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Counting symmetric configurations $v_3$. (English) Zbl 0940.05020
An incidence geometry $(P,{\cal B})$ consists of a set of $v$ points $P= \{p_1,\dots, p_v\}$ and a collection of $b$ blocks ${\cal B}= \{B_1,\dots, B_b\}$ such that $B_i\subseteq P$ for $i= 1,2,\dots, b$. The number of blocks containing a point $p\in P$ is called the degree, denoted by $[p]$. Blocks are also called lines and $|B|$ is called the length of the block $B$. A pair $(p,B)$ with $p\in B\in{\cal B}$ is called a flag. In this case, the point $p$ is said to lie on the line $B$. The line is said to pass through $p$. Additionally, in an incidence geometry any pair of points is joined by at most one line, i.e., $|B_i\cap B_j|\le 1$ for all $i, j\in\{1,\dots, b\}$, $i\ne j$. An incidence geometry $(p,{\cal B})$ is called configuration of type $(v_r,b_k)$ if (1) $|B_j|= k$ for $j= 1,2,\dots, b$ and (2) $[p_i]= r$ for $i= 1,2,\dots, v$. A configuration $(p,{\cal B})$ with $v= b$ is called symmetric. Its type is simply denoted by $v_r$ which is the same as $b_k$ because $k= r$ in this case. A configuration $C= (p,{\cal B})$ is called decomposable if it can be written as the union of two configurations $C_1$, $C_2$ on distinct point sets: $C_1= (p_1,{\cal B}_1)$, $C_2= (p_2,{\cal B}_2)$, and $P= P_1\cup P_2$ and ${\cal B}= {\cal B}_1\cup{\cal B}_2$. Indecomposable configurations are also called connected. The isomorphisms are structure preserving mappings. A triangle of a configuration consists of three points, say $a$, $b$, and $c$, such that the three pairs $\{a,b\}$, $\{b,c\}$, $\{c,a\}$ are contained in different blocks. A configuration that has no triangles is called a triangle-free configuration. An isomorphism $C\to C^d$ is called an anti-automorphism. A configuration which admits an anti-automorphism is called self-dual. An anti-automorphism of order $2$ is called a polarity. A configuration which admits a polarity is called self-polar. The authors give tables of configurations $v_3$ for $v\le 18$ and triangle-free configurations for $v\le 21$ together with some statistics about some properties of structures like transitivity, self-duality or self-polarity.

05B30Other designs, configurations
Full Text: DOI
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