Combinatorial problems from an analytical point of view

点击翻译按钮获取中文摘要

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.