论文标题
与TUTTE相关的随机排名集的期望
Expectations of Tutte-related functions of random ranked sets with multiplicities
论文作者
论文摘要
使用两个模型，我们表明，通过限制或收缩的多种计数定义的随机变量的各种计数函数（例如，经典和算术矩阵）具有相应的多元tutte tutte多项式的期望。第一个模型是基于功夫（2010）卷积公式的概括，该公式从矩形延伸到具有多重性的排名集。该模型使我们能够计算许多熟悉的多项式的期望，例如色，流和ehrhart多项式，从而在随机图上概括了威尔士（1996）的经典结果。第二个模型旨在计算上述多项式通常不评估的不变式的期望，例如在阿贝尔谎言组排列中超出表面的相交的连接组件的数量，以及半开放式的晶格点的晶格数量。特别是，这两个模型都产生了对算术tutte多项式和$ g $ -tutte多项式的新概率解释。也将给出一个简单但似乎是晶格质量ehrhart多项式的新型卷积样公式。
Employing two models, we show that various counting functions of a random variable defined by restriction or contraction of a ranked set with multiplicity (e.g., classical and arithmetic matroids) have expectations given by the corresponding multivariate Tutte polynomial. The first model is based on a generalization of a convolution formula of Kung (2010), extending from matroids to ranked sets with multiplicities. This model enables us to compute the expectations of many familiar polynomials, such as the chromatic, flow and Ehrhart polynomials, generalizing the classical results of Welsh (1996) on random graphs. The second model is designed to compute the expectations of invariants that are generally not evaluations of the polynomials mentioned above, such as the number of connected components of an intersection of hypersurfaces in an abelian Lie group arrangement, and the number of lattice points in a half-open zonotope. In particular, both models yield new probabilistic interpretations of the arithmetic Tutte polynomial and $G$-Tutte polynomial. A simple, but seems to be new convolution-like formula for the Ehrhart polynomials of lattice zonotopes will also be given.
