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Even Set Systems
Andreas Dress1, 2
1Department for Combinatorics and Geometry, CAS-MPG Partner Institute and Key Lab for Computational Biology, Shanghai Institutes for Biological Sciences, Chinese Academy of Sciences, 320 Yue Yang Road, Shanghai 200031, P.R.China
2Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22 -26, D 04103 Leipzig, Germany
dress@sibs.ac.cn, dress@mis.mpg.de
Annals of Combinatorics 13 (2) pp.193-198 June, 2009
AMS Subject Classification: 92B10, 05E99, 05A18
Abstract:
In phylogenetic combinatorics, the analysis of split systems is a fundamental issue. Here, we observe that there is a canonical one-to-one correspondence between split systems on the one, and ``even'' set systems on the other hand, i.e., given any finite set X, we show that there is a canonical one-to-one correspondence between the set \$\p(\sx)\$ consisting of all subsets \$\s\$ of the set \$\sx\$ of all splits of the set X (that is, all 2-subsets {A,B} of the power set \$\px\$ of \$X\$ for which \$A\cup B = X\$ and \$A\cap B = \emptyset\$ hold) and the set \$\p^{even}(\px)\$ consisting of all subsets \$\e\$ of the power set \$\px\$ of \$X\$ for which, for each subset \$Y\$ of \$X\$, the number of proper subsets of Y contained in \$\e\$ is even.
Keywords: splits, split systems, set systems, combinatorics of split systems, combinatorics of set systems, even set systems, phylogenetic analysis, phylogenetic combinatorics

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