cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 24 results. Next

A057156 Number of functions from {0,1}^n to {0,1}^n.

Original entry on oeis.org

1, 4, 256, 16777216, 18446744073709551616, 1461501637330902918203684832716283019655932542976, 39402006196394479212279040100143613805079739270465446667948293404245721771497210611414266254884915640806627990306816
Offset: 0

Views

Author

Henry Bottomley, Aug 15 2000

Keywords

Comments

a(n) is the number of subdivisions of the Brownian motion on the unit interval at the n-th stage of subdivision. - Stephen Crowley, Apr 12 2007

Examples

			a(1)=4 since the possibilities are f(0)=0, f(1)=0; f(0)=0, f(1)=1; f(0)=1, f(1)=0; f(0)=1, f(1)=1.
For n=3: we need to count maps from a set with 8 points to a set with 8 points.  There are 8^8 such functions, that is, a(3) = 8^8 = 2^24 = 16777216. - _N. J. A. Sloane_, Mar 05 2023

References

  • François Robert, Discrete Iterations: A Metric Study, Springer-Verlag, 1986, p. 167.
  • Norbert Wiener, Nonlinear Problems in Random Theory, MIT Press Classic, 1958, Lecture 1.

Links

Crossrefs

Cf. A000079, A000312, A000722, A001146, A036289, A043322, A057157, A092258, A134880.

Programs

  • Mathematica
    lst={};Do[AppendTo[lst,(2^n)^(2^n)],{n,0,8}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 02 2009 *)
  • PARI
    a(n)=1<<(n<Charles R Greathouse IV, Jan 19 2012

Formula

a(n) = (2^n)^(2^n) = A000312(A000079(n)) = A000079(A036289(n)) = A001146(n)^n = A000722(n) + A057157(n).
Sum_{n>=1} 1/a(n) = A134880. - Amiram Eldar, Nov 15 2020

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 02 2009

A000653 Invertible Boolean functions of n variables.

Original entry on oeis.org

2, 7, 1172, 36325278240, 18272974787063551687986348306336, 244766458691906180755079840538506099505695351680436638205950721844523539763881615360
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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

Programs

  • Mathematica
    n = 20; (* the value of n is chosen here *)
e = Table[2, {n}];(*the sequence e*)
Do[
DD = Divisors[k];
e[[k]] = (2^k - Sum[DD[[j]] e[[DD[[j]]]], {j, 1, Length[DD] - 1}])/
   k, {k, 2, n}]
PP = IntegerPartitions[n]; npp =
Length[PP];(*the list of partitions of n*)
(*the maximum length of a cycle in sigma'*)
mlcm = Apply[Max, Table[Apply[LCM, PP[[p]]], {p, npp}]];
(*decompositions of n corresponding to partitions*)
P = Table[0, {i, npp}, {j, n}];
Do[Do[P[[ipp, PP[[ipp, i]]]]++, {i, Length[PP[[ipp]]]}], {ipp, npp}]
EmptyList = Table[0, {j, mlcm}];(*used to initialize spec(sigma')*)
Vn = 0; Do[(*the main loop through all partitions of n*)
PPP = PP[[p]]; np = Length[PPP];(*current partition*)
Spec = EmptyList;(*initialization of spec(sigma')*)
divsets = {};
nd = 1;
Do[(*k is the index of the current Partition element*)
  DD = Divisors[PPP[[k]]];
  AppendTo[divsets, DD];
  nd *= Length[DD], {k, 1, np}];
(*divsets is the list of the sets of divisors of cycle lengths in \
sigma*)
Descartes = Tuples[divsets]; (* nd is the length of Descartes *)
Do[ (*loop through Descartes product *)
  product = Descartes[[id]];
  npr = Length[product];
  lcm = 1; prx = 1; pry = 1;
  (* Theorem 2 *)
  Do[
   lcm = LCM[lcm, product[[ipr]]];
   prx *= product[[ipr]];
   pry *= e[[product[[ipr]]]], {ipr, npr}];
  Spec[[lcm]] += prx*pry/lcm, {id, nd}];
(* Theorem 1 *)
numerator = Product[i^Spec[[i]] Spec[[i]]!, {i, Length[Spec]}];
denominatorr = Product[i^P[[p, i]] P[[p, i]]!, {i, n}];
sum = numerator/denominatorr^2;
Vn += sum, {p, npp}]
Print[{"V_n = ", Vn}] (* Marko Caric, Jan 30 2016 *)

Extensions

a(6) from Sean A. Irvine, Mar 15 2011

A051459 Number of orderings of the subsets of a set with n elements that are compatible with the subsets' sizes; i.e., if A, B are two subsets with A <= B then Card(A) <= Card(B).

Original entry on oeis.org

1, 1, 2, 36, 414720, 189621927936000000, 2156695499113014719143826715127578624000000000000
Offset: 0

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 15 2003

Keywords

Comments

a(7) has 127 digits and too large to include in sequence. - Ray Chandler, Nov 22 2003
From Valentin Bakoev, Nov 20 2017, May 17 2019: (Start)
a(n) is the number of possible orderings of the vectors of the n-dimensional Boolean cube (hypercube) {0,1}^n in accordance with their (Hamming) weights. For arbitrary vectors u, v of {0, 1}^n, if wt(u)
a(n) is also the number of all possible topological orders (sortings) of the directed acyclic graph (DAG) defined by the same poset: {0,1}^n and the relation weight order as it is defined and explained above.
Both comments correspond to the name of the sequence since the corresponding Boolean algebras are isomorphic. (End)

Links

Crossrefs

Cf. A000722, A001142, A022914, A294648.

Programs

  • Maple
    a:= n-> mul(binomial(n, i)!, i=0..n):
seq(a(n), n=0..6);  # Alois P. Heinz, Nov 20 2017
  • Mathematica
    Array[Product[Binomial[#, i]!, {i, #}] &, 7, 0] (* Michael De Vlieger, Nov 20 2017 *)
  • Maxima
    a(n):= prod(binomial(n,k)!,k,0,n); /* Valentin Bakoev, May 17 2019 */
  • PARI
    a(n) = prod(k=0, n, binomial(n, k)!); \\ Michel Marcus, May 18 2019

Formula

a(n) = C(n, 0)! * C(n, 1)! * C(n, 2)! * ... * C(n, n)! = A000722(n) / A022914(n).
log(a(n)) ~ log(2) * n * 2^n. - Vaclav Kotesovec, Nov 24 2023

Extensions

More terms from Ray Chandler, Nov 22 2003
a(0)=1 prepended by Alois P. Heinz, Nov 20 2017

A000654 Invertible Boolean functions of n variables.

Original entry on oeis.org

1, 2, 52, 142090700, 17844701940501123640681816160, 59757436204078657410908164193971330396709572693816353610758085074676243846093824
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

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

Programs

  • 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 *)

Extensions

More terms from Sean A. Irvine, Mar 15 2011

A001537 Invertible Boolean functions with AG(n,2) acting on the domain and range.

Original entry on oeis.org

1, 1, 4, 302, 2569966041123963092, 76230976900860740792605252293646252383143627390965685153124757864
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

A128904 is possibly an erroneous version of this sequence. - Sean A. Irvine, Feb 27 2012

References

  • 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

Extensions

a(6) from Sean A. Irvine, Feb 26 2012

A000652 Invertible Boolean functions of n variables.

Original entry on oeis.org

1, 1, 6, 924, 81738720000, 256963707943061374889193111552000, 30978254928194376001814792318154658399138184007229852126545533479881553257431040000000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

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

Formula

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

Extensions

More terms from Vladeta Jovovic, Feb 23 2000

A046856 a(n) = (2^n)!/4^n, with a(1)=1, a(2)=2.

Original entry on oeis.org

1, 2, 630, 81729648000, 256963707943060088053923840000000, 30978254928194376001814792318154658399137088909801072314160618743948902400000000000000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The next term has 212 digits. - Harvey P. Dale, May 31 2021

Links

Programs

  • Maple
    a:= n-> ceil((2^n)!/4^n):
seq(a(n), n=1..6);  # Alois P. Heinz, May 31 2021
  • Mathematica
    Join[{1,2},Table[(2^n)!/4^n,{n,3,6}]] (* Harvey P. Dale, May 31 2021 *)

A046857 a(n) = (2^n)!/(n!^2).

Original entry on oeis.org

2, 6, 1120, 36324288000, 18272974787062050706056806400000, 244766458691906180755079840538505942906762183978675139272380197483053056000000000000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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

Crossrefs

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

Extensions

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

A057157 Number of non-invertible functions from {0,1}^n to {0,1}^n.

Original entry on oeis.org

0, 2, 232, 16736896, 18446723150919663616, 1461501637330639787366751139186115801643772542976
Offset: 0

Views

Author

Henry Bottomley, Aug 15 2000

Keywords

Links

Crossrefs

Cf. A057156, A000722, A036679, A000079.

Programs

  • Magma
    [(2^n)^(2^n)-Factorial(2^n): n in [0..5]]; // Vincenzo Librandi, Aug 22 2011
  • Mathematica
    Table[(2^n)^(2^n) - (2^n)!, {n,0,5}] (* G. C. Greubel, Nov 08 2018 *)
  • PARI
    vector(6,n,n--; (2^n)^(2^n) - (2^n)!) \\ G. C. Greubel, Nov 08 2018

Formula

a(n) = (2^n)^(2^n) - (2^n)! = A057156(n) - A000722(n) = A036679(A000079(n)).
Showing 1-10 of 24 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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