Combinatorial Species and Tree-like Structures

Multilinear generating functions for Charlier polynomials

“…We assume that the reader is familiar with enumerative applications of exponential generating functions, as described, for example, in [14,Chapter 5] and [3]. The product formula and the exponential formula for exponential generating functions discussed in these references play an important role in the combinatorial proof of the multilinear formula.…”

Analytic functors between presheaf categories over groupoids

“…The concept of multivariate analytic functor on the category Set of sets and functions was introduced by A. Joyal in [11] to provide a conceptual basis for his theory of combinatorial species of structures [10,1].…”

“…An endofunctor on Set is said to be analytic if it has a Taylor series development as in (1) above; that is, if it is naturally isomorphic to P for some species P. One respectively regards species of structures and analytic functors as combinatorial versions of formal exponential power series and exponential generating functions. A. Joyal characterised the analytic endofunctors on Set as those that preserve filtered colimits, cofiltered limits, and quasi-pullbacks (equivalently, weak pullbacks).…”

Abel-Rothe Type Generalizations of Jacobi's Triple Product Identity

“…Further, we utilize 4) to denote the basic hypergeometric s φ s−1 series, and the bilateral basic hypergeometric s ψ s series, respectively.…”

“…However, this, if feasible at all, remains at present an open task. A promising direction would make use of Bergeron, Labelle and Leroux's [4] combinatorial approach using the theory of species.…”