Combinatorial problems from an analytical point of view

这个项目致力于在分析和组合学的交界处出现新的正性性质。完全正性与机械系统的研究有关，并在组合学、统计学和复杂分析中得到应用。Hankel-全正性是Stieltjes矩序列的一个特征性质，正如Gantmacher和Krein所证明的那样。Flajolet在1980年的论文中介绍了Stieltjes连分式在组合学中的一个重要应用：他用组合的方法解释了相关幂函数的系数。艾伦·索卡尔，这个项目的潜在主持人，进一步证明了许多重要的组合多项式序列是系数方向的Hankel-全正的。在某些情况下，严格的证明来自于与连分式的联系。在其他情况下，核查需要更复杂的工具，仍然是一个悬而未决的问题。解决这个问题是本课题的第一个目标。本课题的第二个目标是研究一类特殊的形式幂函数级数的正性。这类问题出现在连通图的计数和粒子的相互作用等问题中。从分析的观点来看，目标在于证明这类整函数的零点的几个经验观察性质。这将包括对艾伦·索卡尔关于零点的简单性的猜想及其泰勒系数关于参数的正性的验证。

This project is devoted to novel positivity properties emerging at the interface between analysis and combinatorics.Total positivity arose in connection with studies of mechanical systems and has applications to combinatorics, statistics, and complex analysis. Hankel-total positivity is a characteristic property of Stieltjes moment sequences, as was shown by Gantmacher and Krein. Flajolet's paper of 1980 introduced an important application of the Stieltjes continued fractions to combinatorics: he interpreted combinatorially the coefficients of the associated power series. Alan Sokal, the prospective host on this project, further identified many important sequences of combinatorial polynomials as coefficientwise Hankel-totally positive. In certain cases, a rigorous proof follows from the connection to continued fractions. In other cases, the verification needs more sophisticated tools and remains an open problem. Dealing with this problem is the first goal of this project.The second goal of this project is the study of positivity for a special class of formal power series. This class arises in such problems as the enumeration of connected graphs and the interaction of particles. From the analytical viewpoint, the goal consists in proving several empirically observed properties of zeros of such entire functions. This will include verification of Alan Sokal's conjectures on the simplicity of zeros and on the positivity of their Taylor coefficients with respect to the parameter.