**1**

vote

**0**answers

61 views

### Presentation of the Rybnikov matroid

In this well celebrated work Gregory Rybnikov exhibit an example of two arrangements with the same underlying matroid, but with fundamental groups which are not isomorphic. This is a key ...

**5**

votes

**0**answers

124 views

### Euclidean Minimum Spanning Trees Restricted to One Vertex Per Grid Cell

Given an $n \times n$ grid with unit grid cells, and one point from the interior
of each cell, what is are best possible lower and upper bounds for lengths of minimum spanning trees? The lower bound ...

**4**

votes

**1**answer

253 views

### Strings with no long runs from proper subalphabets

Let $R_{n,k,b}$ be the number of $b$-ary strings of length $n$ that contain some run of length at least $k$ from some $(b-1)$-ary subalphabet. Let $N_{n,k,b}=b^n-R_{n,k,b}$ be the size of the ...

**6**

votes

**1**answer

94 views

### Sum identities with immanants

For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show
$$
\sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} ...

**5**

votes

**0**answers

61 views

### Uniform generation of Symmetric Plane Partitons

In the conclusion of An Involution Principle-Free Proof of Stanley's Hook-Content Formula Krattenthaler notes that the techniques of the paper might be useful for finding bijective proofs of the ...

**1**

vote

**1**answer

207 views

### A Bernstein-like Combinatorial Sum

Do sums of the "convoluted Bernstein" form
$$
\sum_{j=1}^m {m\choose j} q^j (1-q)^{m-j} {n \choose k} \left(\frac j m\right)^k \left(1- \frac j m\right)^{n-k}
$$
admit closed forms? My attempts to ...

**6**

votes

**0**answers

291 views

### What can I further assume about the speeds of runners of Lonely Runner Conjecture WLOG?

Lonely Runner Conjecture is the following problem, and the conjecture was proven to be true for $k\leq 7$:
Let $V$ be the set of $k$ distinct positive integers with $v_1<v_2<...<v_k$ ...

**5**

votes

**1**answer

343 views

### How many random matrices does it take to generate a matrix algebra?

Let $\mathbb{F}$ be a finite field.
Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra.
Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$
that one needs to take ...

**2**

votes

**0**answers

40 views

### A weaker version of Randell Isotopy Theorem

I am studying a problem in hyperplane arrangement theory related to the homotopy type of the complement manifold of a certain class of hyperplane arrangements.
In a well celebrated paper Richard ...

**23**

votes

**1**answer

1k views

### Algebraic proof of Five-Color Theorem using chromatic polynomials by Birkhoff and Lewis in 1946

I'm guessing everyone is familiar with Four Color Theorem which was proved by Appel and Haken using computers. A weaker version of this theorem is Five Color Theorem which states that a planar graph ...

**5**

votes

**1**answer

255 views

### A claim from “Graph minors - a survey” by Robertson and Seymour

Can someone give me a proof sketch for this:
Let $\mathscr{P}_n$ be the set of all graphs which do not contain a path on $n$ vertices as a subgraph. Define the type of a graph inductively as: the type ...

**31**

votes

**2**answers

945 views

### Random sequence of integers in $\{1, 2, \dots, n \}$ which is “everywhere probably increasing” - how long can it be?

Let $D=(d_1,d_2,\dots,d_k)$ be a sequence of correlated random variables. $D$ is "everywhere $r$-probably increasing" if the event $d_j > d_i$ has probability $\geq r$ for all $j > i$.
Fix $r ...

**4**

votes

**5**answers

335 views

### Table of planar connected graphs

For the other day I need to use the table of planar connected graph with few vertex. In the wolfram's mathworld , they listed only the graph with $4$ vertex.
Does anyone knows webpages or pdf on the ...

**1**

vote

**1**answer

173 views

### Singular homology of the zero loci of polynomials

I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in ...

**1**

vote

**0**answers

92 views

### Are back edges mandatory in Ford Fulkerson algorithm?

Consider the algorithm of Ford Fulkerson where, for each iteration, you add flow along a path equal to the maximum residual capacity along this path.
Does it exist, for every network, a choice of ...

**11**

votes

**1**answer

210 views

### Complexity of planar scissor congruence

Two (not necessarily convex) poygons of equal area are scissor-congruent, i.e. both can be cut along a finite number of straight lines or segments into isometric pieces.
What can be said about the ...

**2**

votes

**1**answer

141 views

### Linear independence of +/- 1 strings/vectors II

Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a ...

**3**

votes

**2**answers

244 views

### Isoperimetric inequality on the Hamming cube

Suppose $X \subseteq \lbrace 0 , 1 \rbrace ^{m}$ such that $|X| \geq 2^{0.8m}$, and $m \geq 2$, then prove that there exists $x,y \in X$ with $||x - y||_{1} \geq m/2$.
My approach to prove this was ...

**7**

votes

**1**answer

784 views

### A Question about Palindromic Numbers and System of Arithmetic Progression

Based from Harminc and Sotak's result, www.fq.math.ca/Scanned/36-3/harminc.pdf
We know that under certain condition, an arithmetic progression can contain an infinitely many palindromes.
My question ...

**1**

vote

**1**answer

121 views

### Linear independence of +/- 1 strings/vectors

Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a ...

**3**

votes

**1**answer

129 views

### Simultaneous approximation by rationals with relatively prime numerators

The following seems hard to me (or perhaps just not true), but perhaps I am mistaken. It is known that given irrational numbers $x_1$ and $x_2$, there are infinitely many simultaneous rational ...

**1**

vote

**1**answer

107 views

### Combinatorial polynomials from general diagram fillings?

There is a plethora of polynomials defined on partition shaped Young diagrams, (Schur, Jack, Grothendieck,...), and skew Young diagrams.
There are also composition shaped diagrams that are responsible ...

**1**

vote

**1**answer

97 views

### Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$?

Feedback vertex set is a set of vertices whose removal leaves an acyclic graph.
It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also ...

**1**

vote

**4**answers

616 views

### Distribution of composite numbers

I have moved this question to math.stackexchange.com. People who are interested in this question can discuss at :http://math.stackexchange.com/questions/1272431/distribution-of-composite-numbers
...

**5**

votes

**1**answer

228 views

### Dimension of the span of all partial derivatives of a given symmetric polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ ...

**4**

votes

**0**answers

287 views

### How is $ \sum_{x \in X(\mathbb{F}_q)} \dots $ a generalization of cardinality?

This quarter Maxim Kontsevich is offering a course on exponential integral. There is not much in the way of notes, but is one page with mysterious comments.
Let $X$ be an algebraic variety over ...

**3**

votes

**0**answers

103 views

### scalar multiple of Young symmetrizer

The following is a lemma from Fulton and Harris' book -Representation theory,a first course (page 53):
Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...

**4**

votes

**2**answers

178 views

### Vertex expansion of the Hamming graph

Let $G$ be a graph, and for every $S \subseteq V$, let $N(S)$ denote the neighborhood of $S$. The vertex expansion of $G$ is
$$ \min_{S\subseteq V, |S|\le |V|/2} \left\{ \frac{|N(S)|}{|S|} ...

**10**

votes

**1**answer

339 views

### Equation with $q$-binomial coefficients

Let $d\ge2$, and let $q$ be a power of a prime. As usual, define $N(d,q)=\sum_{k=0}^d{d\choose k}_q$.
I wonder if there are $d$ and $q$ as above such that $1+N(d,q)=q^{d+1}$.
(If the answer is ...

**1**

vote

**1**answer

173 views

### Random graphs with boundary in a game (Tsuro)

Suppose we have an $n \times n$ board and we have $n^2 - 1$ square tiles. These tiles consist of a 8 vertices, two on each edge, and every vertex is connected to precisely one other vertex. These ...

**2**

votes

**0**answers

140 views

### Largest Fourier coefficient of sparse boolean function

Consider a Boolean function $f: GF(2)^n \rightarrow \{0, 1 \}$. I would like to show that if $f$ is sparse, i.e. $\sum f(i) \leq t$, then $f$ must have a large Fourier coefficient. (A Fourier ...

**8**

votes

**1**answer

160 views

### Extending subsets to supersets in different ways

We are given a collection of sets $A_1,\ldots,A_s$, pairwise different and each of cardinality $k$, and a collection of sets $B_1,\ldots,B_s$, pairwise different and each of cardinality $l>k+1$, ...

**5**

votes

**1**answer

168 views

### Graph of graph homomorphisms

For (finite or infinite) undirected, simple graphs $G, H$, let
$V_{\text{Hom}} = \{f:G\to H:f\text{ is a graph homomorphism}\}$, and
$E_{\text{Hom}} =\big\{\{f,g\}\subseteq V_{\text{Hom}}: ...

**3**

votes

**1**answer

181 views

### Seymour's second neighborhood conjecture

Does anyone out there know if Seymour's second neighborhood conjecture is still open? if not, I would appreciate any references.

**2**

votes

**0**answers

49 views

### Decomposing a weakly chordal graph into disjoint union of co-chordal graphs

A graph G is said to be co-chordal if it is $\bar C_n$-free for any $n \ge 4$. It is weakly chordal if it is $C_n$ and $\bar C_n$ free for all $n\ge 5$. Assume that the induced matching number of $G$ ...

**5**

votes

**1**answer

138 views

### Algorithm that generates a n-simplex that cover n-polytope?

Given an $n$-cube with unit volume, is there any algorithm that generates a $n$-simplex that covers the $n$-polytope?

**5**

votes

**0**answers

62 views

### Approximating a max-cut's intersection with other cuts

(This is a cross-post from the Theoretical Computer Science Stack Exchange.)
For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set ...

**5**

votes

**1**answer

146 views

### Integer decomposition of dilated integral polytopes

For $n > 0$, let $P$ be an integral polytope, that is, the convex hull in $\mathbb{R}^n$ of points in $\mathbb{Z}^n$. Suppose that $\dim(P) = n$.
Question: Given $d > n + 2$ is it true that
$$ ...

**5**

votes

**1**answer

151 views

### A generalization of Erdős–Newman–Mirsky?

Given a sequence $S$ of natural numbers, write ${\bf Gap}(S)$ for the set of differences between consecutive terms. (So $|{\bf Gap}(S)|=1$ precisely for arithmetic progressions, hence the connection ...

**25**

votes

**1**answer

1k views

### Combinatorics of K(Z,2)?

Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this ...

**1**

vote

**0**answers

38 views

### Lattice-isotopic essentialization of arrangements

I'm working on a problem related to
$\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...

**11**

votes

**3**answers

317 views

### Models for graphs representing real-life networks

I am interested in basic models of graphs (stochastic or deterministic) that are offered for real-life networks (like social networks, the Internet, neuron networks).
I will be thankful for answers ...

**3**

votes

**1**answer

143 views

### An upper bound on the number of sets of parallel lines covering points in a finite plane?

Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a ...

**85**

votes

**7**answers

4k views

### Is the set $ AA+A $ always at least as large as $ A+A $?

Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$?
My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly ...

**12**

votes

**2**answers

737 views

### Mysterious identity between numbers of odd/even meander systems

Definitions:
An upper arch system of order $n$ is a subset of the plane consisting of $n$ non-intersecting closed semicircles in the upper half-plane whose endpoints belong to the set $\{(k,0)\mid ...

**12**

votes

**2**answers

657 views

### A combinatorial question about orthonormal bases

Suppose that $F:S^{n-1}\to A$ is a map of sets from the unit sphere in $\mathbb R^n$ to an abelian group, and that the sum $F(v_1)+\dots +F(v_n)$ over an orthonormal basis is independent of the basis. ...

**14**

votes

**0**answers

195 views

### Coloring a Ferrers diagram

I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed.
Say that a coloring of the dots of a Ferrers ...

**4**

votes

**0**answers

134 views

### Connection between connectivity and cohesion of a graph

Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path.
A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, ...

**2**

votes

**2**answers

348 views

### Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions

Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...

**3**

votes

**1**answer

146 views

### A question of terminology regarding integer partitions

I am wondering if there is a standard notation and name for the following. Let $\lambda$ be a partition $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_r\geq 1$ of $n$ into $r$ parts. Then we can ...