# Questions tagged [species]

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### Sum zero problems on the poset of structures of a combinatorial species

Consider a finite graded poset $\Gamma$ and assign to each maximal element $z\in \Gamma$ a variable $\mu(z)$. I want to solve the system of equations (minimally, I want to compute its rank, ideally, ...
197 views

### Is there a nice formula for the “non-crossing substitution” of linear combinatorial species?

Background A linear species is a functor $$F : \mathrm{Lin} \to \mathrm{FinSet},$$ where $\mathrm{Lin}$ is the category of totally ordered sets and bijections and $\mathrm{FinSet}$ is the category ...
206 views

### Anti-arithmetic product of symmetric functions: (why) is it integral?

This is an analogue of MathOverflow question #138148. Indeed it is so analogous that I wrote the following by copypasting said question and making the necessary changes. For every commutative ring $A$...
821 views

### Arithmetic product of symmetric functions: why is it integral?

For every commutative ring $A$, let $\mathbf{Symm}_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}_{\mathbb{Z}}$. We can define a ...
288 views

### On the category of virtual species

In Foncteurs analytiques et espèces de structures, Joyal defines virtual species, as a (quotient) of formal differences of functors $F,G:\mathbb{B}\rightarrow \mathsf{Set}$, and then proceeds to show ...
135 views

### Formal solutions of semiring equations

I am looking for a general theorem which would tell me when a formal series solution exists for an equation over a semiring. One may assume that the semiring is equipped with a (formal) derivative. ...
6k views

### Combinatorial results without known combinatorial proofs

Stanley likes to keep a list of combinatorial results for which there is no known combinatorial proof. For example, until recently I believe the explicit enumeration of the de Brujin sequences fell ...