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9
votes
0answers
106 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 ...
3
votes
0answers
136 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 ...
8
votes
3answers
558 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 ...
7
votes
1answer
227 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 ...
2
votes
0answers
128 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. ...
37
votes
12answers
5k 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 ...