An Extension of the Exponential Formula in Enumerative Combinatorics

An extension of the exponential formula in enumerative combinatorics

Let α be a formal variable and Fw be a weighted species of structures (class of structures closed under weight-preserving isomorphisms) of the form Fw = E(F c w), where E and F c w respectively denote the species of sets and of connected Fw-structures. Multiplying by α the weight of each F c wstructure yields the species Fw(α) = E(F c αw). We introduce a “universal” virtual weighted species, Λ,...

Combinatorics Enumerative Combinatorics 05

NSF Proposal: Automated Enumerative Combinatorics Automated Enumerative Combinatorics

My research students and I continued to practice a new research methodology, that can be loosely called rigorous experimental mathematics. It has something in common with both “mainstream” experimental mathematics (as preached by the Borwein brothers, David Bailey, Victor Moll, and their collaborators, see e.g. the masterpiece [BB], and the recent collection [BBCGLM]), and automated theorem pro...

Enumerative Combinatorics 8: Species

In this lecture I will discuss a very nice unifying principle for a number of topics in enumerative combinatorics, the theory of species, introduced by André Joyal in 1981. Species have been used in areas ranging from infinite permutation groups to statistical mechanics, and I can’t do more here than barely scratch the surface. Joyal gave a category-theoretic definition of species; I will take ...

Complexity problems in enumerative combinatorics

We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.