A390248 Number of reducible simple Venn diagrams with n curves.

1, 1, 1, 1, 11, 157619

Offset

1,5

Comments

See A386795 for the definition of simple Venn diagrams.

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.

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.

References

P. Hamburger and R. E. Pippert, Simple, reducible Venn diagrams on five curves and Hamiltonian cycles. Geom. Dedicata, 68(3):245-262, 1997.

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.

Links

Table of n, a(n) for n=1..6.

Sofia Brenner, Linda Kleist, Torsten Mütze, Christian Rieck, and Francesco Verciani, Counterexamples to two conjectures on Venn diagrams, arXiv:2503.18554 [math.CO], 2025.

Frank Ruskey and Mark Weston, A survey of Venn diagrams, Electron. J. Combin., Dynamic Survey 5, 1997.

Example

For n=3 curves, there is only the simple Venn diagram shown in the following figure, and it is reducible, thus, a(3)=1.
+-----------+
|           |
|       +---|-------+
|       |   |       |
|   +---|---|---+   |
|   |   |   |   |   |
+---|---|---+   |   |
    |   |       |   |
    |   +-------|---+
    |           |
    +-----------+

Crossrefs

Cf. A386795, A390247.

Sequence in context: A384504 A055311 A116622 * A013794 A022009 A201249

Adjacent sequences: A390245 A390246 A390247 * A390249 A390250 A390251

Keyword

nonn,hard,bref

Author

Torsten Muetze, Oct 30 2025

Status

approved