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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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1answer
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 ...
0answers
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. ...
0answers
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 , section B18 and the post Combinatorially ...
1answer
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 ...
0answers
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?
4answers
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 ...
0answers
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 ...
2answers
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 ...
1answer
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 ...
0answers
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 ...
1answer
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 ...
1answer
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 ...
0answers
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
1answer
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 ...
1answer
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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