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Generating Functions Associated to Species of Structures

Speaker: Amritanshu Prasad, Professor of Mathematics, The Institute of Mathematical Sciences, Chennai

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Abstract: Species of structures were introduced by André Joyal and his group in Québec in the 1980s. They provide a way of organizing classes of labeled combinatorial objects that elevate the art of studying their generating functions to a science. 

Combinatorial relationships realized bijectively among such classes are transformed into functional relationships of their generating functions. For example, from the combinatorial interpretation of a set partition as a set of non-empty sets, the exponential generating function for Bell numbers exp(exp(z)-1) becomes blindingly clear; exp(z) is the generating function of sets and exp(z)-1 that of non-empty ones.
I will discuss species of structures and some generating functions that are associated with them. I will explain how algebraic operations on generating functions can be seen to arise from set-theoretic operations on species. I will introduce the Frobenius characteristic generating function of a species of structures, which is a simple variation of the cycle index generating function, landing us in the world of symmetric polynomials.
Lecture Slides

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