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A337015 Number of distinct transitive subgroups of S_n, counting conjugates as distinct.

%I A337015 #71 Aug 17 2025 21:11:59
%S A337015 1,1,2,9,20,279,512,19087,71602,636365,1517042,321965982,240609602,
%T A337015 8809543877,144729615032,26818608209252,6603755558402,2737593592637477
%N A337015 Number of distinct transitive subgroups of S_n, counting conjugates as distinct.
%C A337015 This sequence is the labeled version of A002106. I have proven that A005432(p)-a(p) == 1 (mod p) if p is prime. Based on n<= 18,
%C A337015 I have conjectured that log(A005432(n)/a(n)) > (n-1)/2 for n prime and log(A005432(n)/a(n)) < (n-1)/2 for n composite.
%C A337015 L. Pyber shows c^{n^2*(1+o(1))} <= a(n) <= d^{n^2*(1+o(1))}, c=2^{1/16}, d=24^{1/6}; conjectures lower bound is accurate.
%H A337015 John Erickson, <a href="/A337015/a337015_3.pdf">COUNTING TRANSITIVE SUBGROUPS OF Sn</a>
%H A337015 L. Pyber, <a href="https://www.jstor.org/stable/2946623">Enumerating Finite Groups of Given Order</a>, Ann. Math. 137 (1993), 203-220.
%e A337015 For n = 4 the following 9 subgroups of S_4 are transitive:
%e A337015 Group( [ (1,4)(2,3), (1,3)(2,4) ] )
%e A337015 Group( [ (1,3,2,4), (1,2)(3,4) ] )
%e A337015 Group( [ (1,4,3,2), (1,3)(2,4) ] )
%e A337015 Group( [ (1,2,4,3), (1,4)(2,3) ] )
%e A337015 Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )
%e A337015 Group( [ (1,2)(3,4), (1,3)(2,4), (1,4) ] )
%e A337015 Group( [ (1,2)(3,4), (1,4)(2,3), (2,4) ] )
%e A337015 Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )
%e A337015 Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )
%o A337015 (GAP)
%o A337015 NrTransSubSn:=function(n)
%o A337015 local s,cnt,i,u,no;
%o A337015   s:=SymmetricGroup(n);
%o A337015   cnt:=0;
%o A337015   for i in [1..NrTransitiveGroups(n)] do
%o A337015     u:=TransitiveGroup(n,i);
%o A337015     no:=Normalizer(s,u);
%o A337015     cnt:=cnt+IndexNC(s,no);
%o A337015     Print("Class ",i,", found ",IndexNC(s,no)," new, total: ",cnt,"\n");
%o A337015   od;
%o A337015   return cnt;
%o A337015 end; # _Alexander Hulpke_
%Y A337015 Cf. A005432, A002106.
%K A337015 nonn,more
%O A337015 1,3
%A A337015 _John Erickson_ and _Alexander Hulpke_, Nov 21 2020