

A186202


The maximal set of disjoint prime cycle permutations on n elements which generate unique subgroups of S(n).


4



0, 1, 4, 13, 41, 151, 652, 2675, 10579, 59071, 711536, 6180307, 76629775, 873676259, 7496233396, 49493077951, 1571673343007, 24729597043375, 584039297226784, 8662254974851091, 87570847718549791, 1147293660298060507, 66175019781864421220, 1378758199197350367079
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Given an subgroup g of S(n) that is unknown and an oracle which takes as input a permutation on n elements, and returns true IFF the permutation is a member of the subgroup; a(n) is the minimum number of permutations that need to be queried to prove g consists of only the identity permutation.
a(n) is the size of a minimum dominating set in the permutation detection graph G(n) with loops removed. Let G(n) have n! vertices, each labeled with unique permutation from S(n). There is a directed edge from i>j IFF the permutation label on vertex j is in the group generated by the single permutation designated by the label on i.
a(n) is an exact bound on the worst case complexity of nontrivial automorphism detection for a generic combinatorial object on n elements. For tractable problem sizes this can yield significant savings over the brute force testing of all n!1 nontrivial permutations. [Chad Brewbaker]
Number of cyclic subgroups of prime order in the symmetric group. [Olivier Gérard, Apr 03 2012]


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..452
Chad Brewbaker, The exact classical query complexity of the hidden subgroup detection problem, (2008)


FORMULA

a(n) = n! * Sum_{pp prime, p<=n}
Sum_{i=1..floor(n/p)} 1 /(p^i*i!*(ni*p)!*(p1)).


EXAMPLE

a(2): (0,1).
a(3): (1,2), (0,1), (0,1,2), (0,2).


MAPLE

with(numtheory):
a:= n> n! *add(add(1/(p^i *i! *(ni*p)! *(p1)),
i=1..floor(n/p)), p={ithprime(k) $k=1..pi(n)}):
seq(a(n), n=1..25); # Alois P. Heinz, Apr 07 2011


MATHEMATICA

a[n_] := n!*Sum[ 1/(p^i*i!*(ni*p)!*(p1)), {p, Prime /@ Range[ PrimePi[n] ] }, {i, 1, Floor[n/p]}]; Table[a[n], {n, 1, 24}] (* JeanFrançois Alcover, Aug 20 2013, after Alois P. Heinz *)


PROG

(PARI) a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, n!/(k!*(nk*p)!*p^k))/(p1)))} \\ Andrew Howroyd, Jul 04 2018


CROSSREFS

Cf. A181949, A181951.
Sequence in context: A005002 A085507 A121654 * A036366 A303863 A255836
Adjacent sequences: A186199 A186200 A186201 * A186203 A186204 A186205


KEYWORD

nonn,nice


AUTHOR

Chad Brewbaker, Feb 14 2011


STATUS

approved



