|
|
A051636
|
|
Number of "labeled" cyclic subgroups of alternating group A_n.
|
|
7
|
|
|
1, 1, 2, 8, 32, 167, 947, 6974, 53426, 454682, 4303532, 50366912, 553031624, 6760260236, 90333982832, 1369522152392, 20986020606632, 350528387240264, 5751957395258096, 111685506968916032, 2139383543480892032, 41770889787378732752, 869742098042083451264
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 1/2*Sum_{pi} (1+(-1)^(k_2+k_4+...)) * n!/(k_1!*1^k_1*k_2!*2^k_2*...*k_n!*n^k_n*phi(lcm{i:k_i != 0})), where pi runs through all partitions k_1+2*k_2+...+n*k_n=n and phi is Euler's function.
|
|
MAPLE
|
b:= proc(n, i, m, t) option remember; `if`(n=0, (1+(-1)^t)/numtheory
[phi](m), add(1/j!/i^j*b(n-i*j, i-1, ilcm(m, `if`(j=0, 1, i)),
irem(t+j*irem(i+1, 2), 2)), j=`if`(i=1, n, 0..n/i)))
end:
a:= n-> n!*b(n$2, 1, 0)/2:
|
|
MATHEMATICA
|
f[list_] :=Total[list]!/(Apply[Times, list]*Apply[Times, Map[Length, Split[list]]!])/EulerPhi[Apply[LCM, list]]; Table[Total[Map[f,
Select[IntegerPartitions[n], EvenQ[Length[Select[#, EvenQ[#] &]]] &]]], {n, 1, 21}] (* Geoffrey Critzer, Oct 03 2015 *)
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, (1 + (-1)^t)/
EulerPhi[m], If[i == 0, 0, Sum[1/j!/i^j*b[n - i*j, i - 1, LCM[m,
If[j == 0, 1, i]], Mod[t+j*Mod[i+1, 2], 2]], {j, Range[0, n/i]}]]];
a[n_] := n! b[n, n, 1, 0]/2;
|
|
PROG
|
(PARI) \\ permcount is number of permutations of given type.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
a(n)={my(s=0); forpart(p=n, if(sum(i=1, #p, p[i]-1)%2==0, s+=permcount(p) / eulerphi(lcm(Vec(p))))); s} \\ Andrew Howroyd, Jul 03 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|