# Questions tagged [species]

The species tag has no usage guidance.

9
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### The generating series of the weighted species of fixpoints

I am wondering if the series
$$\sum_{n=0}^\infty \left(\sum_{k=0}^n \frac{D_{n-k}}{k!(n-k)!}t^k\right)X^n$$
where $D_m$ is the number of derangements of $m$ letters, admits a representation in closed ...

3
votes

1
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### Representing PSET as species

In symbolic method, one often considers two operators on ordinary generating functions, namely
$$
\operatorname{PSET}F(x) = \exp\left(F(x)-\frac{F(x^2)}{2}+\frac{F(x^3)}{3}-\dots\right),
$$
and
$$
\...

5
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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, ...

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### 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 ...

5
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0
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### 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 $A$...

10
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answers

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### 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 ...

9
votes

1
answer

382
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### 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 ...

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### 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. ...

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### 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 ...