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%I #26 Nov 12 2025 23:19:51
%S 1,1,1,1,20,3430404
%N Number of simple Venn diagrams with n curves.
%C A Venn diagram with n curves is a collection of n simple closed curves in the plane that intersect in only finitely many points and create exactly 2^n regions, one for every possible combination of being inside or outside of each of the n curves.
%C A Venn diagram is simple if at most two of the n curves intersect in any point.
%C For this counting sequence we consider two Venn diagrams equivalent if the differ only by mirroring and/or stereographic projection. These are also sometimes called Venn classes or spherical Venn diagrams in the literature.
%C The dual graphs are exactly all non-isomorphic planar spanning subgraphs of the n-dimensional hypercube in which every face has length 4, with the additional requirement that for every position i in {1,...,n} and every bit b in {0,1} the corresponding subgraph induced by all vertices x with x_i=b is connected.
%D K. B. Chilakamarri, P. Hamburger, and R. E. Pippert, Analysis of Venn diagrams using cycles in graphs. Geom. Dedicata, 82(1-3):193-223, 2000.
%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 J. Venn. On the diagrammatic and mechanical representation of propositions and reasonings. Phil. Mag. S. 5., 9(59):1-18, 1880.
%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, Petr Gregor, Torsten Mütze, and Francesco Verciani, <a href="https://arxiv.org/abs/2511.09230">On minimum Venn diagrams</a>, arXiv:2511.09230 [math.CO], 2025.
%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, thus, a(3)=1.
%e +-----------+
%e | |
%e | +---|-------+
%e | | | |
%e | +---|---|---+ |
%e | | | | | |
%e +---|---|---+ | |
%e | | | |
%e | +-------|---+
%e | |
%e +-----------+
%e The corresponding dual graph is the 3-cube.
%Y Cf. A390247, A390248.
%K nonn,hard,more,bref
%O 1,5
%A _Torsten Muetze_, Oct 30 2025