A088042
Number of permutations in the symmetric group S_n such that the size of their conjugacy class is odd.
Original entry on oeis.org
1, 2, 4, 4, 16, 76, 232, 106, 946, 5716, 27776, 63856, 272416, 2390480, 10349536, 2027026, 34459426, 344594404, 2618916472, 10475679736, 54997260256, 568305978472, 3132225435824, 1807129471456, 12047128545376, 175289251587776, 1326384554695552
Offset: 1
Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 02 2003
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a:= n-> n!*add((binomial(n-(n mod 2), 2*k) mod 2)/((n-2*k)!*k!*2^k),
k=0..floor(n/2)):
seq(a(n), n=1..30); # Alois P. Heinz, May 01 2013
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a[n_] := n!*Sum[Mod[Binomial[n-Mod[n, 2], 2*k], 2]/((n-2*k)!*k!*2^k), {k, 0, Floor[n/2]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
A088335
Number of permutations in the symmetric group S_n such that the size of their centralizer is even.
Original entry on oeis.org
0, 0, 2, 4, 16, 96, 576, 4320, 31872, 298368, 3052800, 34387200, 404029440, 5339473920, 75893207040, 1139356108800, 18079668633600, 310896849715200, 5654417758617600, 107707364764876800, 2145784566959308800, 45252164164799692800, 1003024255355781120000
Offset: 0
Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 07 2003
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b:= proc(n, i) option remember; `if`(((i+1)/2)^2n, 0, (i-1)!*
b(n-i, i-2)*binomial(n, i))))
end:
a:= n-> n!-b(n, n-1+irem(n, 2)):
seq(a(n), n=0..30); # Alois P. Heinz, Jan 27 2020
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b[n_, i_] := b[n, i] = If[((i + 1)/2)^2 < n, 0, If[n == 0, 1, b[n, i - 2] + If[i > n, 0, (i - 1)! b[n - i, i - 2] Binomial[n, i]]]];
a[n_] := n! - b[n, n - 1 + Mod[n, 2]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 08 2020, after Alois P. Heinz *)
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seq(n)={Vec(serlaplace(1/(1-x) - prod(k=1, n, 1+(k%2)*x^k/k + O(x*x^n))), -(n+1))} \\ Andrew Howroyd, Jan 27 2020
a(0)=0 prepended and terms a(11) and beyond from
Andrew Howroyd, Jan 27 2020
A368867
Number of labeled mappings from n points to themselves with unique square root (endofunctions).
Original entry on oeis.org
1, 1, 2, 2, 56, 544, 5064, 69348, 1210320
Offset: 0
For n = 3, the two 3-cycles are unique square roots of each other.
Note that the identity map has more than one square root (i.e., 1->2, 2->1, 3->3 and itself).
Another non-example: 1->1, 2->2, 3->1 has two square roots: itself and 1->2, 2->1, 3->2.
In fact, the only endofunctions on {1,2,3} with unique square roots are the two 3-cycles, so a(3) = 2.
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function increment(size, t)
t[1] = t[1] + 1
local index = 1
while t[index] > size do
t[index] = 1
index = index + 1
if index > size then return true end
t[index] = t[index] + 1
end
return false
end
function get_initial(size)
local return_value = {}
for i = 1, size do return_value[i] = 1 end
return return_value
end
function compute(size)
candidate = get_initial(size)
return_value = 0
repeat
fun_root = get_initial(size)
fun_root_count = 0
repeat
for i = 1, size do
if candidate[i] ~= fun_root[fun_root[i]] then
goto next_fun_root
end
end
fun_root_count = fun_root_count + 1
if (fun_root_count == 2) then break end
::next_fun_root::
until (increment(size, fun_root))
if (fun_root_count == 1) then
return_value = return_value + 1
end
until (increment(size, candidate))
return return_value
end
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