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A000616
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a(-1)=1 by convention; for n >= 0, a(n) = number of irreducible Boolean functions of n variables.
(Formerly M0819 N0310 N1026)
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27
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OFFSET
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-1,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
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FORMULA
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Harrison gives a simple formula in terms of the cycle index of the appropriate group.
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CROSSREFS
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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.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Terms a(9) and a(10) (given in b-file) from Marcus Ritt, Aug 13 2013
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STATUS
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approved
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