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A373502 Size of a complete set of classical projective planes of prime power order q using a given set of q^2+q+1 points.

%I A373502 #27 Aug 01 2024 11:00:12
%S A373502 30,1108800,422378820864000,22104404984349254886359040000,
%T A373502 7197507570101063450093594584788274920397007398780859842560000000000000,
%U A373502 90399509839271079668491458784005740889517921781547218950513473999637402251071324160000000000000000
%N A373502 Size of a complete set of classical projective planes of prime power order q using a given set of q^2+q+1 points.
%C A373502 Projective planes of order q can be seen as a set of v=q^2+q+1 subsets (the lines) of size q+1 of a set of v points with the property that any two distinct lines have exactly one point in common. Obviously, this also holds for any of the v! permutations of the points. However, some of these permutations map the points of a given line l of the plane to the points of another line l' thereby fixing the set of lines and consequently the whole projective plane. These permutations form a subgroup called the collineation group of the projective plane. The size of this group for classical projective planes is given by A373501. Therefore, a(q) is the index of the collineation subgroup in the symmetric group of the points where q=A246655(n).
%D A373502 A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998, pages 118-132.
%H A373502 Paolo Xausa, <a href="/A373502/b373502.txt">Table of n, a(n) for n = 1..12</a>
%H A373502 IBM Research, <a href="https://research.ibm.com/haifa/ponderthis/challenges/May2024.html">Possible Dobble decks</a>, Ponder This Challenge May 2024, asked for a(3) and a(6).
%H A373502 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Order of PGL(3,q) and PGammaL(3,q)</a>.
%F A373502 a(n) = (q^2+q+1)!/(Omega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1)) where q = A246655(n).
%e A373502 For the Fano plane (q=2) there are 7 points and 7 lines. Of the 7!=5040 permutations of the points 168 fix the set of lines and thereby the whole plane. Consequently, there are 5040/168=30 different such planes for any given set of points. See A373501 for a more elaborate discussion of this example.
%t A373502 Map[(#^2+#-1)!/(PrimeOmega[#]*(#-1)^2*#^2) &, Select[Range[10], PrimePowerQ]] (* _Paolo Xausa_, Aug 01 2024 *)
%o A373502 (PARI) a=(q)->(q^2+q+1)!/(bigomega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1)) \\ q=A246655(n)
%Y A373502 Cf. A373501 for the size of the collineation groups.
%Y A373502 Cf. A335866 for the number of projective planes whose lines are cyclic difference sets.
%K A373502 nonn
%O A373502 1,1
%A A373502 _Ralf Goertz_, Jun 08 2024