Jim Pitman: Exponential formulas in probability and combinatorics

Abstract

The Bernoulli, Sterling and Eulerian numbers and polynomials were first defined in the early 1700s to help evaluate variously weighted sums of powers, and to approximate integrals. Subsequent work in the 1800s on probability and the calculus of finite differences, by Laplace, Poisson, Thiele and others, established close connections between these arrays of numbers and the moments and cumulants of the classical probability distributions, including both the discrete and continuous uniform, the binomial, geometric, negative binomial, Poisson, exponential and gamma distributions, as well some other distributions whose transforms involve trigonometric and hyperbolic functions and the Riemann zeta function. This talk will present the classical number arrays from a contemporary perspective,. The emphasis will be on the strong connections which tie these arrays both to the exponential formula of combinatorics, which enumerates labelled structures composed of primitive combinatorial components, and to the exponential formula of probability, that is the L\'evy-Khintchine of the infinitely divisible distribution of a sum of independent Gaussian and Poisson components. Various formulas for the classical number arrays then acquire double or multiple meanings, which connect polynomials enumerating various kinds of permutations to the moments of associated probability distributions, both discrete and continuous.