**35**

votes

**6**answers

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 ...

**27**

votes

**5**answers

3k views

### How many binary operations are associative?

Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be ...

**21**

votes

**5**answers

911 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 ...

**20**

votes

**1**answer

327 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 ...

**19**

votes

**5**answers

882 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 ...

**19**

votes

**2**answers

667 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 ...

**18**

votes

**2**answers

790 views

### Is there a discrete Cerf theory?

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...

**17**

votes

**1**answer

508 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 ...

**16**

votes

**5**answers

3k 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 ...

**15**

votes

**7**answers

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 ...

**14**

votes

**1**answer

697 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 ...

**14**

votes

**3**answers

809 views

### Partitioning the integers $1$ through $n$ so that the product of the elements in one set is equal to the sum of the elements in the other

I asked this question at math.SE a couple of months ago and only got a partial answer, so I thought I would try here.
It is known that, for $n \geq 5$, it is possible to partition the integers ...

**13**

votes

**7**answers

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 ...

**13**

votes

**2**answers

581 views

### Bit String Bijection

I am searching for a bijection between two types of bit strings (strings of 0's and 1's) both of even length (2n).
The restriction on the first type of bit string is that they must have the same ...

**13**

votes

**1**answer

403 views

### Minimal “sumset basis” in the discrete linear space $\mathbb F_2^n$

For a set $C\subseteq \mathbb F_2^n$, let $2C=C+C:=\{\alpha+\beta\colon \alpha,\beta\in C\}$.
I want to find $C$ of the smallest possible size such that $2C=\mathbb F_2^n$. Let $m(n)$ be the size of a ...

**13**

votes

**1**answer

468 views

### Does erosion mix faster than a riffle shuffle?

It is a famous result of Aldous and Diaconis1 that
seven shuffles are necessary and suffice to approximately
randomize 52 cards.2
Here the shuffles are the standard riffle shuffle, where the ...

**13**

votes

**1**answer

438 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 ...

**11**

votes

**7**answers

675 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 $n$ ...

**11**

votes

**2**answers

332 views

### The most number of points that realize only $k$ distinct distances

For $k \ge 1$, let $f_d(k)$ be the largest possible number of points $p_i$
in $\mathbb{R}^d$ that determine at most $k$ distinct (Euclidean) distances,
$\|p_i-p_j\|$.
Example. For points in the plane ...

**10**

votes

**12**answers

1k views

### Continuous notions with compelling discrete analogues

Following up on the previous MO question "Are there any important mathematical concepts without discrete analogue?", I'd like to ask the opposite: what are examples of notions in math that were not ...

**10**

votes

**2**answers

600 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), ...

**10**

votes

**1**answer

241 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, ...

**10**

votes

**1**answer

153 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 ...

**9**

votes

**5**answers

540 views

### Minimal blocking objects with shadows like a cube

This is a more geometric version of the previous question,
"Lattice-cube minimal blocking sets". I will first specialize to $\mathbb{R}^3$, $d=3$.
View an $n \times n \times n$ cube $C_3(n)$ as ...

**9**

votes

**1**answer

434 views

### Is this (funny) combinatorial optimization problem NP-hard ? (cutting numbers and placing them in urns)

The parameters of the problem are $m$ numbers which are integers (these numbers are denoted $b_i$), $n$ urns and in each urn, we can place $C$ numbers. We assume $nC \geq m$ so that the problem is ...

**8**

votes

**3**answers

1k views

### How to characterize a Self-avoiding path.

I cannot find any answer to that apparently simple problem :
On a square lattice, a path is given by a sequence of relative moves in {"move forward", "turn right" and "turn left"}.
Is there a rule ...

**8**

votes

**2**answers

496 views

### Three half circles on the plane may not meet nicely

Let $H$ denote the union of the northen hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$
...

**8**

votes

**2**answers

194 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 $2^k$ ...

**8**

votes

**2**answers

1k views

### palindromic subsequences

I'd like any insight or references to the following two conjectures (see the glossary below for definitions):
Conjecture 1: For any string $x$, there exists a longest common subsequence of $x$ and ...

**8**

votes

**1**answer

375 views

### Die-rolling Hamiltonian cycles

Let $R$ be a rectangular region of the integer lattice $\mathbb{Z}^2$,
each of whose unit squares is labeled with a number
in $\lbrace 1, 2, 3, 4, 5, 6 \rbrace$.
Say that such a labeled $R$ is ...

**8**

votes

**1**answer

360 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

**9**answers

473 views

### What are some early examples of creation of lists / catalogues of (particularly) combinatorial objects?

A lot of effort in discrete maths / combinatorics is expended in the construction of lists, catalogues or census [sic] of combinatorial objects such as groups, graphs, designs etc. These catalogues ...

**7**

votes

**2**answers

307 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 ...

**7**

votes

**6**answers

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

**1**answer

820 views

### A Question about Palindromic Numbers and System of Arithmetic Progression

Based from Harminc and Sotak's result, www.fq.math.ca/Scanned/36-3/harminc.pdf
We know that under certain condition, an arithmetic progression can contain an infinitely many palindromes.
My question ...

**7**

votes

**1**answer

6k views

### How to compute KL-divergence when PMF contains 0s?

From the Wikipedia page on Kullback-Leibler divergence, the way to compute this metric is to utilize the following formula:
The way I understand this is to compute the PMFs of two given sample sets ...

**7**

votes

**1**answer

847 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 ...

**7**

votes

**2**answers

505 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 ...

**7**

votes

**1**answer

271 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 ...

**7**

votes

**3**answers

544 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 ...

**7**

votes

**1**answer

238 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, ...

**7**

votes

**1**answer

126 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 ...

**6**

votes

**3**answers

553 views

### Cylinders dividing $\mathbb{R}^{3}$

Consider $n$ affine copies of a compact cylinder, say $S^{1}\times [-3,3]$ with top and botom, sitting inside $\mathbb{R}^{3}$.
For each $n$ we may ask ourselves how to arrange the $n$ cylinders so ...

**6**

votes

**3**answers

866 views

### References on techniques for solving equations with discontinuous functions such as floor and ceiling?

Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good ...

**6**

votes

**1**answer

453 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 ...

**6**

votes

**1**answer

280 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 ...

**6**

votes

**1**answer

392 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 ...

**6**

votes

**1**answer

88 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 ...

**6**

votes

**1**answer

553 views

### Calculating a specific joint probability involving sums of binomial distributions

The following might look like a simple problem - but the question has been unanswered for more than a week on math.stackexchange.com, and I have asked quite a few of the Ph.d. students at our ...

**6**

votes

**0**answers

91 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 ...