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_{p|p prime, p<=n}
Sum_{i=1..floor(n/p)} 1 /(p^i*i!*(n-i*p)!*(p-1)).
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! *(n-i*p)! *(p-1)),
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!*(n-i*p)!*(p-1)), {p, Prime /@ Range[ PrimePi[n] ] }, {i, 1, Floor[n/p]}]; Table[a[n], {n, 1, 24}] (* Jean-François 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!*(n-k*p)!*p^k))/(p-1)))} \\ Andrew Howroyd, Jul 04 2018
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Chad Brewbaker, Feb 14 2011
STATUS
approved