Combinatorial Species, is a subject I recently came across when just out of curiosity's sake, looked out for possible interaction between category theory and combinatorics. After awhile I ended up here Learning Combinatorial Species., and later on to this book Combinatorial Species and tree-like structures. For someone comfortable in Category Theory, this may be a very beautiful thing to mull over indeed and creates a flexibility to the theory of generating functions as well, and the latter is of important significance.
Though, in any instance of book/notes I can come up with, didn't find out an "intuitive" application of combinatorial species. Combinatorics, is definitely not about counting anymore, but arguably someone declares that the most funny stuff in it has to do with counting problems (because those sort of problems have more intuition I guess).
So my question has to do with that; Could you give me please an instance of application of Combinatorial Species in Enumerative combinatorics? Any other example of application is welcome too of course!
Last comment: I ain't have problem with it becuse its definition or significance. I'm looking for an intuitive approach of the aforementioned notion (I mentioned the latter, because I want to avoid any possible duplication with other possibly related question on MSE, because I think checked them all and not such an answer/question has been showed up. If such a duplication does exist with an answer though, please feel free to mention it!)