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

0,2

Comments

Bisection of A332759.

Links

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

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).

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)):
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 *)

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

Crossrefs

Cf. A332757, A332759.

Sequence in context: A056744 A344184 A164859 * A261467 A180070 A177834

Adjacent sequences: A332865 A332866 A332867 * A332869 A332870 A332871

Keyword

nonn

Author

Nick Krempel, Feb 27 2020

Extensions

Terms a(17) and beyond from Andrew Howroyd, Feb 27 2020

Status

approved