A332840 Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.
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
Offset: 0
Keywords
Examples
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).
Programs
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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
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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-François Alcover, Apr 10 2020 *)
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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
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
Extensions
Terms a(18) and beyond from Andrew Howroyd, Feb 27 2020
Comments