的多集：渐近枚举
Expansive Multisets: Asymptotic Enumeration
论文作者
论文摘要
考虑一个非负序列$ c_n = h（n）\ cdot n^{α-1} \ cdotρ^{ - n} $，其中$ h $慢慢变化，$α> 0 $，$ 0 <ρ<1 $和$ n \ in \ mathbb {n} $。我们研究了$ g（x，y）= \ prod_ {k \ ge1}（1-x^ky）^{ - c_k} $的系数，这是组合对象的多键构建的双变量生成系列。 By a powerful blend of probabilistic methods based on the Boltzmann model and analytic techniques exploiting the well-known saddle-point method we determine the number of multisets of total size $n$ with $N$ components, that is, the coefficient of $x^ny^N$ in $G(x,y)$, asymptotically as $n\to\infty$ and for all ranges of $N$.我们的结果揭示了计数公式的结构中的相变，该结构取决于$ n/n $的比率，并且证明了从双变量局部限制到单变量的原型段落。
Consider a non-negative sequence $c_n = h(n) \cdot n^{α-1} \cdot ρ^{-n}$, where $h$ is slowly varying, $α>0$, $0<ρ<1$ and $n\in\mathbb{N}$. We investigate the coefficients of $G(x,y) = \prod_{k\ge1}(1-x^ky)^{-c_k}$, which is the bivariate generating series of the multiset construction of combinatorial objects. By a powerful blend of probabilistic methods based on the Boltzmann model and analytic techniques exploiting the well-known saddle-point method we determine the number of multisets of total size $n$ with $N$ components, that is, the coefficient of $x^ny^N$ in $G(x,y)$, asymptotically as $n\to\infty$ and for all ranges of $N$. Our results reveal a phase transition in the structure of the counting formula that depends on the ratio $n/N$ and that demonstrates a prototypical passage from a bivariate local limit to an univariate one.
