# 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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### 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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### Help understanding linear contrasts in randomized block design?

I have been given the following question: "An experiment is conducted to compare the yield from one field, where samples from the field are placed in plant pots with different levels of nitrogen ...
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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 ...
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### Incident Matrix and Concurrance Matrix

How would I find the incident and concurrance matrix for the below? \left(\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 & 2 \\ 1 &...
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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 ...
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### Real World Problem - Groups Visiting Various Stations

Context: This is a real world problem that I am trying to solve that I would appreciate some help with - I'm sure there's a Mathematical concept that will help me, but I'm struggling to remember back ...
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### Minimum swaps necessary for every person to meet every other person at every location

The setting: There are 9 people. There are 3 locations. At any time, there are 3 people at each location. People can only swap locations at the same time. The goal: Every person visits each ...
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### Construction of STS with maximal complete arc

By definition, a Steiner Triple System (STS) is a $t-(v,k,\lambda)$ design with blocks of size 3 ($k=3$) such that each pair of points meets in exactly 1 block ($t=2$ and $\lambda=1$). Let us denote ...
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### Minimum size set to guarantee m-matches on lottery

Consider a lottery where you pick a set $S_{pick}$ of $k$ numbers from a total of $N$ numbers. Then a set $S_{draw}$ of $n$ numbers ($n<=k<N$) is draw and if all the numbers from the $S_{draw}$ ...
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### "Orthogonal" Steiner Systems

Let $1\leq t\leq k\leq v$ be integers. A Steiner system $S:=S(t,k,v)$ is a collection of subsets $K$ of size $k$ of a set $V$ of size $v$ such that for every subset $T\subseteq V$ of size $t$, there ...
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### Number of combinations to introduce people within certain groups

I am trying to calculate a combination of unique people inside of the different groups. Nonetheless, I have not observed any formula to achieve my intention. There are 4 different groups and each ...
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### 10 events, 10 teams

I need help scheduling an event where there are 10 activities and 10 teams. Each team plays against another team for each activity. Each team must play each activity. Ideally, they would not play ...
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### What is the relationship between covering designs (La Jolla Covering Repository) and linear block codes?

In many papers I have been reading, bioengineers are utilizing covering sets from Dan Gordon's La Jolla Covering Repository to design custom Modified Hamming Codes (MHD). These MHDs have custom ...
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### Combinatorial design related to scheduling group activities (everyone tries every activity, no pair is together twice)

Trying to solve this problem led me to consider the following generalization. Let $g$ and $p$ be positive integers. Imagine that you own $g$ distinct board games, where each game requires exactly $p$ ...
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### "Unbalanced" combinatorial designs

A combinatorial design on a set $X$ (which I'll call players) of size $n$ is a collection of subsets of $X$ (which I'll call games) such that: Each player is in exactly $r$ games. Each game contains ...
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### How can I arrange a group of people at tables and switch them around so that no two ever meet twice? [closed]

Say I have 5 tables of 4 people (i.e. 20 people in total) and a different game at every table. We play one round of games (so each table plays one unique game) and then switch tables. Is it possible ...
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### Properties of submodular functions

I was working on submodular set functions, and I came across a property on Wikipedia, that I was not able to prove/find any reference for. On the Wikipedia article on submodular set functions, Under ...
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### For which $n\in\Bbb N$ can we divide $\{1,2,3,...,3n\}$ into $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$?

For which $n\in \mathbb{N}$ can we divide the set $\{1,2,3,\ldots,3n\}$ into $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$? Since $x_i+y_i=3z_i$ for each ...
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### Minimum for combinatorial sortingproblem

I'm stuck with a combinatorial problem, maybe one of you can help me out, thanks in advance. So heres the problem: Consider tuples $(i,j)\in \{1,...,N_1\}\times\{1,....,N_2\}=A$. Let $S_1,...,S_x$ be ...
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