A181951
Number of cyclic subgroups of prime order in the Alternating Group A_n.
Original entry on oeis.org
0, 0, 1, 7, 31, 121, 526, 2227, 9283, 54931, 694156, 6104011, 76333687, 872550043, 7491293356, 49469173951, 1571562887071, 24729107440927, 584036983443568, 8662243014551731, 87570785839885951, 1147293350653737211, 66175018194591458692, 1378758190497550145383
Offset: 1
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a[n_] := Sum[If[PrimeQ[p], Sum[If[p > 2 || Mod[k, 2] == 0, n!/(k!*(n - k*p)!*p^k)/(p - 1), 0], {k, 1, n/p}], 0], {p, 2, n}];
Array[a, 24] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)
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a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, if(p>2||k%2==0, n!/(k!*(n-k*p)!*p^k)))/(p-1)))}
A214003
Number of degree-n permutations of prime order.
Original entry on oeis.org
0, 1, 5, 17, 69, 299, 1805, 9099, 37331, 205559, 4853529, 49841615, 789513659, 9021065871, 70737031469, 420565124399, 22959075244095, 385032305178719, 10010973102879761, 152163983393187399, 1498273284120348539, 15639918041915598815, 1296204202723400597109
Offset: 1
The symmetric group S_5 has 25 elements of order 2, 20 elements of order 3, and 24 elements of order 5. All other elements are of nonprime order (1, 4, or 6), so a(5) = 25 + 20 + 24 = 69.
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b:= proc(n,p) option remember;
`if`(n add(b(n, ithprime(i)), i=1..numtheory[pi](n)):
seq(a(n), n=1..30); # Alois P. Heinz, Feb 16 2013
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
add(b(n-j, ilcm(j, g))*(n-1)!/(n-j)!, j=1..n))
end:
a:= n-> b(n, 1):
seq(a(n), n=1..23); # Alois P. Heinz, Jan 19 2023
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f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n],PrimeQ[Apply[LCM, #]] &]]], {n, 1,23}] (* Geoffrey Critzer, Nov 08 2015 *)
A181954
Weighted sum of all cyclic subgroups of prime order of the symmetric group S_n.
Original entry on oeis.org
0, 2, 9, 30, 110, 450, 2457, 11774, 47910, 264630, 5565065, 56021922, 866143434, 9894742130, 78233264865, 470058202350, 24530748587102, 409761902222094, 10595012400106545, 160826238368038490, 1585844131838898330, 16787211702213659322, 1362379222505265018329
Offset: 1
The symmetric group S_3 has one subgroup of order 3 and three subgroups of order 2, and no other subgroups of prime order. So a(3) = 3 + 2 + 2 + 2 = 9.
Cf.
A181955 (the alternating case).
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a[n_] := Sum[If[PrimeQ[p], Sum[n!/(k!*(n-k*p)!*p^k), {k, 1, n/p}]*p/(p-1), 0], {p, 2, n}];
Array[a, 24] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)
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a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, n!/(k!*(n-k*p)!*p^k))*p/(p-1)))} \\ Andrew Howroyd, Jul 03 2018
A387061
Number of Sylow permutations in S_n.
Original entry on oeis.org
0, 0, 1, 5, 23, 99, 479, 2645, 19599, 154007, 1271519, 11688489, 126123095, 1481833859, 15162417087, 126294191309, 2497347563039, 53642575418415, 937621220224319, 17116389710781136, 281286727706878100
Offset: 0
For a(3) = 5, the solutions are { ((1 2)) ((1 3)) ((2 3)) ((1 2 3)) ((1 3 2)) } (1-length cycles are omitted from the decomposition).
For a(4) = 23, the solutions are (up to permutation) { ((1 2)) ((1 2)(3 4)) ((1 2 3 4)) ((1 2 3)) }.
For a(6) = 479, the solutions include (for example) ((1 2)(3 4 5 6)) but not ((1 2)(3 4 5)).
Showing 1-4 of 4 results.
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