A001532 -id:A001532 - cp's OEIS frontend

A109456 Number of Boolean functions of n variables that are self-dual and regular.

0, 1, 1, 2, 3, 7, 21, 135, 2470, 319124, 1214554343, 1706241214185942

Offset: 0

Don Knuth, Aug 17 2005

Or, number of self-dual 2-monotonic Boolean functions of n or fewer variables.
Agrees with A001532 for n <= 8 but then diverges.
The value for n=10 was calculated with BDD techniques; all solutions are characterized by a binary decision diagram with 30011986 nodes.
The value for n=11 was calculated by Minfeng Wang in May 2012 (see [Hausmann & Rodriguez]). The news was published by J.-C. Hausmann (2015). - Fabián Riquelme, Mar 19 2018
For n from 1 to 9, a(n) agrees with the number of directed zero-sum games with n players in Table 1 of Krohn and Sudhölter. - Peter Bala, Dec 16 2021

D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.1.1 (in preparation).

Jean-Claude Hausmann, Counting polygon spaces, Boolean functions and majority games, arXiv preprint arXiv:1501.07553 [math.CO], 2015.
Jean-Claude Hausmann and Eugenio Rodriguez, The space of clouds in an Euclidean space, Corrections and additional material, 2014.
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; also on ResearchGate.
S. Muroga, T. Tsuboi and C. R. Baugh, Enumeration of threshold functions of eight variables, IEEE Trans. Computers, 19 (1970), 818-825.

a(10) from Don Knuth, Feb 06 2008

a(11) from Fabián Riquelme, Mar 27 2018

A189359 Number of homogeneous games for n players.

Original entry on oeis.org

0, 1, 3, 8, 23, 76, 293, 1307, 6642, 37882

Offset: 0

Fabián Riquelme, Apr 20 2011

I. Krohn and P. Sudhölter, Directed and weighted majority games, Mathematical Methods of Operation Research, 42, 2 (1995), 189-216. See Table 1, p. 213; also on ResearchGate.
P. Sudhölter, Homogeneous games as anti step functions, International Journal of Game Theory Vol. 18, (1989), 433-469.

Subclass of A000617. Cf. A001532, A022493, A109456, A132183.

Conjecture: g.f.: Q(0)*x/(1-x), where Q(k) = 1 + (1-(1-x)^(2*k+2))/(1- (1-(1-x)^(2*k+3))/(1-(1-x)^(2*k+3) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 13 2013

Note that a(n) - a(n-1) = A022493(n) for 1 <= n <= 9. Does this equality hold for n > 9? If so, then we have the g.f. 1/(1 - x)*( Sum_{n >= 1} Product_{k = 1..n} (1 - (1 - x)^k) ). - Peter Bala, Dec 13 2021

A003184 Number of NP-equivalence classes of self-dual threshold functions of exactly n variables.

Original entry on oeis.org

1, 0, 1, 1, 4, 14, 114, 2335, 172958, 52805196

Offset: 1

N. J. A. Sloane

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.
S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 24. (Cases n>7.)
J. von Neumann and O. Morgenstern, Theory of games and economic behavior, Princeton University Press, New Jersey, 1944. Cases n=1 to 5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.
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]
Index entries for sequences related to Boolean functions

Cf. A000619, A001532, A002077-A002080.

a(n) = A001532(n) - A001532(n-1), for n > 1. - Evgeny Luttsev, Sep 09 2014

a(9) from Evgeny Luttsev, Sep 09 2014

Better description and new offset from Alastair King, Mar 17 2023

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A109456 Number of Boolean functions of n variables that are self-dual and regular.

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A189359 Number of homogeneous games for n players.

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A003184 Number of NP-equivalence classes of self-dual threshold functions of exactly n variables.

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