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

Author

Hannes Strass, Jul 11 2014

Keywords

• nonn
• hard
• more

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

Examples

			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.
		

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

• Ringo Baumann and Hannes Strass, On the Number of Bipolar Boolean Functions, Journal of Logic and Computation, exx025. Also available as a Preprint.
• Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
• Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 2, 4, 12.
• Gerhard Brewka and Stefan Woltran, Abstract dialectical frameworks, Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning. Pages 102--111. IJCAI/AAAI 2010.

Crossrefs

Cf. A006126.

Formula

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

Extensions

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