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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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28 views

Model for random graphs where clique number remains bounded

In the Erd?s-Rényi model for random graphs,the clique number is seen to go to infinity al the number of vertices grows. Is anyone aware of models for random graphs with bounded ...
65 views

On the first sequence without collinear triple

Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one. ...
42 views

The scutoid as a noninscribable polyhedron

I have no a good backround in combinatorial geometry but I would like to ask next question, because I think that it is interesting. I know from the book [1], section B18 and the post Combinatorially ...
37 views

Chromatic Polynomial when two disjoint graphs are joined at $2$ distinct points

Consider a graph with chromatic polynomial $P(x)$ joined to a clique of order $k$ in two distinct points (joining here just means interesection of points). Then, what is the chromatic polynomial of ...
48 views

An upper bound on the minimum number of vertices in a girth 5 graph of chromatic number $k$

Is there a known upper bound on the minimum number of vertices in a graph with girth 5 and chromatic number $k$? Could you also give references for this?
436 views

Sums of binomial coefficients weighted by incomplete gamma

I am interested in proving that $$\sum_{k=0}^n\frac{k}{k!}\sum_{l=0}^{n-k}\frac{(-1)^l}{l!}=1$$ and $$\sum_{k=0}^n\frac{k^2}{k!}\sum_{l=0}^{n-k}\frac{(-1)^l}{l!}=2.$$ I verified it numerically ...
424 views

On the first sequence without triple in arithmetic progression

In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence: It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
342 views

Regular subsets of $\text{PSL}(2, q)$

Suppose a group $G$ acts on a set $\Omega$. Call a subset $A \subset G$ regular (or sharply transitive or simply transitive or...) if for every two points $\omega_1, \omega_2 \in \Omega$ there is a ...
153 views

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 ...
56 views

Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?

I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope. I am looking ...
33 views

Induced subgraphs of the line graph of a dense linear hypergraph

Given a hypergraph $H=(V,E)$ we associate to it its line graph $L(H)$ given by $V(L(H)) =E$ and $$E(L(H)) = \big\{\{e_1,e_2\}: e_1\neq e_2 \in E \text{ and } e_1\cap e_2 \neq \emptyset \big\}.$$ We ...
42 views

Rank and edges in a combinatorial graph?

Fix a $d\in\mathbb N$ and consider the matrix $M\in\{0,1\}^{2^d\times d}$ of all $0/1$ vectors of length $d$. Consider the matrix $G\in\{0,1\}^{n\times n}$ whose $ij$ the entry is $0$ if inner product ...
29 views

Somewhat complicated Routing Problem [on hold]

Based on routing problems, in this grid, how many ways can you go from A to B? What methods can be used here? Link to the Image
111 views

Combinatorial problem on periodic dyck paths from homological algebra

edit: I added conjecture 2 that looks much more accessible. Here is the elementary combinatorial translation of the problem (read below for the homological background): Let $n \geq 2$. A Nakayama ...
105 views

From Steiner systems to geometric lattices to matroids

I am looking for a specific matroid. I found a source that claimed to discuss these matroids, but then, only discusses geometric lattice. Even more, in that paper, the geometric lattice that seems to ...

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