Viinistu, Estonia, March 3 - 6, 2025
Professor at University of Iceland, Iceland
The answer to an enumerative combinatorics problem is typically a sequence of natural numbers. However, combinatorialists rarely work directly with this sequence; instead, they find it more convenient to work with a power series whose coefficients form the sequence of interest, known as a generating series. Combinatorial species can be viewed as a "categorification" of generating series. Formally, a species is defined as an endofunctor on the category (or groupoid) of finite sets and bijections. Moreover, operations such as addition, multiplication, composition, and differentiation are defined on species in a way that mimics those on generating series. This allows us to express relationships between combinatorial objects using algebraic expressions and equations (isomorphisms). Another benefit of species is that they inherently carry the necessary information for defining their unlabeled counterparts.
We aim to provide a concrete and accessible introduction to the theory of combinatorial species. No prior knowledge of category theory is assumed or required. We will explore familiar combinatorial families (data structures), such as sets, linear orders (lists), permutations, graphs, and trees, using the framework of combinatorial species. Highlights include a species-based solution to Montmort's (1713) classic hat-check problem and Joyal's (1981) brilliant proof of Cayley's (1889) formula for counting trees.
Last changed
March 6, 2025 9:54 EET (GMT +02:00)
by local organisers