# Questions tagged [species]

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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 ...
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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 $A$...
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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 ...
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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 ...