A001532 - OEIS

A001532

Number of NP-equivalence classes of self-dual threshold functions of n or fewer variables; number of majority (i.e., decisive and weighted) games with n players.
(Formerly M0852 N0324)

1, 1, 2, 3, 7, 21, 135, 2470, 175428, 52980624

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

a(n) is the number of combinatorial types of hyperplane sections of the n-dimensional cube, such that the hyperplane contains the center of symmetry and does not contain any vertex of the cube (Corollary 2.5 of Brandenburg-Meroni). - Chiara Meroni, Dec 11 2025

REFERENCES

H. M. Gurk and J. R. Isbell. 1959. Simple Solutions. In A. W. Tucker and R. D. Luce (eds.) Contributions to the Theory of Games, Volume 4. Princeton, NJ: Princeton University Press, pp. 247-265. (Case n=6.)

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 23. (Cases until n=9.)

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).

J. von Neumann and O. Morgenstern, Theory of games and economic behavior, Princeton University Press, New Jersey, 1944. (Cases n=1 to 5.)

LINKS

Table of n, a(n) for n=1..10.

Marie-Charlotte Brandenburg and Chiara Meroni, Combinatorics of slices of cubes, arXiv preprint arXiv:2510.09265 [math.CO], 2025. See also related code and data.

J.-C. Hausmann, Counting polygon spaces, Boolean functions and majority games, arXiv preprint arXiv:1501.07553 [math.CO], 2015.

J.-C. Hausmann and E. Rodriguez, The space of clouds in Euclidean space, Experiment. Math. 13 (2004), 31-47.

J.-C. Hausmann and E. Rodriguez. The space of clouds in Euclidean space, Corrections and additional material.

J. R. Isbell, On the enumeration of majority games, MTAC, v.13, 1959, pp. 21-28. (Case n=7).

Alastair D. King, Comments on A002080 and related sequences based on threshold functions, Mar 17 2023.

I. Krohn and P. Sudhölter, Directed and weighted majority games, Mathematical Methods of Operation Research, 42, 2 (1995), 189-216. See Table 1, row 4, p. 213.

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.

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]

Erik Ordentlich, Ron M. Roth and Gadiel Seroussi, On q-ary Antipodal Matchings and Applications, 2012.

Index entries for sequences related to Boolean functions

FORMULA

a(n) = Sum_{k=1..n} A003184(k). - Alastair D. King, Oct 26 2023

CROSSREFS

Cf. A000617, A002077, A002078, A002079, A002080, A003184, A109456, A132183, A189359.

Sequence in context: A392965 A262305 A189360 * A109456 A155745 A067738

Adjacent sequences: A001529 A001530 A001531 * A001533 A001534 A001535

KEYWORD

nonn,nice,more

AUTHOR

N. J. A. Sloane