A000654 Invertible Boolean functions of n variables. (Formerly M2173 N0868)

1, 2, 52, 142090700, 17844701940501123640681816160, 59757436204078657410908164193971330396709572693816353610758085074676243846093824

Offset

1,2

Comments

Equivalence classes of invertible maps from {0,1}^n to {0,1}^n, under action of permutation and complementation of variables on domain and range. - Sean A. Irvine, Mar 16 2011

References

M. A. Harrison, The number of classes of invertible Boolean functions, J. ACM 10 (1963), 25-28.

C. S. Lorens, Invertible Boolean functions, IEEE Trans. Electron. Computers, EC-13 (1964), 529-541.

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).

Links

Adam P. Goucher, Table of n, a(n) for n = 1..7

M. A. Harrison, The number of classes of invertible Boolean functions, J. ACM 10 (1963), 25-28. [Annotated scan of page 27 only]

M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12 (1963), 559-561. [Annotated scanned copy]

C. S. Lorens, Invertible Boolean functions, IEEE Trans. Electron. Computers, EC-13 (1964), 529-541.

C. S. Lorens, Invertible Boolean functions, IEEE Trans. Electron. Computers, EC-13 (1964), 529-541. [Annotated scan of page 530 only]

Qing-bin Luo, Jin-zhao Wu, and Chen Lin, Computing the Number of the Equivalence Classes for Reversible Logic Functions, Int'l J. of Theor. Phys. (2020) Vol. 59, 2384-2396.

Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 10.

Index entries for sequences related to Boolean functions

Mathematica

cyclify =
  Function[{x},
   Sort@Tally[Length /@ PermutationCycles[x + 1, Identity]]];
totalweight =
  Function[{c}, Product[(x[[1]]^x[[2]]) ( x[[2]]!), {x, c}]];
perms = Function[{n},
   Flatten[Table[
     FromDigits[Permute[IntegerDigits[BitXor[x, a], 2, n], sigma],
      2], {sigma, Permutations[Range[n]]}, {a, 0, 2^n - 1}, {x, 0,
      2^n - 1}], 1]];
countit =
  Function[{n},
   Sum[totalweight[x[[1]]] (x[[2]]^2), {x,
      Tally[cyclify /@ perms[n]]}]/((2^n) (n!))^2];
Table[countit[n], {n, 1, 5}] (*  Adam P. Goucher, Feb 12 2021 *)

Crossrefs

Sequence in context: A369089 A216354 A079179 * A061306 A249656 A248987

Adjacent sequences: A000651 A000652 A000653 * A000655 A000656 A000657

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Mar 15 2011

Status

approved