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3
votes
1answer
137 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 ...
17
votes
2answers
741 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 ...
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 ...
11
votes
0answers
270 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 ...
20
votes
5answers
640 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 ...
14
votes
1answer
644 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 ...
13
votes
4answers
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 ...
6
votes
1answer
648 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 ...