A332840
Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.
Original entry on oeis.org
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
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).
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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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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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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
A269799
Number of vertices of the fractional perfect matching polytope for the complete graph on n vertices.
Original entry on oeis.org
0, 1, 1, 3, 22, 25, 717, 1057, 39196, 98829
Offset: 1
For n=4 the fractional perfect matching polytope is the convex hull of the 3 perfect matchings of K_4, so a(4)=3. For n=6, in addition to the 15 perfect matchings of K_6, the 10 pairs of disjoint triangles with edge weights 1/2 are vertices of the polytope, so a(6)=25.
- Roger E. Behrend, Fractional perfect b-matching polytopes I: General theory, Linear Algebra and its Applications 439 (2013), 3822-3858.
- Thomas Christof, Sebastian Schenker, PORTA, Ruprecht-Karls-Universität Heidelberg.
- K. Eickmeyer and R. Yoshida, The Geometry of the Neighbor-Joining Algorithm for Small Trees, in: Proc. 3rd Int. Conference on Algebraic Biology, 2008, Castle of Hagenberg, Austria, Springer LNCS5147, arXiv:0908.0098 [math.CO], 2009.
A376922
Variance of n-th power of a standard normal random variable.
Original entry on oeis.org
1, 2, 15, 96, 945, 10170, 135135, 2016000, 34459425, 653836050, 13749310575, 316126087200, 7905853580625, 213439785208650, 6190283353629375, 191894675132160000, 6332659870762850625, 221641908024728441250, 8200794532637891559375, 319830558102716120460000
Offset: 1
If Z ~ N(0, 1), then Var(Z) = 1, Var(Z^2) = 2, Var(Z^3) = 15, etc.
- M. G. Kendall and A. Stuart, The Advanced Theory of Statistics Volume 1, Charles Griffin & Company, 1963, page 229.
A378100
Number of involutions in the symmetric group S_n with at least one fixed point.
Original entry on oeis.org
0, 1, 1, 4, 7, 26, 61, 232, 659, 2620, 8551, 35696, 129757, 568504, 2255345, 10349536, 44179711, 211799312, 962854399, 4809701440, 23103935021, 119952692896, 605135328337, 3257843882624, 17175956434375, 95680443760576, 525079354619951, 3020676745975552
Offset: 0
a(4) = 7: (1,2)(3)(4), (1,3)(2)(4), (1,4)(2)(3), (1)(2,3)(4), (1)(2,4)(3), (1)(2)(3,4), (1)(2)(3)(4).
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a := proc(n)
local k, total, deranged;
total := add(factorial(n)/(factorial(n-2*k)*2^k*factorial(k)), k=0..floor(n/2));
if mod(n, 2) = 0 then
deranged := factorial(n)/(2^(n/2)*factorial(n/2));
else
deranged := 0;
end if;
return total - deranged;
end proc:
seq(a(n), n=1..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [0, 1$2, 4][n+1],
a(n-1)+(2*n-3)*a(n-2)-(n-2)*(a(n-3)+(n-3)*a(n-4)))
end:
seq(a(n), n=0..27); # Alois P. Heinz, Nov 24 2024
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a[n_] := Module[{total, deranged},
total = Sum[n! / ((n - 2 k)! * 2^k * k!), {k, 0, Floor[n/2]}];
deranged = If[EvenQ[n], n! / (2^(n/2) * (n/2)!), 0];
total - deranged
];
Table[a[n], {n, 1, 20}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(x+x^2/2)-exp(x^2/2))) \\ Joerg Arndt, Nov 27 2024
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from math import factorial
def a(n):
total = sum(factorial(n) // (factorial(n - 2 * k) * 2**k * factorial(k))
for k in range(n // 2 + 1))
deranged = factorial(n) // (2**(n // 2) * factorial(n // 2)) if n % 2 == 0 else 0
return total - deranged
print([a(n) for n in range(1, 21)])
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