A245079 Number of bipolar Boolean functions, that is, Boolean functions that are monotone or antimonotone in each argument.

2, 4, 14, 104, 2170, 230540, 499596550, 309075799150640, 14369391928071394429416818, 146629927766168786368451678290041110762316052

Offset

0,1

Comments

A Boolean function is bipolar if and only if for each argument index i, the function is one of: (1) monotone in argument i, (2) antimonotone in argument i, (3) both monotone and antimonotone in argument i.

These functions are variously called "unate functions" or "locally monotone functions". - Aniruddha Biswas, May 11 2024

References

Richard Dedekind, Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Theiler, in Fest-Schrift der Herzoglichen Technischen Hochschule Carolo-Wilhelmina, pages 1-40. Vieweg+Teubner Verlag (1897).

Links

Table of n, a(n) for n=0..9.

Ringo Baumann and Hannes Strass, On the Number of Bipolar Boolean Functions, Journal of Logic and Computation, exx025. Also available as a Preprint.

A. Biswas and P. Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.

G. Brewka and S. Woltran, Abstract dialectical frameworks, Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning. Pages 102--111. IJCAI/AAAI 2010.

Formula

a(n) = Sum_{i=1..n}(2^i * C(n,i) * A006126(i)) + 2.

Example

There are 2 bipolar Boolean functions in 0 arguments, the constants true and false.
All 4 Boolean functions in one argument are bipolar.
For 2 arguments, only equivalence and exclusive-or are not bipolar, 16-2=14.

Crossrefs

Cf. A006126.

Sequence in context: A005737 A219767 A000609 * A167008 A376697 A329234

Adjacent sequences: A245076 A245077 A245078 * A245080 A245081 A245082

Keyword

nonn,hard,more

Author

Hannes Strass, Jul 11 2014

Extensions

a(7)-a(8) corrected by and a(9) from Aniruddha Biswas, May 11 2024

Status

approved