Joyal's [combinatorial species](http://en.wikipedia.org/wiki/Combinatorial_species), endofunctors in the category of finite sets with bijections $\mathbf B$ have found numerous applications. One generalisation is given by so-called "tensor species" (also "tensorial species", or, "linear species" - not to be confused with the species on totally ordered sets in the book by Bergeron, Labelle and Leroux) which are defined as functors from $\mathbf B$ into the category of finite dimensional vector spaces (say, over the complex numbers) with linear transformations $\mathbf{Vect}$.


I wonder whether there have been any "practical" applications of tensor species? I know of a very short list of articles dealing with them (eg. by [Méndez](http://www.pnas.org/content/88/21/9892.full.pdf)) but hardly any spelled out examples. I wonder whether I overlooked something.


Note that for any combinatorial species $F$ we cann regard $F[\{1,2,\dots,n\}]$ as a finite set with an action of the symmetric group. Similarly, if $F$ is a tensor species, we can regard $F[{1,2,\dots,n}]$ as a linear representation of the symmetric group. Thus, I am mostly interested in examples that use the combinatorial operations for greater clarity of a construction.



Has there been any application of tensor species?
In the theory of algebraic operads, the language of "tensor species" is often used, 
see Chapter 5 of "Algebraic Operads, Jean-Louis Loday & Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012).


For example one can define an operad very concise as a monoid in species under a certain monoidal structure. Without this language, it takes quite a while to write down all the compatibilities with the various $S\_n$ actions (Although I find it illuminating to write down an "elementary" definition never the less).


There are in fact many definitions in the theory of operads, which are a bit cumbersome to write down without talking about "tensor species".


And of course things like generating series for operads are of interest and operadic phenomena/constructions like Koszul duality give constraints/relations for their generating series.


