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A186202 The maximal set of disjoint prime cycle permutations on n elements which generate unique subgroups of S(n). 4

%I #45 Jul 04 2018 02:29:46

%S 0,1,4,13,41,151,652,2675,10579,59071,711536,6180307,76629775,

%T 873676259,7496233396,49493077951,1571673343007,24729597043375,

%U 584039297226784,8662254974851091,87570847718549791,1147293660298060507,66175019781864421220,1378758199197350367079

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

%C 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.

%C 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.

%C 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_]

%C Number of cyclic subgroups of prime order in the symmetric group. [_Olivier Gérard_, Apr 03 2012]

%H Alois P. Heinz, <a href="/A186202/b186202.txt">Table of n, a(n) for n = 1..452</a>

%H Chad Brewbaker, <a href="http://sites.google.com/site/brewsite/eprints/minCyclicj.pdf">The exact classical query complexity of the hidden subgroup detection problem</a>, (2008)

%F a(n) = n! * Sum_{p|p prime, p<=n}

%F Sum_{i=1..floor(n/p)} 1 /(p^i*i!*(n-i*p)!*(p-1)).

%e a(2): (0,1).

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

%p with(numtheory):

%p a:= n-> n! *add(add(1/(p^i *i! *(n-i*p)! *(p-1)),

%p i=1..floor(n/p)), p={ithprime(k) $k=1..pi(n)}):

%p seq(a(n), n=1..25); # _Alois P. Heinz_, Apr 07 2011

%t 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_ *)

%o (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

%Y Cf. A181949, A181951.

%K nonn,nice

%O 1,3

%A _Chad Brewbaker_, Feb 14 2011

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