All Questions
Tagged with design-theory or combinatorial-designs
125 questions
0
votes
1
answer
118
views
Configurations of signs in a matrix under certain conditions
I have a combinatorial question which is out of my research area.
Given a $2^k\times 2^k$ matrix $A=[a_{i,j}]$ with entries in $\lbrace0,\pm1\rbrace$, where $k$ is a positive integer. Is it possible ...
- 619
2
votes
0
answers
187
views
Matrix with elementary symmetric polynomials as entries
Let $n\geq 1$, and for each $j=1,\ldots, n+1$ let $\mathbf{X}_{j}=(X_{j1},\ldots, X_{jn})$ be $n$ variables. Let $M$ be the $(n+1)\times (n+1)$ matrix whose $(i,j)$-th entry is $$M_{ij}=(-1)^i e_{i-1}(...
4
votes
1
answer
170
views
About $CW(512,16^2)$
Definitions: A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{ij}$ in $\{0, \pm 1\}$ such that $WW^T=kI$,
where $I$ is the identity matrix. A circulant ...
- 696
5
votes
2
answers
189
views
Bisymmetric Hadamard matrices
Definitions: An $n\times n$ Hadamard matrix is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal.
A symmetric matrix is a square matrix that is equal to its own ...
- 696
6
votes
0
answers
111
views
What is the largest subgraph of the Kneser graph which has a small chromatic number?
While trying to characterize constraint satisfaction problems which can be solved by the Linear Programming relaxation, I've run into a few perplexing puzzles related to the existence of certain ...
- 8,688
15
votes
1
answer
886
views
Scrambling a “Connections” grid
Given a 4-by-4 array of distinct words, is it possible to scramble the array in four different ways in such a fashion that each possible word-pair appears adjacently in one of the five arrays (the ...
- 19.7k
3
votes
1
answer
294
views
Smallest number of subsets whose squares cover the whole square
Let $2 \leq k \leq n$ be integers, let $[n] := \{1,2,\ldots,n\}$, and for a subset $A \subseteq [n]$ let $A^2 := A \times A$ be the Cartesian product of $A$ with itself and let $|A|$ denote the ...
- 6,351
4
votes
0
answers
89
views
Software reference for combinatorial design
If one were to require quick and easy access to sizeable latin squares, room squares, Steiner systems, designs, balanced block designs... where to look, what software to use?
- 1,461
4
votes
0
answers
77
views
Existence of finite 3-dimensional hyperbolic balanced geometry
Together with @TarasBanakh we faced the problem described in the title. Let me start with definitions.
A linear space is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of ...
- 501
2
votes
2
answers
320
views
A graphic representation of classical unitals on 28 points
I would like to understand the geometry of the classical unitals.
They are block designs containing $q^3+1$ points and whose blocks have cardinality $q+1$, where $q$ is a prime power. For $q=2$ (if I ...
- 41.8k
5
votes
5
answers
572
views
Is every uniform hyperbolic linear space infinite?
I start with definitions.
Definition 1. A linear space is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ satisfying three axioms:
(L1) for any distinct ...
- 41.8k
11
votes
1
answer
390
views
Does every finite affine plane have the doubling property?
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
- 41.8k
2
votes
1
answer
98
views
One question about nega-cyclic Hadamard matrices
Let $n$ be a multiple of $4$, is there any $n \times n$ negacyclic Hadamard matrix? If yes - how to construct it? If no - why?
Here an $n \times n$ nega-cyclic matrix is a square matrix of the form:
\...
- 696
5
votes
1
answer
141
views
On the half-skew-centrosymmetric Hadamard matrices
Definition 1: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.
Definition 2: A matrix $A$ is half-skew-centrosymmetric if ...
- 696
0
votes
1
answer
157
views
"JigSaw Puzzle" on Set Family
One of my research problem can be reduced to a question of the following form
Given a set family $\mathcal{F}$ of $[n]$ , such that every element of $[n]$ lies in exactly $K$ sets in $\mathcal{F}$, ...
- 384
0
votes
0
answers
84
views
Bounds for smallest non-trivial designs
Given $s>t\ge 2$, let $N(s,t)$ be the smallest integer $n>s$ such that there exists an “$(n;s;t;1)$-design” (i.e., a collection of $s$-subsets $e_1,\dots,e_m$ of $[n]:=\{1,\dots,n\}$, such that ...
- 3,499
1
vote
1
answer
136
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On the existence of symmetric matrices with prescribed number of 1's on each row
We are considering the following problem:
Given an integer $n$ and a sequence of integers $r_i,\ 1\le i\le n$, with $0\le r_i\le n-1$ does there exists a symmetric matrix $A$ such that the diagonal ...
- 13
6
votes
1
answer
151
views
How to construct a skew Hadamard matrix of order 756?
Where can I find the construction for a skew Hadamard matrix of order 756?
According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard matrices and Seberry - On skew Hadamard ...
- 261
10
votes
1
answer
319
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Construction of skew-Hadamard matrix of order 292
I am currently looking into how to construct a skew-Hadamard matrix of order 292. Where can I find such construction?
According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard ...
- 261
11
votes
2
answers
661
views
$\mathbb Z/p\mathbb Z=A\cup(A-A)$?
$\newcommand{\Z}{\mathbb Z/p\mathbb Z}$
Can one partition a group of prime order as $A\cup(A-A)$ where $A$ is a subset of the group, $A-A$ is the set of all differences $a'-a''$ with $a',a''\in A$, ...
- 23k
3
votes
1
answer
153
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Cycling through a general combinatorial design on $\omega$
This is a generalisation of an older question inspired by a football tournament (which does not have an answer yet).
Let $\frak P$ be a partition of $\omega$ into blocks, that is, pairwise disjoint ...
- 48.9k
1
vote
0
answers
104
views
On a combinatorial design inspired by a football (soccer) tournament
Real-world inspiration. My younger son was playing a micro football (soccer) tournament this afternoon with $3$ other friends. Let's label the $4$ kids $0,1,2,3$. They played $3$ matches:
$\{0,1\} \...
- 48.9k
1
vote
1
answer
136
views
3-partition of a special set
$S_5$ is a set consisting of the following 5-length sequences $s$: (1) each digit of $s$ is $a$, $b$, or $c$; (2) $s$ has and only has one digit that is $c$.
$T_5$ is a set consisting of the following ...
- 47
5
votes
0
answers
911
views
The existence of big incompatible families of weight supports
In 2018 Mario Krenn posed this originated from recent advances in quantum physics question on a maximum number of colors of a monochromatic graph with $n$ vertices. Despite very intensive Krenn’s ...
- 5,409
8
votes
3
answers
433
views
Latin squares with one cycle type?
Cross posting from MSE, where this question received no answers.
The following Latin square
$$\begin{bmatrix}
1&2&3&4&5&6&7&8\\
2&1&4&5&6&7&8&3\\...
- 168
4
votes
1
answer
81
views
k-partite design
Is the following true? For every $n \geq 1, k\geq 2$, there is a set $S \subseteq [n]^k$ of size $|S| = n^2$ such that every two $k$-tuples in $S$ have at most one common entry.
Does anyone know if ...
- 428
1
vote
0
answers
56
views
Are sharper lower bounds known for these potentials on the sphere?
Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that
$$
\sum_{i=1}^n\sum_{j=1}^n(x_i\cdot x_j)^{2\ell}\geq\frac{(2\ell-1)!!(d-2)!!}{(d+2\ell-2)!!}\cdot n^2,
$$
with ...
- 7,633
3
votes
0
answers
54
views
Cliques in Incomplete block designs
I'm interested in inequalities that guarantee the presence of cliques in incomplete block designs. Here's the set-up:
I have an incidence structure $(V, B)$ which is an incomplete block design: $V$ is ...
- 11.2k
3
votes
0
answers
148
views
Linear combinations of special matrices
I am a hobby computer scientist and I have a problem to which I am searching an efficient algorithm.
Given an integer n, we want to combine some square input-matrices of size n in a way that is ...
- 73
4
votes
3
answers
780
views
Does an $(x, bx)$-biregular graph always contain a $x$-regular bipartite subgraph?
I guess a discrete-mathematics-related question is still welcome in MO since I was new to the community and learned from this amazing past post. The following claim is a simplified and abstract form ...
- 43
1
vote
0
answers
51
views
Optimal choice of points to maximize majorities in a $t-(v,k,\lambda)$ design
Let us consider a design $\mathcal{D} = (V,\mathcal{B})$ with points in $V$ and blocks in $\mathcal{B}$. I am interested in the special case of a $t-(v,k,\lambda)$ design for $k=3$, i.e., all blocks ...
- 143
12
votes
1
answer
194
views
Ternary sequences satisfying $ x_i + y_i = 1 $ for some $ i $
Consider a set of strings $ {\mathcal S} \subset \{0, 1, 2\}^n $ satisfying the following two conditions: 1.) every string in $ {\mathcal S} $ has exactly $ k $ symbols from $ \{0, 1\} $ (i.e., $ \...
- 503
11
votes
5
answers
506
views
What are efficient pooling designs for RT-PCR tests?
I realize this is long, but hopefully I think it may be worth the reading for people interested in combinatorics and it might prove important to Covid-19 testing. Slightly reduced in edit.
The ...
- 14.4k
3
votes
1
answer
150
views
Mutually orthogonal Latin hypercubes
A $d$-dimensional Latin hypercube with side length $n$ is a $d$-dimensional array with $n$ symbols such that along any line parallel to an axis, each symbol appears exactly once. Let us call a $(n,d)$ ...
- 675
3
votes
1
answer
83
views
For which sets of $(n, m, k)$ does there exist an edge-labelling (using $k$ labels) on $K_n$, such that every single-labelled subgraph is $K_m$?
Or, equivalently - for what sets of $(n, m, k)$ is it possible, for a group* of $n$ people, to arrange $k$ days of "meetings", such that every day the group is split into subgroups of $m$ people, and ...
- 131
1
vote
1
answer
77
views
Does there exist a non-degenerate symmetric combinatorial 3-design?
Is there a non-degenerate 3-design where the number of blocks equals the number of points?
Non-degenerate in this context means that a point is incident with at least 2 and at most #blocks-2 blocks.
- 11
2
votes
0
answers
85
views
Packings with block size equal to $6$?
In design theory the following is the defintion of a packing :
Definition : A $(v,k)$-packing is a pair $(V, \mathcal{B})$ of a finite set $V$ of cardinality $\vert V \vert = v$ and a finite set $\...
- 223
4
votes
3
answers
240
views
Best strategy for a combinatorial game
Consider the following scenario. We have 20 balls and 100 boxes. We put all 20 balls into the boxes, and each box can contain at most one ball.
Now suppose we are given 5 chances to pick 20 out of ...
- 281
5
votes
2
answers
211
views
Coloring in Combinatorial Design Generalizing Latin Square
I have a question about a combinatorial design very similar to a Latin Square, which is arising out of an open problem in graph theory. The design is an $n \times n$ matrix whose entries we want to ...
- 439
3
votes
0
answers
130
views
Graeco-Latin squares and outer-automorphisms
It is well known that $n=6$ is the only number greater than two in which there is no Graeco-Latin square of order $n$. It is also well known that $n=6$ is the only number greater than two in which ...
- 2,649
10
votes
2
answers
362
views
Lower bound for a combinatorial problem ($N$ students taking $n$ exams)
We have $N$ students and $n$ exams. We need to select $n$ out of the students using the grade of those exams. The procedure is as follows:
1- We set some ordering on the exams.
2- Going through this ...
- 628
2
votes
1
answer
60
views
Constructing Group Divisible Designs - Algorithms?
I am starting my research on group divisible designs this year and I wonder if there are any algorithms/software that help with constructions.
Thank you
- 29
4
votes
1
answer
112
views
Bounding the number of orthogonal Latin squares from above
As is usual, let $N(n)$ denote the maximum size of a set of mutually orthogonal Latin squares of order $n$. I am wondering what results hold that bound $N(n)$ from above; the only ones I can think of ...
- 513
3
votes
1
answer
120
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On the existence of a certain graph/hypergraph pair
Let $V$ be a finite set, $G$ a simple graph with vertex set $V$, and $H$ a hypergraph (i.e., set of subsets) with vertex set $V$ satisfying the following three conditions:
each pair of elements of $V$...
- 24.2k
2
votes
1
answer
130
views
Distinguishing points by sets of given size
The problem is:
Given a finite set $X$ with size $x$ and let $B$ denote a family of $k$-element subsets of $X$, called blocks. What is the smallest possible number $n$ of blocks such that every ...
- 9,501
8
votes
2
answers
577
views
Pfaffian representation of the Fermat quintic
It is known (see for instance Beauville - Determinantal hypersurfaces) that a generic homogeneous polynomial in $5$ variables of degree $5$ with complex coefficients can be written as the Pfaffian of ...
- 7,300
4
votes
2
answers
262
views
Known result about existence of $n$-vertex $k$-uniform $r$-hypergraphs?
Are there known results about $n,k,r$ such that $n$-vertex $k$-uniform $r$-regular hypergraphs exist? If this is too large a class of hypergraphs, what if $k=\tilde{\theta}(\sqrt{n})$? What if an ...
- 281
7
votes
2
answers
434
views
Minimum covers of complete graphs by $4$-cycles
I am interested in coverings of the (edge set of the) complete graph $K_n$ by cycles of length $4$. It is clear that such coverings exist for each $n \ge 4$. I need to find the minimum number of $4$-...
- 337
2
votes
1
answer
100
views
Relative difference sets?
Do you know how to find order of known groups with RDS or without? Or Known groups which have RDS or not? If there is a references to survey all researches?
user91523
3
votes
1
answer
222
views
Has this kind of design been studied before?
Consider a design $(X,\mathcal{B})$, satisfying:
Each block in $\mathcal{B}$ has the same size
The intersection of every two blocks has the same size
Of course, it is easy to find many examples of ...
- 63