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Combinatorial Intricacies of Labeled Fano Planes

Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
Academic Editors: Antonio M. Scarfone and Kevin H. Knuth
Entropy 2016, 18(9), 312; https://doi.org/10.3390/e18090312
Received: 20 July 2016 / Revised: 5 August 2016 / Accepted: 19 August 2016 / Published: 23 August 2016
(This article belongs to the Collection Advances in Applied Statistical Mechanics)
Given a seven-element set X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that is to say an unordered triple from X—ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order s, 0 s 3 , if there are s defective lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown that no labeled Fano plane can have all points of zero-th order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out. View Full-Text
Keywords: labeled Fano planes; ordinary/defective lines; Steiner triple systems labeled Fano planes; ordinary/defective lines; Steiner triple systems
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Saniga, M. Combinatorial Intricacies of Labeled Fano Planes. Entropy 2016, 18, 312.

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