A303728 -id:A303728 - cp's OEIS frontend

A051636 Number of "labeled" cyclic subgroups of alternating group A_n.

1, 1, 2, 8, 32, 167, 947, 6974, 53426, 454682, 4303532, 50366912, 553031624, 6760260236, 90333982832, 1369522152392, 20986020606632, 350528387240264, 5751957395258096, 111685506968916032, 2139383543480892032, 41770889787378732752, 869742098042083451264

Offset: 1

Vladeta Jovovic

Alois P. Heinz, Table of n, a(n) for n = 1..140
L. Naughton and G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8.

Row sums of A303728.

Cf. A000010, A051625.

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:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 03 2018

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;
Array[a, 25] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)

\\ 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

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.

A181950 Weighted sum of all cyclic subgroups of the Alternating Group A_n.

Original entry on oeis.org

1, 1, 4, 19, 91, 571, 4096, 38599, 370399, 3771751, 40020916, 486887611, 6457566259, 97397627419, 1566407932636, 25622476773391, 416792928270751, 7346982309720079, 141863542111338124, 2968348473040595971, 65223378275792128771, 1460499016109864574691, 32600807940657384926884

Offset: 1

Olivier Gérard, Apr 03 2012

nonn

Sum of the order of all cyclic subgroups of Alt_n.
Each permutation is counted as many times as it appears in a cyclic subgroup.
a(7) = 2^12 is remarkable as a power of 2.

			a(5) = 1*1 + 2*15 + 3*10 + 5*6 = 1 + 30 +30 +30 = 91.

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A051593, A051636, A303728.

\\ 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, my(d=lcm(Vec(p))); s+=d*permcount(p)/eulerphi(d))); s} \\ Andrew Howroyd, Jul 03 2018

a(n) = Sum_{k=1..A051593(n)} k*A303728(n, k). - Andrew Howroyd, Jul 03 2018

Terms a(9) and beyond from Andrew Howroyd, Jul 03 2018

cp's OEIS Frontend

A051636 Number of "labeled" cyclic subgroups of alternating group A_n.

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