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0
votes
1answer
125 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) = ...
19
votes
5answers
756 views

Three-halves-free words (analogous to square-free)

A square-free word is a string of symbols (a "word") that avoids the pattern $XX$, where $X$ is any consecutive sequence of symbols in the string. For alphabets of two symbols, the longest square-free ...
6
votes
3answers
379 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 ...
15
votes
5answers
2k views

Has any attempt been made to classify finite groupoids?

I recently stumbled upon the Mathieu groupoid and I found them fascinating. It appears as a subset of $S_{13}$ which is not closed under multiplication, but it turns out to be a groupoid with 13 ...
0
votes
1answer
463 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 ...
2
votes
1answer
96 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 ...
5
votes
1answer
289 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
144 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
262 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 ...
2
votes
1answer
80 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
132 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. ...
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 ...
2
votes
1answer
207 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
202 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
181 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 ...
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 ...
7
votes
1answer
236 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
142 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 ...
13
votes
1answer
279 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 ...
7
votes
6answers
2k views

Discrete Mathematics textbooks for undergraduates

For the first time, I will be teaching a course on Discrete Mathematics for electrical and computer undergraduates students. I intend to focus on practical applications. I would be grateful if ...
7
votes
2answers
402 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
270 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
126 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 ...
0
votes
1answer
288 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
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
163 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 ...
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 ...
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 ...
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
145 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
179 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
294 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 ...
7
votes
1answer
353 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 ...
7
votes
2answers
288 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

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 ...
14
votes
1answer
664 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 ...
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 ...
19
votes
1answer
272 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 ...
2
votes
1answer
112 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 ...
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 ...
1
vote
1answer
304 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 $a_n$ ...
2
votes
2answers
296 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
847 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 ...
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 ...
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
152 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 ...
-1
votes
1answer
311 views

Number of blocks in a t-(v,k,l) design with empty intersection with a given set U [closed]

Question Given a $t-(v,k,\lambda)$ design $(X,\mathcal{B})$ and a set $U\subset X$ with $|U|=u\leq t$, what is the number of blocks $B\in\mathcal{B}$ such that $B\cap U=\emptyset$? The answer is: ...