A002078 -id:A002078 - cp's OEIS frontend

A002077 Number of N-equivalence classes of self-dual threshold functions of exactly n variables.

1, 0, 1, 4, 46, 1322, 112519, 32267168, 34153652752

Offset: 1

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 10.
S. Muroga and I. Toda, Lower bound on the number of threshold functions, IEEE Trans. Electron. Computers, 17 (1968), 805-806.
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).

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. A002078, A002079, A002080.

A002080(n) = Sum_{k=1..n} a(k)*binomial(n,k). Also A000609(n-1) = Sum_{k=1..n} a(k)*binomial(n,k)*2^k. - Alastair D. King, Mar 17 2023.

Better description from Alastair King, Mar 17 2023

A000609 Number of threshold functions of n or fewer variables.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

A002080 Number of N-equivalence classes of self-dual threshold functions of n or fewer variables.

Original entry on oeis.org

1, 2, 4, 12, 81, 1684, 122921, 33207256, 34448225389

Offset: 1

N. J. A. Sloane

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
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).

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. A000609, A002077, A002078.

a(n) = Sum_{k=1..n} A002077(k)*binomial(n,k) = (1/2^n)*Sum_{k=1..n} A000609(k-1)*binomial(n,k). - Alastair D. King, Mar 17 2023.

Better description and corrected value of a(7) from Alastair King (see link) - N. J. A. Sloane, Oct 24 2023

A002079 Number of N-equivalence classes of threshold functions of exactly n variables.

Original entry on oeis.org

2, 1, 2, 9, 96, 2690, 226360, 64646855, 68339572672

Offset: 0

N. J. A. Sloane

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 8.
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).

Alastair D. King, Comments on A002080 and related sequences based on threshold functions, Mar 17 2023.
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. [Annotated scanned copy]
Index entries for sequences related to Boolean functions

Cf. A002077, A002078, A002080.

A002078(n) = Sum_{k=0..n} a(k)*binomial(n,k). A000609(n) = Sum_{k=0..n} a(k)*binomial(n,k)*2^k. - Alastair D. King, Mar 17 2023.

Better description from Alastair King, Mar 17 2023.

A001529 NPN-equivalence classes of threshold functions of n or fewer variables.

Original entry on oeis.org

1, 2, 3, 6, 15, 63, 567, 14755, 1366318

Offset: 0

N. J. A. Sloane

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 19.
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).

Goto, Eiichi, and Hidetosi Takahasi, Some Theorems Useful in Threshold Logic for Enumerating Boolean Functions, in Proceedings International Federation for Information Processing (IFIP) Congress, 1962, pp. 747-752. [Annotated scans of certain pages]
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]

Cf. A002078, A002079, A001530.

cp's OEIS Frontend

A002077 Number of N-equivalence classes of self-dual threshold functions of exactly n variables.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A000609 Number of threshold functions of n or fewer variables.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A002080 Number of N-equivalence classes of self-dual threshold functions of n or fewer variables.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A002079 Number of N-equivalence classes of threshold functions of exactly n variables.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A001529 NPN-equivalence classes of threshold functions of n or fewer variables.

Views

Author

Keywords

References

Links

Crossrefs

A109455 Number of equivalence classes of threshold functions under permutations of the variables.

Views

Author

Keywords

References

Crossrefs