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 1-3 of 3 results.

A332758 Number of fixed-point free involutions in the n-fold iterated wreath product of C_2.

Original entry on oeis.org

0, 1, 3, 17, 417, 206657, 44854599297, 2021158450131287670017, 4085251621720569336520310526902208564886017, 16689280870666586360302304039420036318743515355074220606298783584912362351240766944257
Offset: 0

Views

Author

Nick Krempel, Feb 22 2020

Keywords

Comments

Also the number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2^n.
Also the number of fixed-point free involutory automorphisms of the complete binary tree of height n.

Examples

			For n=2, the a(2)=3 fixed-point free involutions in C_2 wr C_2 (which is isomorphic to the dihedral group of degree 4) are (12)(34), (13)(24), and (14)(23).

Crossrefs

Cf. A332757.

Programs

  • Mathematica
    Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {0}, 9] (* Michael De Vlieger, Feb 25 2020 *)

Formula

a(n) = a(n-1)^2 + 2^(2^(n-1)-1), a(0) = 0.
a(n) ~ C^(2^n) for C = 1.467067423065535412629251121186749718727038915553188083467...
a(n) = 2^(2^(n-1)) * b(n), where b(0) = 0, b(n+1) = b(n)^2 + 1/2. - Jianing Song, Apr 09 2025

A332759 Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree n.

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 12, 12, 44, 44, 88, 88, 264, 264, 528, 528, 2064, 2064, 4128, 4128, 12384, 12384, 24768, 24768, 90816, 90816, 181632, 181632, 544896, 544896, 1089792, 1089792, 4292864, 4292864, 8585728, 8585728, 25757184, 25757184, 51514368, 51514368
Offset: 0

Views

Author

Nick Krempel, Feb 22 2020

Keywords

Comments

As the Sylow 2-subgroups of S_(2n) are isomorphic to those of S_(2n+1), the terms of this sequence come in pairs.
Also the number of involutory automorphisms (including identity) of the full binary tree with n leaves (hence 2n-1 vertices) in which all left children are complete (perfect) binary trees.

Examples

			For n=4, the a(4)=6 elements satisfying x^2=1 in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are the identity and (13), (24), (12)(34), (13)(24), (14)(23).

Crossrefs

Cf. A000085.

Programs

  • Maple
    b:= proc(n) b(n):=`if`(n=0, 1, b(n-1)^2+2^(2^(n-1)-1)) end:
a:= n-> (l-> mul(`if`(l[i]=1, b(i-1), 1), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 27 2020
  • Mathematica
    Join[{1}, Block[{nn = 33, s}, s = Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {1}, Ceiling@ Log2@ nn]; Array[Times @@ s[[Position[Reverse@ IntegerDigits[#, 2], 1][[All, 1]] ]] &, nn]]] (* Michael De Vlieger, Feb 25 2020 *)

Formula

a(n) = Product(A332757(k)) where k ranges over the positions of 1 bits in the binary expansion of n.
a(n) = big-Theta(C^n) for C = 1.6116626399..., i.e., A*C^n < a(n) < B*C^n for constants A, B (but it's not the case that a(n) ~ C^n as lim inf a(n)/C^n and lim sup a(n)/C^n differ).
Conjecture: B=1 and A=0.409091077245262341747187571213565366725933766222357989... - Vaclav Kotesovec, Feb 26 2020

Extensions

More terms from Alois P. Heinz, Feb 27 2020

A332868 Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.

Original entry on oeis.org

1, 2, 6, 12, 44, 88, 264, 528, 2064, 4128, 12384, 24768, 90816, 181632, 544896, 1089792, 4292864, 8585728, 25757184, 51514368, 188886016, 377772032, 1133316096, 2266632192, 8860471296, 17720942592, 53162827776, 106325655552, 389860737024, 779721474048, 2339164422144
Offset: 0

Views

Author

Nick Krempel, Feb 27 2020

Keywords

Comments

Bisection of A332759.

Examples

			For n=2, the a(2)=6 elements satisfying x^2=1 in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are the identity and (13), (24), (12)(34), (13)(24), (14)(23).

Links

Crossrefs

Cf. A332757, A332759.

Programs

  • Maple
    b:= proc(n) b(n):=`if`(n=0, 1, b(n-1)^2+2^(2^(n-1)-1)) end:
a:= n-> (l-> mul(`if`(l[i]=1, b(i), 1), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 27 2020
  • Mathematica
    b[n_] := b[n] = If[n == 0, 1, b[n - 1]^2 + 2^(2^(n - 1) - 1)];
a[n_] := Function[l, Product[If[l[[i]] == 1, b[i], 1], {i, 1, Length[l]}]][ Reverse @ IntegerDigits[n, 2]];
a /@ Range[0, 35] (* Jean-François Alcover, Apr 10 2020, after Alois P. Heinz *)
  • PARI
    a(n)={my(v=vector(logint(max(1,n), 2)+1)); v[1]=2; for(n=2, #v, v[n]=v[n-1]^2 + 2^(2^(n-1)-1)); prod(k=1, #v, if(bittest(n,k-1), v[k], 1))} \\ Andrew Howroyd, Feb 27 2020

Formula

a(n) = A332759(2*n).
a(n) = Product(A332757(k+1)) where k ranges over the positions of 1 bits in the binary expansion of n.
a(n) = big-Theta(C^n) for C = 2.59745646488..., i.e., A*C^n < a(n) < B*C^n for constants A, B (but it's not the case that a(n) ~ C^n as lim inf a(n)/C^n and lim sup a(n)/C^n differ).

Extensions

Terms a(17) and beyond from Andrew Howroyd, Feb 27 2020
Showing 1-3 of 3 results.

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