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

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

$\mathcal{P}_{B}=\kappa$ points in each block $B$, and such that
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 $0\leq\iota\leq\tau$ (on the same $\nu$ points and with the same block size $\kappa$), with $\lambda_{{\,\iota}}$ given by $\lambda_{{\,\tau}}=\lambda$ and recursively
$\lambda_{{\,\iota}}\,=\;{\nu\iota\over\kappa\iota}\,\lambda_{{\,\iota+1}}$ 
from which we get the number of blocks as
$\lambda_{0}\,=\;{\nu!\,/\,(\nu\tau)!\over\kappa!\,/\,(\kappa\tau)!}\;=\;{\nu% \choose\tau}\bigg/{\kappa\choose\tau}$ 
Being a 0design says nothing more than all blocks having the same size. As soon as we have $\tau\geq 1$ 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 $\iota+o\leq\tau$. 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 $\tau\geq 2$) 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.

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 $\tau\geq 3$ no designs exist with $\mathcal{P}=\mathcal{B}$ other than trivial ones (where any $\kappa=\nu1$ points form a block).
Mathematics Subject Classification
62K10 no label found51E30 no label found51E05 no label found05B25 no label found05B07 no label found05B05 no label found Forums
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