A000653 -id:A000653 - cp's OEIS frontend

A000722 Number of invertible Boolean functions of n variables: a(n) = (2^n)!.

1, 2, 24, 40320, 20922789888000, 263130836933693530167218012160000000, 126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000

Offset: 0

N. J. A. Sloane

These are invertible maps from {0,1}^n to {0,1}^n, or in other words permutations of the 2^n binary vectors of length n.

2^n-th order derivative of n-th Mandelbrot iterate. Example: a(2) = 24, after one iterate in the Mandelbrot(z(n+1) = z(n)^2 + c) we have the function z(2) = z^4 + 2*c*z^2 + c^2 + c, for which the 4th-order derivative is 24. - Bert van den Bosch (zeusooooo(AT)hotmail.com), Sep 07 2003

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).

Delbert L. Johnson, Table of n, a(n) for n = 0..8
M. A. Harrison, The number of classes of invertible Boolean functions, J. ACM 10 (1963), 25-28. [Annotated scan of page 27 only]
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]
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
Index entries for sequences related to Boolean functions

Cf. A000652, A000653, A000654, A001038, A001537, A046856, A046857, A244061.

a[n_] := Factorial[2^n]; Table[a[n],{n,0,6}] (* James C. McMahon, Dec 06 2023 *)

atonfact(a,n) = {sr=0; for(x=1,n, y =(a^x)!; sr+=1.0/y; print1(y" "); ); print(); print(sr) }

a(n) = (2^n)!.

Sum of reciprocals = 0.54169146825401604874... - Cino Hilliard, Feb 08 2003

A000654 Invertible Boolean functions of n variables.

Original entry on oeis.org

1, 2, 52, 142090700, 17844701940501123640681816160, 59757436204078657410908164193971330396709572693816353610758085074676243846093824

Offset: 1

N. J. A. Sloane

nonn

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

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).

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

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 *)

More terms from Sean A. Irvine, Mar 15 2011

A000652 Invertible Boolean functions of n variables.

Original entry on oeis.org

1, 1, 6, 924, 81738720000, 256963707943061374889193111552000, 30978254928194376001814792318154658399138184007229852126545533479881553257431040000000

Offset: 0

N. J. A. Sloane

Equivalence classes of invertible maps from {0,1}^n to {0,1}^n, under action of (C_2)^n on both domain and range.

M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 154, problem 12.
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).

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

Cf. A001038, A000653, A000654, A000722, A001537, A046856, A046857.

A000652: n->2^(-2*n)*( (2^n)! + (2^n-1)^2 * ( (2^(n-1))! )*2^(2^(n-1)));

More terms from Vladeta Jovovic, Feb 23 2000

A000723 Invertible Boolean functions of n variables.

Original entry on oeis.org

1, 3, 840, 54486432000, 68523655451482690147713024000000, 2753622660283944533494648206058191857701074569760095316814277221684346880000000000000

Offset: 1

N. J. A. Sloane

nonn

Equivalence classes of invertible maps from {0,1}^n to {0,1}^n, under action of (C_2)^n on domain and permutation of variables on range. - Sean A. Irvine, Mar 15 2011

Also the number of distinct adjacency matrices of the n-hypercube graph Q_n. - Eric W. Weisstein, Mar 31 2017

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

Vincenzo Librandi, Table of n, a(n) for n = 1..8
M. A. Harrison, The number of classes of invertible Boolean functions, J. ACM 10 (1963), 25-28. [Annotated scan of page 27 only]
Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Hypercube Graph
Index entries for sequences related to Boolean functions

[Factorial(2^n - 1)/Factorial(n): n in [1..10]]; // Vincenzo Librandi, Mar 28 2012

f[n_] := (2^n - 1)!/n!; Array[f, 6] (* Robert G. Wilson v, Mar 14 2011 *)
Table[Gamma[2^n]/n!, {n, 6}] (* Eric W. Weisstein, Mar 31 2017 *)

a(n) = (2^n-1)!/n!. - Sean A. Irvine, Mar 15 2011

More terms from Sean A. Irvine, Mar 14 2011

A001038 Invertible Boolean functions with GL(n,2) acting on the domain and range.

Original entry on oeis.org

2, 2, 10, 52246, 2631645209645100680144, 312242081385925594286511113384607360432260178128338777217975928751832

Offset: 1

N. J. A. Sloane

nonn

The Lorens paper gives the incorrect value a(5)=2631645209645100680142. - Sean A. Irvine, Feb 27 2012

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).

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, Chen Lin, Computing the Number of the Equivalence Classes for Reversible Logic Functions, Int'l J. of Theor. Phys. (2020) Vol. 59, 2384-2396.
Index entries for sequences related to Boolean functions

Cf. A000652, A000653, A000654, A000722, A001537, A046856, A046857.

Corrected and extended by Sean A. Irvine, Feb 26 2012

A000724 Invertible Boolean functions of n variables.

Original entry on oeis.org

1, 3, 196, 3406687200, 2141364232858913975435172249600, 43025354066936633335853878219659247776604712057098163541301459387254457761792000000

Offset: 1

N. J. A. Sloane

nonn

Equivalence classes of invertible maps from {0,1}^n to {0,1}^n, under action of (C_2)^n on domain and F_n=[S_2]^(S_n) on range. - Sean A. Irvine, Mar 16 2011

Technical report version of Harrison's paper contains incorrect value for a(4). - Sean A. Irvine, Mar 16 2011

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

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

Table[((2^n)! + (2^n - 1) (2^(n - 1))! 2^(2^(n - 1)) * (n! * Sum[ (2^(n - 2 k) - 1)/((n - 2 k)!*k!), {k, 0, Floor[(n - 1)/2]}]))/(n! 2^(2 n)), {n, 6}] (* Michael De Vlieger, Aug 20 2017 *)

a(n) = ((2^n)! + (2^n-1) * (2^(n-1))! * 2^(2^(n-1)) * b(n)) / (n! * 2^(2*n)) where b(n) = n! * Sum_{k=0..floor((n-1)/2)} (2^(n-2*k)-1) / ((n - 2*k)! * k!). - Sean A. Irvine, Aug 20 2017

More terms from Sean A. Irvine, Mar 15 2011

A000725 Invertible Boolean functions of n variables.

Original entry on oeis.org

1, 2, 154, 2270394624, 571030462095782973206774552784, 3824475917061034074298122508414160251634847335755905881951011420229530501911521280

Offset: 1

N. J. A. Sloane

nonn

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

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

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

More terms from Sean A. Irvine, Mar 15 2011
a(6) corrected by Sean A. Irvine, May 29 2013
a(5) corrected by Sean A. Irvine, Jun 03 2013

cp's OEIS Frontend

A000722 Number of invertible Boolean functions of n variables: a(n) = (2^n)!.

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A000654 Invertible Boolean functions of n variables.

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A000652 Invertible Boolean functions of n variables.

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A000723 Invertible Boolean functions of n variables.

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A001038 Invertible Boolean functions with GL(n,2) acting on the domain and range.

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A000724 Invertible Boolean functions of n variables.

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A000725 Invertible Boolean functions of n variables.

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