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4
votes
3answers
175 views

Relation between diametral path and regularity of a graph

Let $G(V,E)$ be a graph. A path whose length is equal to the diameter of a graph is called a diametral path. In a cycle graph every vertex has $2$ diametral paths. Now I need to prove that this: ...
2
votes
1answer
53 views

Calculating greatest common divisor series [closed]

How to compute the value of [G(1,x)+G(2,x)+G(3,x)+....+G(x,x)] efficiently? When x can be as large as million.G = greatest common divisor.
0
votes
0answers
36 views

Separation on discrete set

Consider the set $L = \prod_{i=1}^n\{1,0\}$, i.e. L consists of the element of n-tuples whose entries are 0 or 1. Also we can regard $L$ as a subset of $R^n$. Define linear functions $f(x)= a_1x_1+ ...
1
vote
0answers
55 views

bounded degree graph colouring.

I was wondering if anyone could provide references on the following: Is determining the chromatic number of a bounded degree graph APX-complete? 2.I've seen the result that states it is NP-hard ...
2
votes
0answers
45 views

Hypergraph edge colouring

I'm interested in knowing if finding the edge-chromatic number of a $k$-uniform $k$-partite hypergraph is NP-hard for $k\geq 3$ Could anyone provide a reference for the result? By edge-chromatic ...
15
votes
2answers
774 views

A set of integers whose factorial can be written as a product of two factorials

I am trying to collect informations concerning the set $$\mathcal{A}=\left\{n\in\mathbb{N} \mid (\exists k,l\in\{2,3,\dots,n-2\})(n!=k!l!)\right\}.$$ It seems not much is known about the set ...
4
votes
1answer
65 views

Separate a special poset by function

Assume $A = \prod_{i=1}^n\{0,1\}$, i.e. element $(a_1,\cdots,a_n)=a\in A$ is n-tuples like $(1,0,1,\cdots)$. There is an obvious partial order on the $A$: say $a < b$ for $a,b\in A$ if and only ...
4
votes
0answers
154 views

The properties of Pos

Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as: $$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$ When given $n\in\mathbb{N}$, this function gives the 'position' of ...
1
vote
0answers
46 views

Discrete Mathematics Uses [closed]

I am trying to explain how and why discrete maths is used in areas such as programming, correctness, data types, state transistion and conditionals. I'm having a really hard time articulating it ...
2
votes
0answers
87 views

Growth of inner products between two random vectors on the sparse hypercube

We define the $s$-sparse hypercube in $\mathbb{R}^d$ as \begin{align} \mathbb{H}_s = \bigl \{ {\bf{v}} \in \{ -1, 0 , 1\}^d \colon \| {\bf{v}} \|_0 = s \bigr\}, \end{align} where $ \| {\bf v} \|_0 $ ...
2
votes
0answers
53 views

What do you call the collection of all sets shattered by $F$?

The proof of Pajor's lemma uses the collection of all sets $S\subseteq X$ shattered by some $F\subseteq 2^X$. Is there a standard term for the former object? I've been privately referring to it as the ...
11
votes
1answer
220 views

Generating function of a sequence is not algebraic

Let we have a sequence $\{a_{n}\}$, such that $\forall n \,\, a_{n}>0$ and $a_{n} \rightarrow\infty, n\rightarrow\infty$. Also let's suppose that we have a subsequence $\{a_{n_{k}}\}$ such that ...
1
vote
1answer
114 views

Building the string on $\{0,1\}$ alphabet with $\Omega(n^{2})$ different substrings [closed]

As we know the number of different substrings has the upper bound $O(n^{2})$. Consider the strings on $\{0,1\}$ alphabet. Can I build a string with $\Omega(n^{2})$ different substrings? Actually I ...
0
votes
0answers
45 views

Approximation to colouring for bounded degree graphs

I have already asked one question on colouring, this question is more specific. Given a bounded degree graph $G$ with $\Delta(G)=2d$, is there a well know algorithm to achieve an approximation ratio ...
1
vote
1answer
76 views

Graph colouring for bounded degree graphs

I'm fairly new to colourings on bounded degree graphs i'm interested in the following questions, For planar graphs with bounded degree $4$ is finding the colouring number $NP$-hard? So is ...
40
votes
1answer
2k views

Why is the Frankl conjecture hard?

This is a naive question that could justifiably be quickly closed. Nevertheless: Q. Why is Péter Frankl's conjecture so difficult? If any two sets in some family of sets have a union that also ...
7
votes
1answer
812 views

Bounding the entropy of a convolution

Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...
3
votes
0answers
110 views

Find the number of boolean functions of n variable that satisfy the following condition

For how many boolean functions is this true? The length of the shortest disjunctive normal form of that functions is equal to 2^(n-1). And the the number of variable entries in the minimal dnf of that ...
2
votes
1answer
69 views

Directed edge-colouring

I'm interested to know whether the following problem is NP-complete or if there is an algorithm to solve it. Suppose we are given a directed graph $G=(V,E^{\rightarrow})$ and we want to colour the ...
1
vote
0answers
37 views

Is the complement of a vertex figure in an (abstract) polytope connected?

I consider an (abstract) regular polytope $P$, and $H$ a vertex figure of $P$. Is the complement $P \setminus H$ connected (as a poset, in the sense that its Hasse diagram, ignoring the improper ...
1
vote
0answers
14 views

Maximization of the difference of a monotone submodular function and a linear function with a cardinality constraint

Maximizing a monotone submodular function with a cardinality constraint can be solved by using a simple greedy heuristic. However, if the submodular function is non-monotone, the greedy heuristic can ...
7
votes
1answer
147 views

Factoring a multiset into a product of two multisets

Given a multiset $S$ of $mn$ numbers, how hard is to find multisets $A$ and $B$, of $m$ and $n$ numbers respectively, such that $$ S = \{ xy \mid x\in A, y\in B \}~~~~\text{(multiset sense)}, $$ or ...
11
votes
2answers
655 views

What is the probability that two random permutations have the same order?

I am interested in the orders of random permutations. Since the law of the logarithm of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), ...
2
votes
1answer
86 views

Orthogonal embeddings and edge lengths

I'm interested in orthogonal embeddings of graphs into the 2-dimensional, i.e where vertices are placed at integer co-ordinates and edges are routed along the grid lines and are not allowed to ...
1
vote
0answers
26 views

Some confusion regarding the definition of NPO reduction

I've seen the following definition in a paper on approximation preserving reductions. Definition:Let $\pi_{1}$ and $\pi_{2}$ be two NPO maximization problems. Then we say that $\pi_{1} \leq_{R} ...
1
vote
1answer
115 views

Determinant of discrete Laplacian

It can easy be shown by induction that the determinant of the $(N-1)\times (N-1)$ matrix $$\begin{pmatrix} 2 & -1 & & \\ -1 & 2 & \ddots & \\ & ...
5
votes
3answers
331 views

Fundamental solution of Discrete Laplace in the plane

We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator Then, it is natural to consider its fundamental solution $u$, i.e. ...
13
votes
7answers
2k views

Special arithmetic progressions involving perfect squares

Prove that there are infinitely many positive integers $a$, $b$, $c$ that are consecutive terms of an arithmetic progression and also satisfy the condition that $ab+1$, $bc+1$, $ca+1$ are all perfect ...
1
vote
0answers
61 views

Building an orthogonal embedding for a 4-planar graph

I'm interested in the following paper http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf In particular i'm interested in the construction Valiant describes to prove that it is possible to ...
7
votes
1answer
250 views

Recursive sequence of binomial random variables

Fix $p>0$ and define a recursive sequence of random variables with $X_1 =1$ and $$X_{k+1} = X_k + \text{Bin}(X_k,p).$$ Thus, $\mathbf E [ X_k ] = (1+p)^k$. I would like a left tail bound. Perhaps, ...
11
votes
7answers
714 views

Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...
2
votes
2answers
350 views

Minimally intersecting subsets of fixed size

The question I have, is how to generate the following collection of subsets: Given a set $S$ of size $n$. I want to find a sequence of $k$ subsets of fixed size $m$, $0<m<n$, such that at each ...
2
votes
1answer
146 views

Alternating sign binomial identity [closed]

I recently noticed that for a triple of integers $k \geq 2$, $k \geq m \geq t \geq 1$, the following identity seems to hold $\sum_{j=0}^{m-t} (-1)^{m-t-j}{k \choose j}{m-1-j \choose t-1}={k-t \choose ...
5
votes
4answers
196 views

Selecting subsets with size $\frac{n}{2}$ covering every pair of the elements

Given a set $S$ of $n$ elements. Let $T$ be the set of all subsets of $S$, with size $\frac{n}{2}$ ($n$ is even). We want to select a subset $T'$ of $T$, with the property that for any pair of the ...
4
votes
1answer
279 views

Number of different positions of rooks on chessboard

I know that this topic as been mentioned before, but no accurate answer has been provided. Suppose we have to place $n$ rooks on $n \times n$ chessboard so that no one attacks another. How to count ...
2
votes
3answers
350 views

An identity involving a product of two binomial coefficients

I'm trying to find a closed formula (in the parameters $q,N$) for the following sum: $$ \sum_{k=q}^{N} {{k-1}\choose{q-1}} {{k}\choose {q}} $$ Can anybody give me a lead? Lior
5
votes
1answer
160 views

Intersection of rotating regular polygons

This question has a recreational flavor, but may not be entirely uninteresting. Let $P_k$ be a unit-radius regular polygon of $k$ sides, and $P_n$ a unit-radius regular polygon of $n \ge k$ sides. ...
1
vote
1answer
90 views

Discrete summation of Gaussian functions. Decay time problem

I am facing the following problem. I have a function which is defined through a discrete sum of Gaussians $$F_M(t) = 2\sum\limits_{n=1}^{M}e^{-t^2 \sigma^2 n^2}\times \sum\limits_{k=n}^{M}p_k p_{k-n} ...
0
votes
0answers
63 views

Factorial Sums over Compositions or ``Unlabeled Permutations"

Let $C_n$ denote subset of integer compositions of $n$ and $c=(c_1,c_2,\dots c_n)$ In a divergent sum, the sequence $$ a_n=\sum_{c\in C_n} \prod_{c_i\in c} c_i! $$ frequently shows up and one ...
6
votes
1answer
655 views

Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$ such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...
2
votes
1answer
633 views

Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...
3
votes
0answers
126 views

Generating function of a sequence involving reciprocals of binomial coefficients

Question: Is there a closed-form expression for the following sum $$ F(z,k,r)=\sum_{n=0}^{r} \frac{z^n}{{n+k} \choose {k}}\label{sum}\tag{1} $$ where $z\in\mathbb{C}$, and $r$, $k$ are non-negative ...
0
votes
0answers
92 views

Summing up costs over a Markov chain

I apologize in advance if this question is too simplistic to be appropriate for MathOverflow. I have inquired in multiple places but have found little to indicate that this is a previously studied ...
6
votes
1answer
94 views

Above/below directed graph on cells of arrangement of lines

This question concerns the structure of a directed graph built on the cells of an arrangement of lines. My basic question is whether this graph has been studied before, perhaps in another guise. I ...
1
vote
0answers
42 views

Finding optimal set of permutations [closed]

I have the following data set of a human population. The data set captures households and relationships of the persons living in those households. My problem is how to group the individuals into ...
3
votes
0answers
66 views

What is the maximal number of partitions with this maximal intersection property?

Let $X = \{ 1, \dots, n = sk \}$ be a finite set. Let $\mathscr P, \mathscr Q$ be equi-partitions of $X$ into $k$ sets of size $s$. Denote by $V(\mathscr P, \mathscr Q)$ the maximum size of ...
3
votes
3answers
140 views

Simplifying a Taylor polynomial that involves Stirling numbers of the second kind

I am currently trying to evaluate a complicated function $f$ at a point $x+a$ using a high order Taylor polynomial about the point $a$. The polynomial has the standard form: $$\sum_{n=1}^k ...
8
votes
2answers
201 views

Repeats of all binary strings of length k

The question seems like it should be known, but I was not able to find it anywhere. How many binary strings of length $n$ are required so that for every $k$ positions in these strings, all $2^k$ ...
10
votes
1answer
250 views

Finding combinatorial models / statistics

In many cases in combinatorics and especially algebraic combinatorics with some representation theory, the main problem is about finding the correct statistic on a mathematical object. For example, ...
7
votes
1answer
885 views

Integer solution to special system of linear equations

This problem appear in my research, but I am unable to solve it. There should be an easy argument, but I have not yet found it. Informal version An integer $k\geq 2$ is fixed. We are given a matrix ...