Questions tagged [combinatorial-species]

For questions concerning the theory and applications of combinatorial species, the endofunctors of the category of finite sets and bijections, and the calculus thereof.

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1answer
88 views

Are valued-quiver species and combinatorial species related?

In this paper, Joel Lemay defines a species as a generalization of a valued quiver $Q$, where to each node $i \in Q_0$ we assign a division ring $\mathbf{k}_i$, and to each arrow $(a \colon i \to j) \...
10
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2answers
225 views

Bijection between perfect matchings permutations with even cycles

It is possible to proof that the number of perfect matching on a set of $2n$ elements is $n!!$, and on the other hand, it is also possible to proof that the number of permutations $\varphi$ of a set ...
0
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1answer
36 views

Changes in generating function expansion corresponding to changes of GF

Is there systematic review of how various transformations of GF affect its expansion? For EGF part of this is theory of species, but how about OGFs?
8
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1answer
239 views

Combinatorial Species, significance and problems can be solved with it.

Combinatorial Species, is a subject I recently came across when just out of curiosity's sake, looked out for possible interaction between category theory and combinatorics. After awhile I ended up ...
1
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1answer
340 views

Number of ordered, unlabeled binary rooted trees with n nodes and k leafs

I want to find the number of ordered, unlabeled binary rooted trees with $n$ nodes and $k$ leafs as an exercise. To be more precise. I am interested in objects like this ((c) 2015 M. Fulmek, PS ...
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0answers
55 views

Combinatorial Reasoning For Identity of Species

Given a combinatorial species $\mathcal{X},$ let $\mathcal{X}^{\bullet}$ be the pointing species that distinguishes a certain element (e.g. if $\mathcal{X}$ were the species of trees then $\mathcal{X}^...
0
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1answer
79 views

Finding the number of surjective distributions

I have been stuck with the following problem for hours, and I was hoping someone could give me a hint to attack it. Thanks! We define a distribution as a function together with a linear ordering on ...
1
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1answer
117 views

A Closed Form Lower Bound Approximating $p_{n,m,s} = n![z^n]\left(\sum_{k=0}^s\frac{z^k}{k!}\right)^m$

Here, I found $p_{n,m,s} = n![z^n]\left(\sum_{k=0}^s\frac{z^k}{k!}\right)^m = \sum\limits_{\substack{k_1 + \cdots + k_m=n\\0\leq k_i \leq s}} \frac{n!}{k_1!\cdots k_m!}$ as the number of ways to ...
8
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2answers
282 views

What are the group objects in the category of finite sets and bijections, and its functor category?

An object $G$ in a category $\mathcal{C}$ is called a group object if, given any object $X$ in $\mathcal{C}$, there is a group structure on the morphisms $\operatorname{hom}\left(X,G\right)$ such that ...
1
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1answer
67 views

Induced bijections of combinatorial species

I'm doing this exercise: Prove that $\mathcal{S}[\beta]$ is a bijection. Here, $\beta:M\rightarrow N$ is a bijection of finite sets, $\mathcal{S}$ is a species, and $\mathcal{S}[\beta]:\mathcal{S}[...
0
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1answer
86 views

Relating the number of partitions with ordered blocks

Studying the theory of combinatorial species from this book I came across the notion of the spices of linearly ordered sets with $k$ blocks $\rm{Bal}^k$ as seen in Example 2.12 on the bottom of the ...
2
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0answers
96 views

Testing combinatorial species for isomorphism

Given a system of species equations that specifies two species, is there an algorithm to test if they are isomorphic? Testing for isomorphism can be done by testing the equality of the coefficients ...
5
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3answers
633 views

Learning Combinatorial Species.

I have been reading the book conceptual mathematics(first edition) and I'm also about halfway through Diestel's Graph theory (4th edition). I was wondering if I was able to start learning about ...
14
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4answers
1k views

What is the difference between analytic combinatorics and the theory of combinatorial species?

Yesterday I asked the question Why should a combinatorialist know category theory?, where Chris Taylor suggested me to have a look at combinatorial species. I had heard the term before but I haven't ...
4
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1answer
155 views

A question about the definition of species.

A species $F$ is defined as an endofunctor of the category of finite sets. What if our combinatorial structure is not defined for sets of arbitrary size. More precisely, can we define a species from a ...