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A248107
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Number of isomorphism classes of affine Mendelsohn triple systems of order n.
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22
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1, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 5, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 5, 0, 0, 2, 0
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OFFSET
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1,7
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COMMENTS
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A Mendelsohn triple system is affine if the associated quasigroup is affine, i.e, given by x*y=(1-f)(x)+f(y) over an abelian group (A,+) with an automorphism f.
For Steiner triple systems, the enumeration is settled by the following observation: a Steiner triple system is affine if and only if A=Z_3^n and f(x)=-x.
The existence spectrum (i.e., n such that a(n)>0) is A003136.
This is the sequence a(n) defined in the Donovan et al. paper.
The b(n) sequence defined there gives the number of non-affine systems.
The first 728 values of b(n) are now known: they are all zeros, except b(81) = 2, b(243) = 6, b(324) = 2, b(567)=4. We do not know b(729).
The reason is the following: it follows from the Galkin-Fischer-Smith theorem that, for n = m * 3^d, m not divisible by 3, we have b(n) = a(m) * b(3^d).
At the time of writing the paper, we could use known data about commutative Moufang loops to determine b(1) = b(3) = b(9) = b(27) = 0, and b(81) = 2. Later we managed to develop smarter enumeration methods that allowed us to determine b(243)=6: see Jedlička et al. (2007).
Since so many of the initial values of b(n), this does not have its own OEIS entry. (End)
Conjecture: This is the same sequences as A352561.(Note that A352561 has an explicit Dirichlet generating function.) - N. J. A. Sloane, Mar 21 2022
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LINKS
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PROG
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(GAP)
# For brevity, I do not exploit multiplicativity of a(n) here.
a := function(n)
local count, gg, g, autg, conj, f, b, x;
count := 0;
for gg in AllGroups(Size, n, IsAbelian, true) do
g := Image(IsomorphismPermGroup(gg), gg);
autg := AutomorphismGroup(g);
conj := List(ConjugacyClasses(autg), x->Representative(x));
for f in conj do
b := true;
for x in g do
if not
Image(f, Image(f, x))*Image(f, x^-1)*x = ()
then b := false; break;
fi;
od;
if b then count := count + 1; fi;
od;
od;
return count;
end;
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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