- Detailed program:
- download [pdf]
- Mireille Bousquet-Mélou
- “Enumeration with "catalytic" parameters: a survey”
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Many results in combinatorial enumeration are based on a recursive
description of the objects one tries to count. It often happens
that, in order to translate this description into a recurrence relation
(for the counting sequence) or a functional equation (for the
generating function), one is forced to take into account certain
additional parameters, which are not otherwise of particular
interest: following Zeilberger, we call such parameters "catalytic".
For instance, if the objects are walks with up and down steps on a half-line {0,1,2,...}, and the recursive description is "delete the last step to obtain a shorter walk", one has to take into account the position of the final point of the walk. Admittedly, there are more clever ways of addressing this problem, but it sometimes take a long time before clever constructions are found: about thirty years in the case of planar maps, for example. Moreover, working with naive -- but robust -- recursive descriptions allows one to solve several problems in a unified ans systematic way.
We will survey results and techniques that have been developed in the past few years to solve enumerative problems involving catalytic parameters. The associated functional equations typically involve divided differences, like (F(t;x)-F(t;1))/(x-1). Generic results begin to emerge, depending on whether the equation is linear, or not, and whether it involves one catalytic variable, or more.
The talks will be based on examples involving classical combinatorial objects: lattice paths, maps, permutations. - Elvira Di Nardo
- “Symbolic methods in probability and in statistics ”
- In the last ten years, the employment of symbolic methods has substantially extended both the theory and the applications of statistics. Here, by symbolic methods we refer to the set of manipulation techniques arising from the classical umbral calculus as introduced by Rota and Taylor in 1994. The purpose of these symbolic techniques in statistics and in probability is twofold: to find new algebraic identities, adding new insights in the theory, and to set up efficient mechanical processes performing algebraic calculations through an algorithmic approach. In particular, to find efficient symbolic algorithms challenges with new problems involving both computational and conceptual issues. In this picture, the combinatorics has no doubt a preeminent role. But, what we regard as symbolic computation is now evolving towards an algebraic language which aims to combine syntactic elegance and computational efficiency. Experience have shown that syntactic elegance often requires the acquisition of innovative methods and to climb this steep learning curve can be a deterrent to pursue the goal. But, having got a different and sometimes deeper viewpoint, the efficiency is obtained as by product and the result can be surprisingly better of what you expected. Evidences of efficiency in applying these symbolic methods will be show within statistical inference, parameter estimation, Levy processes theory and more in general problems involving multivariate functions. Recent connections with free probability and its applications, within random matrices and other satellite area, is extending the boundaries of applicability of these symbolic methods.
