# Questions tagged [combinatorial-designs]

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 of experiments.

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### How many locks and keys: combinatorics problem [duplicate]

A village keep all their most precious belongings in a vault. The vault has a certain number of locks, each lock with an individual and specific key. The people in the village want to make sure that ...
1 vote
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### List of 3 unique triplets from a group of 9 where every row is unique as well.

Say we have 9 people. Every week, they will form into groups of 3. The condition is that the triplet that is formed must never have been seen together before. So let's say for week 1, we have: ABC, ...
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### Combinatorial scheduling problem

In optimising a parallel computer program, I need to solve what looks like a simple combinatorial problem, but it's driving me nuts. There are N*(N-1)/2 (i,j) tuples of all combinations of integer 0..(...
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### Generating groups without repetition

I got curious about this when making groups for a tournament. We have have 11 players, and the tournament consists of 4 rounds. In every round, there will be three groups. Two groups of 4 people, and ...
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### How to arrange numbers on grid to satisfy a minimum condition?

Take an $N \times M$ rectangular grid and arrange the integers from $1$ to $N M$ so that all grid point gets an assignment without repetition, and let the integer number on location (grid point) $n,m$...
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### Question dealing with partitions of a set with N elements into classes with 2 elements

I stumbled upon a maths problem wich I need To solve for a current paper I am writing. Have you seen this before? Is it solved and more concretely is there an efficient algorithm for this problem? If ...
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### Can we characterize the “associate classes” of a unipotent quasi-commutative quasigroup as some combinatorial design?

$I_n$ is the $n \times n$ or order $n$ identity matrix, $J_n$ is the order $n$ matrix of all ones, and $n \in \mathbb{Z}^+$. We define a Latin square $\mathcal{L_n}$ to be a set of $n$ permutation ...
1 vote
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### Can there be more than $\log_2(n)$ mutually orthogonal $(\pm1)$-vectors $x$ in $\mathbb{R}^n$ such that $x_{2k-1} \neq x_{2k}$ for all $k$?

A Hadamard matrix of order $n$ is an $n \times n$ matrix with entries $\pm1$ such that any two rows are mutually orthogonal. Any Hadamard matrix must necessarily have order equal to $1$, $2$, or a ...
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### Generate groups for multiple match rounds while minimizing the number of times two participants are in the same match [closed]

We are hosting an event with 100 participants. $20$ Participants can participate in one Match. $5$ Matches with $20$ Participants each will be called one Round. (Each participant is only playing one ...
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### Need to split 2n teams into n different games with no team meeting another team or playing a game more than once

I've found multiple similar questions, but none answering this case, and I can't manage to extrapolate from the other answers. The closest I've found is this one. Here is my problem: I have 10 ...
1 vote
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### Is there a name for arrangements of n sets consisting of unique single values for the intersection of each k of them?

For example the sets {1,2},{1,3} and {2,3} are such that each 2 of them intersect in unique values. This can be done for any n and k where n is the number of sets and k is the number of intersections....
1 vote
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### For a Steiner system, how many blocks intersect exactly one position of a specific block?

Consider Steiner system $S(2,k,v)$ with $2 = t < k < v$, a family of $k$-subsets of finite set $S$ with $|S|=v$ such that each $t$-subset of $S$ is contained in exactly one block. A paper I'm ...
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### maximal number of independent subsets (bitstrings with uniform distribution)?

Let $B=\{0,1\}^n$ be the set of length $n$ bit-strings with the uniform distribution, and let $\mathcal{F}=\{f:B\to \{0,1\}\}$ be the set of all binary functions on $B$. What is the maximal size of an ...
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### Organizing Meetups Where People Meet Exactly Once [duplicate]

Okay, I've seen this question asked multiple ways, but I cannot get my head wrapped around it! With 9 people, I can have a schedule where each person participates in 4 meetings of 3 people per meeting,...
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### How to sort into X bins Y times with minimum overlap?

Let's say I'm hosting a series of dinner parties for a total of $N$ guests. Each night, there are $X$ tables, and we are meeting for a total of $Y$ nights. I want to preassign the guests to tables ...
1 vote
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### Latin square analogy - finding a balanced arrangement of elements

I've come across a practical problem in discrete mathematics, and I suspect that some clever mind knows a better solution than brute force. Imagine that we are hosting a competition in which each of ...
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### Showing that whether a construction is the maximum /minimum or not

16 friends decided to form clubs. Each club will have 4 members, and any two clubs may have at most two member in common. What is the greatest/least possible number of clubs they can form. Got 7 ...
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### Coloring a Generalized Latin Square

Suppose we have an $n \times n$ array, and there is a decomposition $\mathcal{A}$ of its coordinates $a_{i,j}$ into sets $A_m$ as follows: If $a_{i,j} \in A_m$, then $a_{j,i} \in A_m$. So they're ...