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A332840 Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2n. 1
1, 1, 3, 3, 17, 17, 51, 51, 417, 417, 1251, 1251, 7089, 7089, 21267, 21267, 206657, 206657, 619971, 619971, 3513169, 3513169, 10539507, 10539507, 86175969, 86175969, 258527907, 258527907, 1464991473, 1464991473, 4394974419, 4394974419, 44854599297, 44854599297, 134563797891 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

As a Sylow 2-subgroup of S_(4n+2) is isomorphic to a Sylow 2-subgroup of S_(4n) direct product C_2, the terms of this sequence come in equal pairs.

Also the number of fixed-point free involutory automorphisms of the full binary tree with 2n leaves (hence 4n-1 vertices) in which all left children are complete (perfect) binary trees.

LINKS

Table of n, a(n) for n=0..34.

FORMULA

a(n) = Product(A332758(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 = 2.1522868238..., 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).

a(n) = A332869(floor(n/2)). - Andrew Howroyd, Feb 27 2020

EXAMPLE

For n=2, the a(2)=3 fixed-point free involutions in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are (12)(34), (13)(24), and (14)(23).

MAPLE

b:= proc(n) b(n):=`if`(n=0, 0, 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..40);  # Alois P. Heinz, Feb 27 2020

MATHEMATICA

A332758[n_] := A332758[n] = If[n == 0, 0, A332758[n-1]^2 + 2^(2^(n-1)-1)];

a[n_] := Product[A332758[k], {k, Flatten@ Position[ Reverse@ IntegerDigits[ n, 2], 1]}];

a /@ Range[0, 34] (* Jean-Fran├žois Alcover, Apr 10 2020 *)

PROG

(PARI) a(n)={my(v=vector(logint(max(1, n), 2)+1)); v[1]=1; 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. A123023, A001147, A332758, A332869.

Sequence in context: A278627 A231908 A226610 * A279172 A268339 A224750

Adjacent sequences:  A332837 A332838 A332839 * A332841 A332842 A332843

KEYWORD

nonn

AUTHOR

Nick Krempel, Feb 26 2020

EXTENSIONS

Terms a(18) and beyond from Andrew Howroyd, Feb 27 2020

STATUS

approved

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Last modified November 28 13:47 EST 2021. Contains 349413 sequences. (Running on oeis4.)