# Tagged Questions

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votes

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20 views

### Asymptotic analysis of a sum of complex summands using integral

I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, ...

**8**

votes

**2**answers

372 views

### A game of stones

How I arrived at this question is a rather long story having to do with the honors calculus class I am teaching. At this point it's sheer curiosity on my part. Here is the game.
...

**0**

votes

**0**answers

36 views

### Bounding a ratio by its complement [on hold]

Given some integers $n > \beta + \alpha, \beta > \alpha$ and $\alpha > 0$, is there a real number $\delta$ for which $\frac{n-\alpha}{n-\beta} \geq (\frac{\beta}{\alpha})^{\delta}$, where ...

**0**

votes

**0**answers

50 views

### Number of graphs with M edges that does not contain K-clique [on hold]

If we consider the space of graphs $G(n,M)$ where $M$ denotes the number of edges. Is there any known way of calculating the number of graphs within this space that does not contain any k-cliques? Can ...

**2**

votes

**1**answer

173 views

### Counting Ribbon graphs

Let $G$ be a ribbon graph (sometimes called fat graph) with $v$ vertices and $e$ edges. Furthermore each vertex is of degree $d$.
Q) What is the number of $G$ with the above properties? I mean does ...

**12**

votes

**1**answer

419 views

### Is it possible to define higher cardinal arithmetics

In number theory there are several operators like addition, multiplication and exponentiation defined from $\omega\times\omega$ to $\omega$. Each of them is defined as an ...

**0**

votes

**0**answers

134 views

### On degrees of polynomials with matching zeros in a subset

Let $S\subsetneq \Bbb R^n$ such that $|S|<\infty$ and for all partitions $S_1$ and $S_2=S\backslash S_1$ of $S$, there exits a multilinear polynomial $h$ such that $h(s)=1-h(s'),\mbox{ }\forall ...

**3**

votes

**1**answer

144 views

### Determinant of the oriented adjacency matrix of a tree

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries
$$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\
-1 ...

**2**

votes

**1**answer

166 views

### Simplest form for sum of Binomial Expressions

How difficult is the problem of reducing the number of terms in a sum of binomial expressions? Formally:
Given $a_1, a_2, a_3, … a_n$, and $b_1, b_2, b_3, ... , b_n$, where $a_i, b_i \in \mathbb{Z}$, ...

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votes

**3**answers

528 views

### Koebe–Andreev–Thurston theorem - where can I find a proof?

Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two ...

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votes

**3**answers

560 views

### Conjecture regarding closest point inside a discrete ball to a line

I'm a PhD student in image processing, where I've stumbled into a problem that seems to be essentially number theory. I've hunted around online and while I've found many results on similar problems, ...

**-1**

votes

**0**answers

56 views

### A combinatorial and number theoretical problem [on hold]

Given N positive integers, not necessarily distinct, how many ways you can take 4 integers from the N numbers such that their GCD is 1.
For example,N=10 and the positive integers are ...

**2**

votes

**1**answer

114 views

### A perfect $(n,k)$ shuffle function

Suppose you have a deck of $n$ cards; e.g., $n{=}12$:
$$
(1,2,3,4,5,6,7,8,9,10,11,12) \;.
$$
Cut the deck into $k$ equal-sized pieces, where $k|n$;
e.g., for $k{=}4$, the $12$ cards are partitioned ...

**3**

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**0**answers

73 views

### Partitions with each part dividing the original number

I have a question on partitions that I have not seen being discussed. It deals with those related to divisors.
My definition of partitions I am working with is as follow: a sequence of weakly ...

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votes

**0**answers

38 views

### How can i simplify the sum of modified partial bell polynomials [on hold]

I am trying to prove my conjecture that uses partial bell polynomials as well as modified partial bell polynomials. Putting these bell polynomials into a workable form is a huge problem for me. The ...

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68 views

### Reduction argument from a general vertex set V(G) to a prime power in Prof. Keevash's proof on the Existence of Designs

The proof flow of the paper "On the Existence of Designs" by Prof. Keevash as I understand it is the following:
-- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3)
-- Covering ...

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votes

**1**answer

42 views

### Minimal hypergraphs with respect to separation

Let $H = (V, E)$ be a hypergraph, that is $V$ is a set and $E \subseteq \mathcal{P}(V)$. We say that $H$ is $T_1$ if for $v\neq w$ there are $e_v, e_w \in E$ such that $v\in e_v, w\notin e_v, w\in ...

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**0**answers

96 views

### A question about the duality principle

Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by ...

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74 views

### independent subset problems [on hold]

I'm interested in the following which i suspect is probably a well studied problem.
Given a set $N=\{1,2,...,n\}$ and $M=\{1,2,...,m\}$ consider a map $$f:N\rightarrow 2^{M}$$ (elements of $N$ to ...

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votes

**2**answers

244 views

### Is there any relationship between the topologies of the clique complex and the independence complex?

Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial ...

**25**

votes

**1**answer

607 views

### “Nyldon words”: understanding a class of words factorizing the free monoid increasingly

BACKGROUND.
Let me first introduce some classical definitions, which appear, e.g., in §5 of Lothaire's Combinatorics on Words, in §5.1 of Reutenauer's Free Lie algebras, and in §6.1 of Victor ...

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**2**answers

164 views

### Decomposing polyhedral cones into “direct sums” and a polynomial

This question consists of two parts. I'm not breaking it up into two separate ones because posing the second question would essentially require me two rewrite the first one. Also, to some extent, the ...

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vote

**1**answer

101 views

### NP hard problems on UD graphs

I'm reading up on NP hard problems in Unit Disk graphs. I'd like to point out i'm fairly new to this NP hard stuff so i'm trying to get around how to prove something is NP hard.
...

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votes

**2**answers

364 views

### Infected square

I saw the following problem in Mathematical Puzzles from Peter Winkler (very good book, by the way): imagine you infect k cases of a chessboard nxn and the infection spreads to a case if it has at ...

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**0**answers

114 views

### Polytopes with few vertices and few facets

I recently realized that, for fixed $\alpha$ and $\beta$, the number of (combinatorial types of) $d$-polytopes with $\leq d+1+\alpha$ vertices and $\leq d+1+\beta$ facets is bounded by a constant that ...

**6**

votes

**1**answer

176 views

### Origin of the numbers game

The numbers game is a (one-player) game played on a finite graph with an initial assignment of numbers to its vertices, studied by Alon, Bj\"orner, Brenti, Donnelly, Eriksson, Krasikov, Mozes, Peres, ...

**1**

vote

**1**answer

62 views

### Regular unimodular triangulation for a certain simplex

Consider an $n$-simplex with vertices given by
$(0,0,\dots,r_i,r_{i+1},\dots,r_n)$ where $r_1,\dots,r_n$ are given natural numbers,
and $i=0,1,\dots,n+1$.
Does this simplex admit a regular, ...

**3**

votes

**1**answer

75 views

### Stable equilibria of points on the 2-sphere

Suppose $n$ points lie on the sphere $S^2=\{x\in\mathbb{R}^3\mid \|x\|=1\}$ and are subjected to a repulsive acceleration that pushes away a point from each other point with an intensity proportional ...

**13**

votes

**9**answers

1k views

### Combinatorial Databases

At one point, I remember being excited by seeing the website Encyclopedia of Combinatorial Structures as an extension of Sloane's Online Integer Sequence Database site. Unfortunately, the site (ECS) ...

**8**

votes

**0**answers

144 views

### Maximum number of vectors in a hypercube satisfying given conditions

$\mathcal{C}$ is a collection of binary vectors of length $n$, i.e. $\mathcal{C}\subseteq\{0,1\}^n$. For arbitrary $x,y,z\in\mathcal{C}$ and $x\neq z$, $y\neq z$, there always holds that the Euclidean ...

**0**

votes

**0**answers

198 views

### A hard combinatorial identity

I asked this questions on the http://math.stackexchange.com/questions/1016713/ But I don't get answer.
I try to prove the following hypothesis
$$\sum_{i=0}^{min\{k, n-1\}}(-1)^i { n+i-1 \choose ...

**0**

votes

**0**answers

79 views

### Does the weak Hadwiger conjecture imply the Hadwiger conjecture?

For any cardinal $\kappa$, let $K_\kappa$ denote the complete graph on $\kappa$. We consider the following statements:
(H) If $G$ is a graph and $\chi(G) = \kappa$ then $K_\kappa$ is a minor of $G$.
...

**0**

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**0**answers

85 views

### Proof of Faà di Bruno's formula with convolution identity

Faa di bruno's formula can be expressed as follows
$$
\frac{d^n}{dx^n}[f(g(x))] = \sum_{k=1}^{n}f^{(k)}(g(x)) B_{n,k}(g'(x),....,g^{(n-k+1)}(x))
$$
On this Wikipedia page, there is a convolution ...

**-1**

votes

**1**answer

64 views

### Combinatorial Interpretation of Generalized Stirling numbers

I know the combinatorial interpretation of first, and second order Stirling numbers (#of k cycles of n items, and #of partitions n items into k subsets). Is there an interpretation for the generalized ...

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vote

**0**answers

33 views

### Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ ...

**2**

votes

**1**answer

82 views

### irregular pairs in half graphs - Szemeredi regularity

Szemeredi's regularity lemma is a well-known result about partitioning large graphs into pieces such that most pairs of pieces are "regular". The precise statement takes a bit of detail so I'll just ...

**0**

votes

**0**answers

85 views

### Best possible concentration inequality in high dimensions

Let $X_1,\ldots,X_n$ be independent random variables in $\mathbb{R}^d$ with $EX_i=0$ and $||X_i||_{2}\leq 1$. What is the best known exponential upper bound for
$$P(||X_1+\cdots+X_n||_{2}>x)?$$
In ...

**12**

votes

**0**answers

163 views

### Coloring the edges of a torus graph

Question:Consider the $2k \times 2k$ grid graph on a torus. Is it true that for every $2$-coloring of the edges, there is an antipodal pair of vertices connected by a path that changes colors at most ...

**2**

votes

**1**answer

118 views

### A property of monomials in a Schubert polynomial

I came across a property of monomials in a Schubert polynomial in Lascoux's book:
http://www-igm.univ-mlv.fr/~al/ARTICLES/CoursYGKM.pdf
page 62, footnote 4. The property is as follows.
Let us ...

**12**

votes

**1**answer

188 views

### Number of orders of $k$-sums of $n$-numbers

Suppose we have a $n$-element set $S$. Denote the set of its $k$-element subsets by $K$ ($|K|=\binom{n}{k}$).
If the elements of $S$ are real numbers then to each $k$-element subset we can associate ...

**2**

votes

**1**answer

160 views

### Random walks with exponential decreasing steps

Let $g$ be the golden number (or another algebraic integer in $(0,1)$ that fullfills an equation with coefficients $\pm 1$). Consider the random walk on $\mathbb{R}$ starting with $0$ and walking ...

**10**

votes

**2**answers

364 views

### Number of paths through infinite trees with given “growth rates”

(Preface: This may be a naive or easy question for experts....)
Consider an infinite tree, rooted on the left, where each node has two children; the number of nodes at each level (distance from the ...

**0**

votes

**1**answer

89 views

### All $2$-designs arising from the action of the affine linear group on the field of prime order

Let $p$ be a prime and $\mathbb{Z}_p$ denote as usual the field of order $p$. Let $AL(p)$ be the affine linear group $\{x\mapsto ax+b \;|\; a\in \mathbb{Z}_p\setminus \{0\}, b\in\mathbb{Z}_p\}$. For a ...

**5**

votes

**1**answer

136 views

### Combinatorial proof of the Cauchy identity for double Schubert polynomials

The Cauchy identity for double Schubert polynomials states
$$ \mathfrak{S}_w(x;-y) = \sum_{\substack{u,v \in S_n \\ w=v^{-1}u \\ l(w) = l(v) + l(u)}} \mathfrak{S}_u(x)\mathfrak{S}_v(y).$$
Is there a ...

**2**

votes

**0**answers

175 views

### Minimizing $\{0,1\}$-sequence permutations

Explanation: For a given bit sequence $f$, reposition the bits as to minimize $G$ which can be thought of as a measure of how poorly proportional $f$ is to each of its subsequences.
Let $p \in ...

**6**

votes

**2**answers

224 views

### Counting valid coordinates

We are given a matrix $D = (d(i,j))_{1 \leq i,j \leq n}$ such that $d(x,z) \leq d(x,y) + d(y,z)$ for each $1 \leq x,y,z \leq n$. It is also known that $d(x,y) \in \mathbb{N}$ (In this question $0 \in ...

**5**

votes

**1**answer

177 views

### Bijective proof of an Abel-Hurwitz-type identity

Can anyone sketch for me a bijective proof of the fact that the number of spanning trees of the complete graph on $n$ vertices, $K_n$ (given by the formula $ t_n = n^{n-2}$), satisfies $ t_n = ...

**2**

votes

**1**answer

104 views

### Maximum number of hyperedges on a hypergraph without a spanning tree

Although every connected graph has a spanning tree, the same is not true for hypergraphs: consider the hypergraph on 4 vertices with all possible edges of size 3. You need to pick at least two edges ...

**11**

votes

**0**answers

93 views

### On some special spanning trees of grid graphs

I would like to know if there are existing results on the following objects:
spanning trees of a grid graph, with no corridor
where a corridor is a vertex having exactly two neighbors, on ...

**1**

vote

**1**answer

136 views

### Probability of each edge in K-clique [closed]

For $c \in R$ and $k \in N$, $k \geq 3$ let
$p_{k,c} := n^{\frac{−2}{k+1}}log^c(n)$.
I would like to prove that exists $c\in R$ such that every edge in the random graph $G(n,p_{k,c})$ lies in a copy ...