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Number of fixed-point free involutions in the n-fold iterated wreath product of C_2.
+10
3
0, 1, 3, 17, 417, 206657, 44854599297, 2021158450131287670017, 4085251621720569336520310526902208564886017, 16689280870666586360302304039420036318743515355074220606298783584912362351240766944257
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.
FORMULA
a(n) = a(n-1)^2 + 2^(2^(n-1)-1), a(0) = 0.
a(n) ~ C^(2^n) for C = 1.467067423065535412629251121186749718727038915553188083467...
EXAMPLE
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).
MATHEMATICA
Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {0}, 9] (* Michael De Vlieger, Feb 25 2020 *)
Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree n.
+10
1
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
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.
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
EXAMPLE
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).
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)):
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 *)
Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.
+10
1
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
FORMULA
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).
EXAMPLE
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).
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)):
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]];
PROG
(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
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