A000617 Number of NP-equivalence classes of threshold functions of n or fewer variables. (Formerly M0727 N0272)

2, 3, 5, 10, 27, 119, 1113, 29375, 2730166, 989913346

Offset

0,1

Comments

From Fabián Riquelme, Jun 01 2012: (Start)

NP-equivalence classes of threshold functions are equivalent to weighted games, in simple game theory.

The number for n=9 was first documented in Tautenhahn's thesis. (End)

References

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 14.

S. Muroga, T. Tsuboi, and C. R. Baugh, Enumeration of threshold functions of eight variables, IEEE Trans. Computers, 19 (1970), 818-825.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. Tautenhahn, Enumeration einfacher Spiele mit Anwendungen in der Stimmgewichtsverteilung, 2008. Master's thesis, University of Bayreuth, 269 pages (in German).

Links

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

S. Bolus, A QOBDD-based Approach to Simple Games, Dissertation, Doktor der Ingenieurwissenschaften der Technischen Fakultaet der Christian-Albrechts-Universitaet zu Kiel, 2012. - From N. J. A. Sloane, Dec 22 2012

I. Krohn and P. Sudhölter, Directed and weighted majority games, Math. Methods Operat. Res. 42 (2) (1995) 189-216, Table 1.

S. Kurz, On minimum sum representations for weighted voting games, arXiv:1103.1445 [math.CO], 2011-2018.

S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971 [Annotated scans of a few pages]

S. Muroga, T. Tsuboi, and C. R. Baugh, Enumeration of threshold functions of eight variables, IEEE Trans. Computers, 19 (1970), 818-825. [Annotated scanned copy]

Eda Uyanık, Olivier Sobrie, Vincent Mousseau, and Marc Pirlot, Enumerating and categorizing positive Boolean functions separable by a k-additive capacity, Discrete Applied Mathematics, Vol. 229, 1 October 2017, p. 17-30. See Table 3.

Formula

a(n) = Sum_{k=0..n} A000619(k). - Alastair D. King, Oct 26, 2023.

Crossrefs

Cf. A000619.

Sequence in context: A336991 A223545 A088938 * A132183 A259878 A003504

Adjacent sequences: A000614 A000615 A000616 * A000618 A000619 A000620

Keyword

nonn,hard,more,nice

Author

N. J. A. Sloane

Status

approved