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Number of bipolar Boolean functions, that is, Boolean functions that are monotone or antimonotone in each argument.
7

%I #54 May 11 2024 17:10:25

%S 2,4,14,104,2170,230540,499596550,309075799150640,

%T 14369391928071394429416818,

%U 146629927766168786368451678290041110762316052

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

%C 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.

%C These functions are variously called "unate functions" or "locally monotone functions". - _Aniruddha Biswas_, May 11 2024

%D 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).

%H Ringo Baumann and Hannes Strass, <a href="https://doi.org/10.1093/logcom/exx025">On the Number of Bipolar Boolean Functions</a>, Journal of Logic and Computation, exx025. Also available as a <a href="https://iccl.inf.tu-dresden.de/w/images/a/ab/HS1112620917_2017_JLC-16-40.pdf">Preprint</a>.

%H A. Biswas and P. Sarkar, <a href="https://arxiv.org/abs/2304.14069">Counting unate and balanced monotone Boolean functions,</a> arXiv:2304.14069 [math.CO], 2023.

%H G. Brewka and S. Woltran, <a href="http://aaai.org/ocs/index.php/KR/KR2010/paper/view/1294">Abstract dialectical frameworks</a>, Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning. Pages 102--111. IJCAI/AAAI 2010.

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

%e There are 2 bipolar Boolean functions in 0 arguments, the constants true and false.

%e All 4 Boolean functions in one argument are bipolar.

%e For 2 arguments, only equivalence and exclusive-or are not bipolar, 16-2=14.

%Y Cf. A006126.

%K nonn,hard,more

%O 0,1

%A _Hannes Strass_, Jul 11 2014

%E a(7)-a(8) corrected by and a(9) from _Aniruddha Biswas_, May 11 2024