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%I #15 Nov 11 2025 17:20:48
%S 1,1,1,1,11,157619
%N Number of reducible simple Venn diagrams with n curves.
%C See A386795 for the definition of simple Venn diagrams.
%C A Venn diagram with n curves is reducible if one of its curves can be removed so that the remaining n-1 curves form a Venn diagram, i.e., if there is a curve that touches every region.
%C A reducible diagram with n curves is extendable, i.e., we can add a curve to make it a Venn diagram with (n+1) curves. Winkler conjectured that every simple Venn diagram with n curves (reducible or not) is extendable to a simple Venn diagram with n+1 curves, but this conjecture is false.
%D P. Hamburger and R. E. Pippert, Simple, reducible Venn diagrams on five curves and Hamiltonian cycles. Geom. Dedicata, 68(3):245-262, 1997.
%D P. Winkler, Venn diagrams: some observations and an open problem, in Proceedings of the Fifteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, pages 267-274, 1984.
%H Sofia Brenner, Linda Kleist, Torsten Mütze, Christian Rieck, and Francesco Verciani, <a href="https://arxiv.org/abs/2503.18554">Counterexamples to two conjectures on Venn diagrams</a>, arXiv:2503.18554 [math.CO], 2025.
%H Frank Ruskey and Mark Weston, <a href="https://www.combinatorics.org/files/Surveys/ds5/VennEJC.html">A survey of Venn diagrams</a>, Electron. J. Combin., Dynamic Survey 5, 1997.
%e For n=3 curves, there is only the simple Venn diagram shown in the following figure, and it is reducible, thus, a(3)=1.
%e +-----------+
%e | |
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%e +---|---|---+ | |
%e | | | |
%e | +-------|---+
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%e +-----------+
%Y Cf. A386795, A390247.
%K nonn,hard,bref
%O 1,5
%A _Torsten Muetze_, Oct 30 2025