design
A $\tau $$(\nu ,\kappa ,\lambda )$ design, aka $\tau $design or block design^{}, is an incidence structure $(\mathcal{P},\mathcal{B},\mathcal{I})$ with

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$\mathcal{P}=\nu $ points in all,

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${\mathcal{P}}_{B}=\kappa $ points in each block $B$, and such that

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any set $T\subseteq \mathcal{P}$ of $T=\tau $ points occurs as subset $T\subseteq {\mathcal{P}}_{B}$ in exactly $\lambda $ blocks.
The numbers $\tau ,\nu ,\kappa ,lambda$ are called the parameters of a design. They are often called $t$, $v$, $k$, $\lambda $ (in mixed Latin and Greek alphabets) by some authors.
Given parameters $\tau ,\nu ,\kappa ,lambda$, there may be several nonisomorphic designs, or no designs at all.
Designs need not be simple (they can have repeated blocks), but they usually are (and don’t) in which case $B$ can again be used as synonym for ${\mathcal{P}}_{B}$.

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0designs ($\tau =0$) are allowed.

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1designs ($\tau =1$) are known as tactical configurations.

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2designs are called balanced incomplete block designs or BIBD.

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3, 4, 5… designs have all been studied.
Being a $\tau $$(\nu ,\kappa ,\lambda )$ design implies also being an $\iota $$(\nu ,\kappa ,{\lambda}_{\iota})$ design for every $0\le \iota \le \tau $ (on the same $\nu $ points and with the same block size $\kappa $), with ${\lambda}_{\iota}$ given by ${\lambda}_{\tau}=\lambda $ and recursively
$${\lambda}_{\iota}=\frac{\nu \iota}{\kappa \iota}{\lambda}_{\iota +1}$$ 
from which we get the number of blocks as
$${\lambda}_{0}=\frac{\nu !/(\nu \tau )!}{\kappa !/(\kappa \tau )!}=\left(\genfrac{}{}{0pt}{}{\nu}{\tau}\right)/\left(\genfrac{}{}{0pt}{}{\kappa}{\tau}\right)$$ 
Being a 0design says nothing more than all blocks having the same size. As soon as we have $\tau \ge 1$ however we also have a 1design, so the number ${\lambda}_{1}={\mathcal{B}}_{P}$ of blocks per point $P$ is constant throughout the structure^{}. Note now
$${\lambda}_{0}\kappa ={\lambda}_{1}\nu $$ 
which is also evident from their interpretation^{}.
As an example: designs (simple designs) with $\kappa =2$ are multigraphs^{} (simple graphs), now

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$\tau =0$ implies no more than that,

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$\tau =1$ gives regular graphs^{}, and

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$\tau =2$ gives complete graphs^{}.
A more elaborate “lambda calculus^{}” (pun intended) can be introduced as follows. Let $I\subseteq P$ and $O\subseteq P$ with $I=\iota $ and $O=o$. The number of blocks $B$ such that all the points of $I$ are inside $B$ and all the points of $O$ are outside $B$ is independent of the choice of $I$ and $O$, only depending on $\iota $ and $o$, provided $\iota +o\le \tau $. Call this number ${\lambda}_{\iota}^{o}$. It satisfies a kind of reverse Pascal triangle^{} like recursion
$${\lambda}_{\iota}^{o}={\lambda}_{\iota +1}^{o}+{\lambda}_{\iota}^{o+1}$$ 
that starts off for $o=0$ with ${\lambda}_{\iota}^{0}={\lambda}_{\iota}$. An important quantity (for designs with $\tau \ge 2$) is the order ${\lambda}_{1}^{1}={\lambda}_{1}^{0}{\lambda}_{2}^{0}={\lambda}_{1}{\lambda}_{2}$.
Finally, the dual of a design can be a design but need not be.

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A square design aka symmetric design is one where $\tau =2$ and $\mathcal{P}=\mathcal{B}$, now also ${\mathcal{P}}_{B}={\mathcal{B}}_{P}$. Here the dual is also a square design.
Note that for $\tau \ge 3$ no designs exist with $\mathcal{P}=\mathcal{B}$ other than trivial ones (where any $\kappa =\nu 1$ points form a block).
Title  design 
Canonical name  Design 
Date of creation  20130322 19:14:09 
Last modified on  20130322 19:14:09 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  5 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 62K10 
Classification  msc 51E30 
Classification  msc 51E05 
Classification  msc 05B25 
Classification  msc 05B07 
Classification  msc 05B05 
Synonym  block design 
Synonym  taudesign 
Synonym  $\tau $design 
Synonym  BIBD 
Defines  block 
Defines  simple design 
Defines  square design 
Defines  symmetric design 
Defines  tactical configuration 
Defines  balanced incomplete block design 