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-2
votes
1answer
44 views

Show that among any 6 non-negative integers one can find 2 integers so that their difference is divisible by 5 [on hold]

I have the following question in homework I have been assigned in Discrete Mathematics relating the pigeonhole principle: "Show that among any 6 non-negative integers one can find 2 integers so that ...
-3
votes
0answers
57 views

Permutations with fixd points [closed]

I am trying to write a java program that counts permutations of a string, I would like to check my results by hand, but I can remember (or find) the formula to count the number of permutations, and ...
2
votes
1answer
92 views

On a problem about $GF(2)^n$

For $A\subseteq {\mathbb F}_2^n$ let $$ Q(A)=\{\alpha+\beta\mid \alpha,\beta \in A,\ \alpha\neq\beta \}. $$ I want to prove or disprove that if $|A|=2^k+1$ for some integer $k$, then $$ ...
0
votes
0answers
14 views

On the stability analysis of a discrete difference system with multiplicative noise

If we assume that \begin{equation*} \rho \{\phi \otimes \phi+\psi \otimes \psi\}<1 \end{equation*} where $\rho$ denotes the spectral radius, then can we verify the following inequality holds ...
0
votes
0answers
37 views

Lattice isotopy type of uniform hyperplane arrangements

I am working on a problem related to the isotopy type of a certain class of hyperplane arrangements in $\mathbb{C}^{d}.$ For more references, compare Randell's work "Lattice-isotopic arrangements are ...
5
votes
1answer
270 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 ...
9
votes
1answer
143 views

Optimal shape for stabbing balls in $\mathbb{R}^3$

I have radius $r < \frac{1}{2}$ congruent balls with centers randomly distributed uniformly within a region, say, within a unit-radius sphere $S$. I shoot a ray/path through $S$, hoping to ...
3
votes
1answer
260 views

Number of semi-standard tableau

What is the number of semi-standard tableau (weakly increasing on rows and strictly increasing on columns) for the partition $2n=n+n$ with entries $\{1,2, \cdots ,n\}$ such that each $i$ appears ...
0
votes
1answer
106 views

Number of squares in a grid under certain conditions

Consider an $(n+1)\times (n+1)$ grid of lattice points in the plane. $A(n):$ # of squares with vertices on the grid. It's relatively well-known that $A(n)=\frac{n(n+1)^2(n+2)}{12}$. Now, $A(n) = ...
2
votes
1answer
77 views

Complex lines arrangements from a given wiring diagram

I'm working on some presentation of the fundamental group of the complement of a complex lines arrangement in $\mathbb{C}^{2}.$ In particular, in Arvola's article "The fundamental group of the ...
3
votes
1answer
130 views

Braid wiring diagrams and matroids

recently I started reading some articles about the presentation of the fundamental group of lines arrangements in $\mathbb{C}^{2}$ via Wiring diagrams. I also found some relation with matroid theory. ...
2
votes
1answer
203 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 ...
4
votes
1answer
199 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 ...
7
votes
0answers
180 views

Minimal “basis” in $n$ dimensional unit cube

Let's $$ B^n=\{\bar\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)|\alpha_i\in \{0,1\}\};~~~~n=1,2,\ldots $$ and let's $$ C\subseteq B^n, $$ $$ S(C)=\{\bar\alpha\oplus\bar\beta\ | \bar\alpha,\bar\beta\in ...
7
votes
1answer
233 views

Number of median graphs?

What is the number of $n$-vertex median graphs? These graphs generalize hypercubes and trees, and have many applications. It seems unlikely that a closed form expression is known, so I would also be ...
2
votes
1answer
140 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 ...
6
votes
0answers
89 views

Checking if a multiplication table represents a group [duplicate]

Given an $n \times n$ multiplication table, can one check if it represents a group in $o(n^3)$ time? All properties can be checked by mindless try-all possibilities loops: Whether there is an ...
7
votes
2answers
393 views

A continuous function for defining unique values to a 1024x1024 image with a 24 bit 3 color channel image

I need to generate a color map which I am not sure exist. I have a 1024x1024 image which would contain 2^20 pixels. I have 3 color channels which each have 8 bits which would leave us with 2^24 ...
6
votes
1answer
266 views

A variation on Bulgarian solitare

It appears that a variation on Bulgarian solitare has a fixed point regardless of the starting $n$. For example, let $n=69$, and consider this partition: $$ (8,8,7,7,5,5,5,5,5,4,3,3,2,2) $$ In ...
0
votes
1answer
117 views

Probability of k overlapping subsets in N trials

Ok, here is what I am attempting to find an answer to: I draw M uniformly random subsets of size K from the set of numbers $\Omega=\{1, \dots, N\}$ (where uniformly random means that each unique ...
1
vote
0answers
52 views

Cross-correlation of two functions which are not fixed

I am trying to cross-correlate two functions, but one of which is changing each 'step' of the cross-correlation. I want to cross-correlate T(f) and ...
0
votes
4answers
154 views

about the structure of components of tensor product if more than one bipartite graph is taken

I was reading about tensor product of graphs. We know that if we take tensor product of n graphs and want this product to be a connected graph then at most one graph should be bipartite. In the book ...
0
votes
0answers
81 views

Rank function and closure operator for a set system

I would like to trace the concepts "rank function" and "closure operator" back to some structures as primitive as possible. For a set system $(E,F)$ which is an independence system or a greedoid, I ...
16
votes
1answer
451 views

Longest of random worm-like paths in $\mathbb{Z}^2$

Imagine at each lattice point of $\mathbb{Z}^2$ within $[1,3n]^2$, with coordinates $\equiv 2 \bmod 3$, we place, with equal probability, one of these six patterns:       The result ...
34
votes
6answers
3k views

Is it easy to produce hard-to-color graphs?

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each ...
1
vote
0answers
35 views

Find a common expression (parameterized by y = 1/2, 1 and 2) uniting three (rational) polynomials in k [closed]

I am seeking a common "simple" expression (preferably/presumably a sum of products of linear factors, or sum of products of low-degree factors) uniting the three polynomials (parameterized by ...
1
vote
1answer
142 views

Calculate the discrete set of points B which are in the convex hull of the set of points A

This problem is likely best described with the following picture: Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...
1
vote
1answer
178 views

A square-squareroot integer race sequence involving primes

I wonder what is the expected behavior of this process? Let $f^2_{\mathrm{next}}(n) =$ the next prime after $n^2$. $g_{\mathrm{sqrt}}(n) = \lfloor \sqrt{n} \rfloor$. Now iterate as ...
11
votes
0answers
287 views

Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$. If there are $g(n)$ non-isomorphic groups of order $n$, ideally each group would occur with probability ...
0
votes
1answer
462 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 ...
7
votes
1answer
352 views

Is this a new formula? $\Delta^d x^n/d! = \sum_k \left[ x \atop k\right]{ k+n \brace x + d}(-1)^{x+k}$

$$\frac{\Delta^d x^n}{d!} = \sum_k \left[ x \atop k\right]{ k+n \brace x + d}(-1)^{x+k}$$ Where $x$, $n$ and $d$ are non-negative integers, $\Delta^d$ is the $d$-th difference with respect to ...
2
votes
1answer
110 views

Family of sets with unique subsets

I am given a set $M\subseteq\{1,\dots,n\},\,|M|=m$ and a famliy of $k$-sets ($k<m$) $\mathcal{U}=\{U_1,\dots,U_p\},\,U_{i}\subset M$. For this family, I would like to check one of the following ...
13
votes
1answer
270 views

Is there an unambiguous CFL whose complement is not context-free?

I'm doing a little bit of research about context-free languages. A question that's popped up is whether or not there exists an unambiguous context-free language whose complement is not a context-free ...
14
votes
1answer
658 views

Is every graph the center of some other graph?

The center of a graph $G$ is the set of vertices that minimize the largest distance to vertices in $G$, e.g., in the graph below, that radius is $4$:           Define the ...
7
votes
2answers
285 views

Iteration of a 2D map involving absolute value: phase transition?

I was looking at this map: $f(x,y) \mapsto (|x-y|,x)$, starting from some point with coordinates $(x,y) \in [0,1]^2$, and iterating: $(x,y),\, f(x,y), \, f^2(x,y), \,f^3(x,y), \ldots$. It displays ...
2
votes
1answer
110 views

Subdividing toward a unit distance graph in the plane

I just want to ask, what is the significance of subdividing a graph toward a unit distance graph? I have seen several studies of it but I cannot find what its importance. I mean, like the study of ...
6
votes
0answers
732 views

How many 2L-bit numbers are the product of two L-bit numbers?

If I multiply two integers $x, y $ in $[0,2^L)$, I get an integer in $[0,2^{2L})$. Clearly, this map from $[0,2^L) \times [0,2^L) \to [0,2^{2L})$ is not bijective. I am interested in the size of ...
15
votes
8answers
1k views

Two questions from combinatorics on words

Question 1. Assume that an infinite word $u\in\{0,1\}^{\mathbb Z}$ is not balanced. Is it true that there exists a finite 0-1 word $w$ such that $0w01w1$ or $1w10w0$ is a factor of $u$? Is it true ...
3
votes
2answers
286 views

Cardinality of intersection of a random subset with a fixed subset

How can I simply prove the following fact: Let $A := \{1, \dots n \}$ and $B := \{1, \dots, \lfloor \frac{n}{4} \rfloor \}$. Let $d \in (0,1)$ and let $R$ be a randomly choosen (with uniform ...
3
votes
1answer
271 views

Sequences with integral variances

This is a companion to my earlier question, Sequences with integral means. This new question is, frankly, not as interesting, but it feels necessary to complete the thought. Let $V(n)$ be the ...
21
votes
5answers
845 views

Sequences with integral means

Let $S(n)$ be the sequence whose first element is $n$, and from then onward, the next element is the smallest natural number ${\ge}1$ that ensures that the mean of all the numbers in the sequence is ...
19
votes
1answer
270 views

Hidden points in polygons

Let $h(n)$ be the largest number of mutually invisible points that can be located in a polygon $P$ of $n$ vertices. Two points $x$ and $y$ are mutually invisible if the segment $xy$ contains a point ...
1
vote
1answer
303 views

a conjecture about a specific subset of $S_n$

Let $n>3$ be a positive integer.We denote the symmetric group of $n$ elements by $S_n$ and the identity mapping by $id$. For every $f\in S_n$, $f(1,2,\ldots,n)=(a_1,a_2,\ldots,a_n)$, denote $a_n$ ...
1
vote
1answer
72 views

the “compact good array” in a “good $N^+$-cycle”

We use $N^+$ to denote the set of positive integers.For any finite array $A:(a_1,b_1),...,(a_k,b_k)$,where every $(a_i,b_i)\in N^+\times N^+$,we call $A$ is good if for every $i\in ...
5
votes
3answers
346 views

Computing the lexicographic indices of integer partition

If we order all the partitions of a integer in a lexicographic order, how can we compute the position of each partition in this order without having to explicitly list all other partitions that ...
0
votes
1answer
283 views

Indexing combinations with repetition

Let $\Sigma$ be a finite set of symbols with total order. Let $C_k$ be the set of all $k$-multiset (unordered collection of $k$ elements from $S$, with repetition allowed). We can order all the ...
1
vote
1answer
107 views

Transform $a\mathbf x+\mathbf b$, then make it $k$-sparse, resulting least modification?

Consider vector $\mathbf x = (x_1,x_2,\cdots,x_n)$, $a$ a scalar, $\mathbf b = (b_0,\cdots,b_0)$ and $k < n$. I want to transform $\mathbf x$ ($\mathbf y = a\mathbf x+\mathbf b$) such that if I ...
3
votes
1answer
108 views

Obtaining each set (hyperedge) in a set system (hypergraph) as a union of sets in a smaller set sytem

Let $V$ be a set and $E$ a set of subsets of $V$. I'd like to know the proper terminology for the following concept. Let me call it "generator". A generator is a set $F$ of subsets of $V$ such that ...
3
votes
1answer
149 views

Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
0
votes
0answers
140 views

Lattice basis reductions and finding minimal values

While reading several articles about lattice basis reduction I am left with a few questions. For one, I came across this piece of text Let $\alpha$ and $\beta \in \mathbb{R}$. Also let $X>0$ and ...