A109456 -id:A109456 - cp's OEIS frontend

A000609 Number of threshold functions of n or fewer variables.

2, 4, 14, 104, 1882, 94572, 15028134, 8378070864, 17561539552946, 144130531453121108

Offset: 0

a(n) is also equal to the number of self-dual threshold functions of n+1 or fewer variables. - Alastair D. King, Mar 17 2023.

Sze-Tsen Hu, Threshold Logic, University of California Press, 1965 see page 57.
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 3.
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).
C. Stenson, Weighted voting, threshold functions, and zonotopes, in The Mathematics of Decisions, Elections, and Games, Volume 625 of Contemporary Mathematics Editors Karl-Dieter Crisman, Michael A. Jones, American Mathematical Society, 2014, ISBN 0821898663, 9780821898666

Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
Nicolle Gruzling, Linear separability of the vertices of an n-dimensional hypercube, M.Sc Thesis, University of Northern British Columbia, 2006. [From W. Lan (wl(AT)fjrtvu.edu.cn), Jun 27 2010]
Samuel C. Gutekunst, Karola Mészáros, and T. Kyle Petersen, Root Cones and the Resonance Arrangement, arXiv:1903.06595 [math.CO], 2019.
Alastair D. King, Comments on A002080 and related sequences based on threshold functions, Mar 17 2023.
Isaac K. Martin, Andrew G. Moore, John T. Daly, Jess J. Meyer, and Teresa M. Ranadive, Design of General Purpose Minimal-Auxiliary Ising Machines, arXiv:2310.16246 [math.OC], 2023. See p. 7.
Chris Mingard, Joar Skalse, Guillermo Valle-Pérez, David Martínez-Rubio, Vladimir Mikulik, and Ard A. Louis, Neural networks are a priori biased towards Boolean functions with low entropy, arXiv:1909.11522 [cs.LG], 2019.
Guido F. Montufar and Jason Morton, When Does a Mixture of Products Contain a Product of Mixtures?, arXiv preprint arXiv:1206.0387 [stat.ML], 2012-2014.
S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971 [Annotated scans of a few pages]
Muroga, Saburo, Iwao Toda, and Satoru Takasu, Theory of majority decision elements, Journal of the Franklin Institute 271.5 (1961): 376-418. [Annotated scans of pages 413 and 414 only]
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]
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Stephen Wolfram, A New Kind Of Science. page 1102.
Wikipedia, Linear separability [From W. Lan (wl(AT)fjrtvu.edu.cn), Jun 27 2010]
R. O. Winder, Enumeration of seven-argument threshold functions, IEEE Trans. Electron. Computers, 14 (1965), 315-325.
Index entries for "core" sequences
Index entries for sequences related to Boolean functions

Cf. A000615, A002077-A002080, A109456, A116986.

a(n) = Sum_{k=0..n} A000615(k)*binomial(n,k) = Sum_{k=0..n} A002079(k)*binomial(n,k)*2^k. Also A002078(n) = (1/2^n)*Sum_{k=0..n} a(k)*binomial(n,k), a(n-1) = Sum_{k=1..n} A002077(k)*binomial(n,k)*2^k, and A002080(n) = (1/2^n)*Sum_{k=1..n} a(k)*binomial(n,k). - Alastair D. King, Mar 17 2023.

a(9) from Minfeng Wang, Jun 27 2010

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.

Original entry on oeis.org

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

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.)
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.)

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

Cf. A000617, A002077-A002080, A003184, A109456, A132183, A189359.

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

a(10) added by W. Lan (wl(AT)fjrtvu.edu.cn), Jun 27 2010

Better description from Alastair King, Mar 17 2023.

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

A189360 Number of self-dual homogeneous games for n players.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 21, 132, 2188

Offset: 0

Fabián Riquelme, Apr 20 2011

Found by F. Riquelme

Intersection of A189359 and A109456.

A290946 Number of chambers of Hassett's decomposition of the weight domain of the moduli space of genus 0 n-pointed curves.

Original entry on oeis.org

1, 27, 1087, 105123, 31562520, 33924554539, 140306938682875

Offset: 3

Connor Halleck-Dube, Oct 20 2017

a(n) is also the number of goldilocks linear threshold functions on n variables. Asymptotically equal to the total number of linear threshold functions on n variables, divided by 2^n.

K. Ascher, C. Dube, D. Gershenson, E. Hou, Enumerating Hassett's wall and chamber decomposition of the moduli space of weighted stable curves, arXiv:1709.03663 [math.AG], 2017.

Cf. A000609, A000615, A109456.

a(n) ~ 2^(n^2 - n log n + O(n)).

cp's OEIS Frontend

A000609 Number of threshold functions of n or fewer variables.

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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.

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A189359 Number of homogeneous games for n players.

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A189360 Number of self-dual homogeneous games for n players.

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A290946 Number of chambers of Hassett's decomposition of the weight domain of the moduli space of genus 0 n-pointed curves.

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