# Questions tagged [co.combinatorics]

Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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### When does a graph have a minimally strong orientation?

Given any asymmetric relation $A\subseteq V^2$ a digraph $D=(V,A)$ is minimally strong iff $D$ is strongly connected and for every arc $a\in A$ the digraph $D−a=(V,A\setminus\{a\})$ is not strongly ...
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### Bound on the number of unlabeled tree on n vertices

By the Cayley's Theorem, the number of labeled tree on n vertices is at most n^{n-2}. On the other hand, what is the bound on the number of unlabeled tree on n vertices?
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### Cyclic action on Kreweras walks

A Kreweras walk of length $3n$ is a word consisting of $n$ $A$'s, $n$ $B$'s, and $n$ $C$'s such that in any prefix there are at least as many $A$'s as $B$'s, and at least as many $A$'s as $C$'s. For ...
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### Request for an exact formula related to a partition in number theory

The Frobenius equation is the Diophantine equation $$a_1 x_1+\dots+a_n x_n=b,$$ where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$ must consist of non-...
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### To find a longer path with fixed endvertices in a graph satisfies the following property

Suppose that $G=(V,E)$ is a simple graph and $P=(V_1,E_1)$ is a path in $G$ where $$V_1=\{v_0,v_1,\cdots,v_n\},\ E_1=\{v_0v_1,v_1v_2,\cdots,v_{n-1}v_n\}.$$ I found that if the path $P$ satisfies: ...
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### What generalizes symmetric polynomials to other finite groups?

Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are ...
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### An extension of Erdos' distinct distances problem based on circles of various radii

Consider a collection $C_1,C_2, \dots, C_n$ of circles in the plane and suppose that the center of $C_i$ is $o_i$ and the radius of $C_i$ is $r_i$. We will define the relative distance between the ...
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### Counting self avoiding walks in a strip

Consider the strip $\{0,1,\ldots n\}\times\{0,1,2\}$ in $\mathbb{N}^2.$ Is a formula known for the total number of self avoiding walks in this strip starting at $(0,0)$ in terms of the parameter $n$? ...
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### Where can I find Gonthier's Coq code proving the four color theorem?

In a 2008 article in the Notices, Georges Gonthier announced a computer-checked proof of the four color theorem using Coq: Gonthier, Georges. Formal proof—the four-color theorem. Notices Amer. ...