Cover monoidal categories on species and free cover monoids

Abstract

We study a new product on the category of combinatorial species, which we call the cover Cauchy product. This product is defined as a relaxed version of the classical Cauchy product on species. We prove that the category of species together with the cover Cauchy product has the structure of a monoidal category. We define a cover Cauchy product on exponential generating functions and show that the exponential generating function of the cover Cauchy product of two species is precisely the cover Cauchy product of their exponential generating functions. Finally, we show that the free cover Cauchy monoid in one generator is given by the species of packed words, which are in bijective correspondence with surjections.