**3**

votes

**0**answers

37 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 ...

**-1**

votes

**0**answers

22 views

### How can i simplify the sum of modified partial bell polynomials

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 ...

**0**

votes

**0**answers

37 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 ...

**0**

votes

**1**answer

33 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 ...

**0**

votes

**0**answers

85 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 ...

**0**

votes

**0**answers

66 views

### independent subset problems

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 ...

**9**

votes

**2**answers

234 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 ...

**19**

votes

**0**answers

384 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 ...

**5**

votes

**2**answers

156 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 ...

**1**

vote

**1**answer

98 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.
...

**8**

votes

**2**answers

357 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 ...

**10**

votes

**0**answers

111 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

174 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, ...

**0**

votes

**1**answer

155 views

### Closed form of the sum involving product of 4 binomial coefficients [closed]

I'm trying to find closed form of the expression: $$\sum_{k\in\mathbb{N}}{{a \choose k}{b \choose k}{a \choose c-k}{b \choose c-k}},$$
where $a,b,c\in \mathbb{N}$, and ${x \choose y}=0$ for $y>x$ ...

**1**

vote

**1**answer

61 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

72 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

140 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

196 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

76 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**

votes

**0**answers

84 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

63 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 ...

**-2**

votes

**0**answers

85 views

### Calculation of Shannon and Simpson Index as a function of time [closed]

Suppose a_i(t) for i=1,2,...,n, represents species density at time t
As the system evolves, smaller a_i's, than a critical value, get still smaller and ultimately vanish while the others keep ...

**1**

vote

**0**answers

31 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

81 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

81 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

159 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

185 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

159 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

360 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

88 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

134 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

159 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

223 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

175 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

102 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

92 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

133 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 ...

**0**

votes

**1**answer

68 views

### Using upper bound information in graph search

I am using A* (A-Star) to search a graph. A* algorithm takes advantage of the information $h(x)$, which is a lower bound of the distance between a vertex $x$ and the destination vertex.
In other ...

**4**

votes

**1**answer

266 views

### The formula for a perhaps basic identity (move from stackexchange)

The following question is moved from math stackexchange. It seems that this is not a popular question, but I really want to know the answer so I moved it to here. The question reads as follows.
We ...

**1**

vote

**0**answers

116 views

### Simultaneous vanishing of convolutions of Mertens function with itself

By Landau's theorem on Dirichlet series, we know that all the step functions ($k\geq 1$)
$$M_k(x)=\frac{1}{2\pi i}\int^{2+i\infty}_{2-i\infty}\frac{x^sds}{\zeta^k(s)s}=\sum_{n\leq ...

**7**

votes

**5**answers

432 views

### Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...

**5**

votes

**0**answers

151 views

### The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$

Let $N=\{1,2,\ldots ,n\},n>1$. We wish to construct a set $A\subseteq N$ with the property:
The sum of all the elements of every non empty subset of $A$ is not a
multiple of $n$.
...

**1**

vote

**1**answer

202 views

### Probability of connected graph on torus

Let $G = (V, E)$ be a graph on n vertices constructed in the following way:
Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$.
Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...

**1**

vote

**0**answers

48 views

### clustering in a graph on boolean functions

Fix some n and k. Consider the following directed graph: vertices are all functions $2^n\rightarrow 2^n$ and a vertex f has an edge to a vertex g for every h such that $f=h\circ g$ and h depends only ...

**5**

votes

**2**answers

133 views

### Positivity of Ehrhart polynomial coefficients

Are there any results stating that a given family of convex polytopes have Ehrhart polynomials with non-negative coefficients?
What methods are available for proving such a property for some family ...

**0**

votes

**0**answers

133 views

### Problem regarding orthogonal vectors

Suppose $C_{1},C_{2},...,C_{n}$ are $0-1$ vectors of length $m$. Given $C_{i} \in \{0,1\}^{m}$ with $C_{i}=x_{i1}x_{i2}...x_{im}$ we say $C_{i}'=x_{i1}'x_{i2}'...x_{im}'$ is a subvector of $C_{i}'$ if ...

**3**

votes

**2**answers

253 views

### Graph game minimum vertex degree

Consider the following graph game, given a graph $G=(V,E)$ on $n$ vertices with minimum degree $ >> log(n)$. Players are BR and MA (BR moves first):
BR claims an unclaimed edge from $E$, adds ...

**2**

votes

**1**answer

142 views

### Solving assignment problem using Hungarian method vs min cost max flow problem

The traditional solution for the assignment problem is the Hungarian method - it's complexity is O(V^4) or O(V^3) if using Edmonds method.
However, it can also be reduced to a min cost max flow ...