Tagged Questions

For questions about Combinatorial design theory, a part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. The theory has applications in the area of the design ...

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Existence of a (40,13,4)BIBD (Balanced Incomplete Block Design)

I have been asked to prove that there exists a (40,13,4)BIBD. I admittedly have no idea where to start with this. Checking some of the necessary conditions for BIBDs shows me that if such a BIBD ...
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Points necessary to intersect all lines in finite projective geometry

I'm reading about finite geometries, projective and affine. I wonder what the smallest set of points is, given a geometry $PG(d,q)$, that intersects all lines. (or hyperplanes.) For example in the ...
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covering subsets

Let $A=\left\{ {1, 2, \ldots, n}\right\}$. Let $B$ be the set of all size $m$ subsets of $A$. $B=\left\{{B_1,B_2, \ldots , B_{\binom{n}{m}} } \right\}$, $|B_i|=m$ then we want to find $k$ subsets ...
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Rotation of 15 people at five tables

I have five tables of four people each. At each table is a table leader who remains stationery. How do I rotate the 15 participants so that they get to meet new people each time they rotate?
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Integrality conditions and proof by double counting.

Theorem $\mathbf{3.4.}$ In a block design of type $2-(v,k,\lambda)$ every element lies in precisely $r$ blocks, where $$r(k-1)=\lambda(v-1)\textit{ and }bk=vr\;.$$ The letter $r$ stands for â€œ...
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Constructing $\lambda$-difference sets. Please help.

Given a set say $A=${$0,1,4,16,r$} which is a subset of $\mathbb{Z}_{21}$. How do I find r, such that $A$ is a $\lambda$-difference set for some $\lambda$? Is there some methodical way to solve ...
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I stock in middle of proving that if $A$ is matrix of symmetric block design and $B = 2A - J$ that $J$ is ones matrix then B is a Hadamard matrix if and only if $v = 4(k-\lambda)$. I need to prove ...
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Construction of a (3v-2,3,1)-BIBD

I want to construct a $(3v - 2,3,1)$-BIBD from a $(v,3,1)$-BIBD and a quasigroup of order $v-1$. Attempt: For new BIBD, firstly, I need $3v-2$ points. Let $A$ be a block in $(3v - 2,3,1)$-BIBD and ...
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Is there a formula to rotate users through 2 positions without repeating weeks, repeating positions, or being paired with the same user?

I need to rotate through users to do 2 jobs without doing the same job for consecutive times, without doing the same job twice in a row, and without being paired with the same person twice. Is there ...