Mosaics of combinatorial designs were defined in [3]. Some interesting small examples are constructed in [5]. This web page contains files with the examples in a format suitable for GAP [2], where they can be analyzed using the PAG package [4].
These are the first nontrivial examples of inhomogenous mosaics, comprising designs with distinct parameters. The example from [5] is given in the first file, and the second file contains more mosaics with these parameters.
The mosaics were constructed from difference families in the
cyclic group Z13. The files can be read into GAP
by typing:
gap> LoadPackage("PAG");
gap> m:=ReadMat("13-346ex.txt");;
gap> MosaicParameters(m[1]);
"2-(13,3,1) + 2-(13,4,2) + 2-(13,6,5)"
In [3], a construction of mosaic from resolvable designs was described. Here are some examples of homogenous mosaics containing designs that are not resolvable, which cannot be obtained by the construction from [3]. The first two files are examples from [5]. The third file contains more 2-(9,3,2) mosaics with an automorphism of order 3. The fourth file contains 2-(12,4,3) mosaics with an automorphism of order 11.
Properties of these mosaics can be explored in the PAG package [3]:
gap> m1:=ReadMat("9-3-2ex1.txt")[1];;
gap> d1:=MosaicToBlockDesigns(m1);;
gap> Size(BlockDesignFilter(d1));
1
gap> MakeResolutionsComponent(d1[1]);
gap> d1[1].resolutions.list;
[ ]
gap> m2:=ReadMat("9-3-2ex2.txt")[1];;
gap> d2:=MosaicToBlockDesigns(m2);;
gap> Size(BlockDesignFilter(d2));
3
gap> MakeResolutionsComponent(d2[1]);
gap> MakeResolutionsComponent(d2[2]);
gap> MakeResolutionsComponent(d2[3]);
gap> d2[1].resolutions.list;
[ rec( autGroup := Group([ (1,5,8)(2,6,9)(3,4,7), (1,7,6)(2,8,4)(3,9,5), (1,2)(4,5)(7,9) ]), partition := [ rec( blocks := [ [ 1, 2, 4 ], [ 3, 8, 9 ], [ 5, 6, 7 ] ], isBlockDesign := true, v := 9 ), rec( blocks := [ [ 1, 2, 5 ], [ 3, 7, 8 ], [ 4, 6, 9 ] ], isBlockDesign := true, v := 9 ), rec( blocks := [ [ 1, 3, 4 ], [ 2, 7, 9 ], [ 5, 6, 8 ] ], isBlockDesign := true, v := 9 ), rec( blocks := [ [ 1, 3, 6 ], [ 2, 7, 8 ], [ 4, 5, 9 ] ], isBlockDesign := true, v := 9 ), rec( blocks := [ [ 1, 5, 8 ], [ 2, 6, 9 ], [ 3, 4, 7 ] ], isBlockDesign := true, v := 9 ), rec( blocks := [ [ 1, 6, 7 ], [ 2, 4, 8 ], [ 3, 5, 9 ] ], isBlockDesign := true, v := 9 ), rec( blocks := [ [ 1, 7, 9 ], [ 2, 3, 5 ], [ 4, 6, 8 ] ], isBlockDesign := true, v := 9 ), rec( blocks := [ [ 1, 8, 9 ], [ 2, 3, 6 ], [ 4, 5, 7 ] ], isBlockDesign := true, v := 9 ) ] ) ]
gap> d2[2].resolutions.list;
[ ]
gap> d2[3].resolutions.list;
[ ]
This is a mosaic of projective planes of order 3. It has full automorphism group of order 3 and cannot be constructed by tiling groups with difference sets, as in [1].
gap> m:=ReadMat("13-4-1.txt")[1];;
gap> aut:=MatAut(m);
Group([ (1,3,2)(4,6,5)(7,9,8)(10,12,11)(14,16,15)(17,19,18)(20,22,21)(23,25,24)(28,30,29) ])
gap> Size(aut);
3
More PAG commands for mosaics of combinatorial designs are described in the package manual [6].
Vedran Krcadinac,