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.

30, 1108800, 422378820864000, 22104404984349254886359040000, 7197507570101063450093594584788274920397007398780859842560000000000000, 90399509839271079668491458784005740889517921781547218950513473999637402251071324160000000000000000

Offset: 1

Author

Ralf Goertz, Jun 08 2024

Keywords

• nonn

Comments

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).

Examples

			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.
		

References

• A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998, pages 118-132.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..12
• IBM Research, Possible Dobble decks, Ponder This Challenge May 2024, asked for a(3) and a(6).
• Wikipedia, Order of PGL(3,q) and PGammaL(3,q).

Crossrefs

Cf. A373501 for the size of the collineation groups.

Cf. A335866 for the number of projective planes whose lines are cyclic difference sets.

Programs

Mathematica

Map[(#^2+#-1)!/(PrimeOmega[#]*(#-1)^2*#^2) &, Select[Range[10], PrimePowerQ]] (* Paolo Xausa, Aug 01 2024 *)

PARI

a=(q)->(q^2+q+1)!/(bigomega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1)) \\ q=A246655(n)

Formula

a(n) = (q^2+q+1)!/(Omega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1)) where q = A246655(n).