A000616 a(-1)=1 by convention; for n >= 0, a(n) = number of irreducible Boolean functions of n variables. (Formerly M0819 N0310 N1026)

1, 2, 3, 6, 22, 402, 1228158, 400507806843728, 527471432057653004017274030725792, 11218076601767519586965281984173341005925142853855481024470471657123840

Offset

-1,2

Comments

Number of NP-equivalence classes of switching functions of n or fewer variables.

Number of inequivalent binary nonlinear codes of length n (and all sizes).

a(n+1) = number of NPN-equivalence classes of canalizing functions (see A102449) with n variables. NPN-equivalence allows complementing the function value as well as the individual variables. E.g., the 6 inequivalent canalizing functions when n=3 are 0, x, x AND y, x AND y AND z, x AND (y OR z), x AND (y XOR z). - Don Knuth, Aug 24 2005, Aug 06 2006

Functions' truth tables are colorings of the vertices of n-dimensional hypercubes, where each axis is an input. Actions of reduction (by exchanging pairs of inputs and mapping NOT to them) correspond with invariance under the hypercube's symmetry group, so it is column k=2 of A361870. - Nathan L. Skirrow, Jun 24 2023

References

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 112.

M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 149.

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 11.

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).

I. Tomescu, Introducere in Combinatorica. Editura Tehnica, Bucharest, 1972, p. 129.

Links

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

B. Elspas, Self-complementary symmetry types of Boolean functions, IEEE Transactions on Electronic Computers 2, no. EC-9 (1960): 264-266. [Annotated scanned copy]

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.

R. K. Guy, Letter to N. J. A. Sloane, Mar 1974.

M. A. Harrison, The number of transitivity sets of Boolean functions, J. Soc. Indust. Appl. Math., 11 (1963), 806-828.

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]

J. Sklansky, General synthesis of tributary switching networks, IEEE Trans. Elect. Computers, 12 (1963), 464-469.

I. Toda, On the number of types of self-dual logical functions, IEEE Trans. Electron. Comput., 11 (1962), 282-284.

I. Toda, On the number of types of self-dual logical functions. [Annotated scanned copy]

Index entries for sequences related to Boolean functions

Formula

Harrison gives a simple formula in terms of the cycle index of the appropriate group.

a(n) ~ 2^(2^n)/(n!*2^n), and converges from above. (See A361870 for proof.) - Nathan L. Skirrow, Jun 24 2023

Crossrefs

Row sums of A039754, column k=2 of A361870.

Compare A003180 for equivalence under permutation of inputs without NOTs, A000231 for NOTs without permutation, A000618 for the number of NP-equivalence classes for exactly n variables.

Cf. A102449, A109460, A109462.

Sequence in context: A173249 A303584 A261014 * A233217 A261963 A183179

Adjacent sequences: A000613 A000614 A000615 * A000617 A000618 A000619

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic

Entry revised by N. J. A. Sloane, Aug 07 2006

Terms a(9) and a(10) (given in b-file) from Marcus Ritt, Aug 13 2013

Status

approved