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

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

Author

Nick Krempel, Feb 27 2020

Keywords

• nonn

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

• Alois P. Heinz, Table of n, a(n) for n = 0..2412

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