[design] 9 Partitioning block designs

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# 9 Partitioning block designs

### Sections

1. Partitioning a block design into block designs
2. Computing resolutions

This chapter describes the function `PartitionsIntoBlockDesigns` which can classify partitions of (the block multiset of) a given block design into (the block multisets of) block designs having user-specified properties. We also describe `MakeResolutionsComponent` which is useful for the special case when the desired partitions are resolutions.

## 9.1 Partitioning a block design into block designs

- `PartitionsIntoBlockDesigns(` param`)`

  Let D equal ``param`.blockDesign`. This function returns a list PL of partitions of (the block multiset of) D. Each element of PL is a record with one component `partition`, and, in most cases, a component `autGroup`. The `partition` component gives a list P of block designs, all with the same point set as D, such that the list of (the block multisets of) the designs in ``P`.partition` forms a partition of (the block multiset of) D. The component ``P`.autGroup`, if bound, gives the automorphism group of the partition, which is the stabilizer of the partition in the automorphism group of D. The precise interpretation of the output depends on param, described below.

  The required components of param are `blockDesign`, `v`, `blockSizes`, and `tSubsetStructure`.

  ``param`.blockDesign` is the block design to be partitioned.

  ``param`.v` must be a positive integer, and specifies that for each block design in each partition in PL, the points are 1,...,``param`.v`. It is required that ``param`.v` be equal to ``param`.blockDesign.v`.

  ``param`.blockSizes` must be a set of positive integers, and specifies that the block sizes of each block design in each partition in PL will be contained in ``param`.blockSizes`.

  ``param`.tSubsetStructure` must be a record, having components `t`, `partition`, and `lambdas`. Let t be equal to ``param`.tSubsetStructure.t`, partition be ``param`.tSubsetStructure.partition`, and lambdas be ``param`.tSubsetStructure.lambdas`. Then t must be a non-negative integer, partition must be a list of non-empty sets of t-subsets of `[1..`param`.v]`, forming an ordered partition of all the t-subsets of `[1..`param`.v]`, and lambdas must be a list of distinct non-negative integers (not all zero) of the same length as partition. This specifies that for each design in each partition in PL, each t-subset in ``partition`[`i`]` will occur exactly ``lambdas`[`i`]` times, counted over all blocks of the design. For binary designs, this means that each t-subset in ``partition`[`i`]` is contained in exactly ``lambdas`[`i`]` blocks. The `partition` component is optional if lambdas has length 1. We require that t is less than or equal to each element of ``param`.blockSizes`, and that each block of ``param`.blockDesign` contains at least t distinct elements.

  The optional components of param are used to specify additional constraints on the partitions in PL, or to change default parameter values. These optional components are `r`, `b`, `blockNumbers`, `blockIntersectionNumbers`, `blockMaxMultiplicities`, `isoGroup`, `requiredAutSubgroup`, and `isoLevel`. Note that the last three of these optional components refer to the partitions and not to the block designs in a partition.

  ``param`.r` must be a positive integer, and specifies that in each design in each partition in PL, each point must occur exactly ``param`.r` times in the list of blocks.

  ``param`.b` must be a positive integer, and specifies that each design in each partition in PL has exactly ``param`.b` blocks.

  ``param`.blockNumbers` must be a list of non-negative integers, the i-th element of which specifies the number of blocks whose size is equal to ``param`.blockSizes[`i`]` (in each design in each partition in PL). The length of ``param`.blockNumbers` must equal that of ``param`.blockSizes`, and at least one entry of ``param`.blockNumbers` must be positive.

  ``param`.blockIntersectionNumbers` must be a symmetric matrix of sets of non-negative integers, the `[`i`][`j`]`-element of which specifies the set of possible sizes for the intersection of a block B of size ``param`.blockSizes[`i`]` with a different block (but possibly a repeat of B) of size ``param`.blockSizes[`j`]` (in each design in each partition in PL). In the case of multisets, we take the multiplicity of an element in the intersection to be the minimum of its multiplicities in the multisets being intersected; for example, the intersection of `[1,1,1,2,2,3]` with `[1,1,2,2,2,4]` is `[1,1,2,2]`, having size 4. The dimension of ``param`.blockIntersectionNumbers` must equal the length of ``param`.blockSizes`.

  ``param`.blockMaxMultiplicities` must be a list of non-negative integers, the i-th element of which specifies an upper bound on the multiplicity of a block whose size is equal to ``param`.blockSizes[`i`]` (for each design in each partition in PL). The length of ``param`.blockMaxMultiplicities` must equal that of ``param`.blockSizes`.

  ``param`.isoGroup` must be a subgroup of the automorphism group of ``param`.blockDesign`. We consider two elements of PL to be equivalent if they are in the same orbit of ``param`.isoGroup` (in its action on multisets of block multisets). The default for ``param`.isoGroup` is the automorphism group of ``param`.blockDesign`.

  ``param`.requiredAutSubgroup` must be a subgroup of ``param`.isoGroup`, and specifies that each partition in PL must be invariant under ``param`.requiredAutSubgroup` (in its action on multisets of block multisets). The default for ``param`.requiredAutSubgroup` is the trivial permutation group.

  ``param`.isoLevel` must be 0, 1, or 2 (the default is 2). The value 0 specifies that PL will contain at most one partition, and will contain one partition with the required properties if and only if such a partition exists; the value 1 specifies that PL will contain (perhaps properly) a list of ``param`.isoGroup` orbit-representatives of the required partitions; the value 2 specifies that PL will consist precisely of ``param`.isoGroup`-orbit representatives of the required partitions.

  For an example, we first classify up to isomorphism the 2-(15,3,1) designs invariant under a semi-regular group of automorphisms of order 5, and then use `PartitionsIntoBlockDesigns` to classify all the resolutions of these designs, up to the actions of the respective automorphism groups of the designs.

      gap> DL:=BlockDesigns(rec(
      >    v:=15,blockSizes:=[3],
      >    tSubsetStructure:=rec(t:=2,lambdas:=[1]),
      >    requiredAutSubgroup:=
      >       Group((1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15))));;
      gap> List(DL,D->Size(AutGroupBlockDesign(D)));
      [ 20160, 5, 60 ]
      gap> PL:=PartitionsIntoBlockDesigns(rec(
      >       blockDesign:=DL[1],
      >       v:=15,blockSizes:=[3],
      >       tSubsetStructure:=rec(t:=1,lambdas:=[1])));
      [ rec(
            partition := [ rec( isBlockDesign := true, v := 15, blocks := [ [ 1, 2,
                            6 ], [ 3, 4, 8 ], [ 5, 7, 14 ], [ 9, 12, 15 ],
                        [ 10, 11, 13 ] ] ),
                rec( isBlockDesign := true, v := 15, blocks :=
                      [ [ 1, 3, 11 ], [ 2, 4, 12 ], [ 5, 6, 8 ], [ 7, 13, 15 ],
                        [ 9, 10, 14 ] ] ),
                rec( isBlockDesign := true, v := 15, blocks :=
                      [ [ 1, 4, 14 ], [ 2, 5, 15 ], [ 3, 10, 12 ], [ 6, 7, 11 ],
                        [ 8, 9, 13 ] ] ),
                rec( isBlockDesign := true, v := 15, blocks :=
                      [ [ 1, 5, 10 ], [ 2, 9, 11 ], [ 3, 14, 15 ], [ 4, 6, 13 ],
                        [ 7, 8, 12 ] ] ),
                rec( isBlockDesign := true, v := 15, blocks :=
                      [ [ 1, 7, 9 ], [ 2, 8, 10 ], [ 3, 5, 13 ], [ 4, 11, 15 ],
                        [ 6, 12, 14 ] ] ),
                rec( isBlockDesign := true, v := 15, blocks :=
                      [ [ 1, 8, 15 ], [ 2, 13, 14 ], [ 3, 6, 9 ], [ 4, 7, 10 ],
                        [ 5, 11, 12 ] ] ),
                rec( isBlockDesign := true, v := 15, blocks :=
                      [ [ 1, 12, 13 ], [ 2, 3, 7 ], [ 4, 5, 9 ], [ 6, 10, 15 ],
                        [ 8, 11, 14 ] ] ) ],
            autGroup := Group([ (1,10)(2,11)(3,8)(6,13)(7,14)(12,15),
                (1,13)(2,11)(3,14)(4,5)(6,10)(7,8),
                (1,13,7)(2,11,5)(6,10,14)(9,12,15),
                (2,11,5,15,4,9,12)(3,10,8,14,7,13,6) ]) ),
        rec( partition := [ rec( isBlockDesign := true, v := 15,
                    blocks := [ [ 1, 2, 6 ], [ 3, 4, 8 ], [ 5, 7, 14 ],
                        [ 9, 12, 15 ], [ 10, 11, 13 ] ] ),
                rec( isBlockDesign := true, v := 15,
                    blocks := [ [ 1, 3, 11 ], [ 2, 4, 12 ], [ 5, 6, 8 ],
                        [ 7, 13, 15 ], [ 9, 10, 14 ] ] ),
                rec( isBlockDesign := true, v := 15,
                    blocks := [ [ 1, 4, 14 ], [ 2, 5, 15 ], [ 3, 10, 12 ],
                        [ 6, 7, 11 ], [ 8, 9, 13 ] ] ),
                rec( isBlockDesign := true, v := 15,
                    blocks := [ [ 1, 5, 10 ], [ 2, 13, 14 ], [ 3, 6, 9 ],
                        [ 4, 11, 15 ], [ 7, 8, 12 ] ] ),
                rec( isBlockDesign := true, v := 15,
                    blocks := [ [ 1, 7, 9 ], [ 2, 8, 10 ], [ 3, 14, 15 ],
                        [ 4, 6, 13 ], [ 5, 11, 12 ] ] ),
                rec( isBlockDesign := true, v := 15,
                    blocks := [ [ 1, 8, 15 ], [ 2, 9, 11 ], [ 3, 5, 13 ],
                        [ 4, 7, 10 ], [ 6, 12, 14 ] ] ),
                rec( isBlockDesign := true, v := 15,
                    blocks := [ [ 1, 12, 13 ], [ 2, 3, 7 ], [ 4, 5, 9 ],
                        [ 6, 10, 15 ], [ 8, 11, 14 ] ] ) ],
            autGroup := Group([ (1,15)(2,9)(3,4)(5,7)(6,12)(10,13),
                (1,12)(2,9)(3,5)(4,7)(6,15)(8,14),
                (1,14)(2,5)(3,8)(6,7)(9,12)(10,13),
                (1,8,10)(2,5,15)(3,14,13)(4,9,12) ]) ) ]
      gap> List(PL,resolution->Size(resolution.autGroup));
      [ 168, 168 ]
      gap> PL:=PartitionsIntoBlockDesigns(rec(
      >       blockDesign:=DL[2],
      >       v:=15,blockSizes:=[3],
      >       tSubsetStructure:=rec(t:=1,lambdas:=[1])));
      [  ]
      gap> PL:=PartitionsIntoBlockDesigns(rec(
      >       blockDesign:=DL[3],
      >       v:=15,blockSizes:=[3],
      >       tSubsetStructure:=rec(t:=1,lambdas:=[1])));
      [  ]

  ## 9.2 Computing resolutions

- `MakeResolutionsComponent(` D`)`
- `MakeResolutionsComponent(` D`,` isolevel`)`

  This function computes resolutions of the block design D, and stores the result in ``D`.resolutions`. If ``D`.resolutions` already exists then it is ignored and overwritten. This function returns no value.

  A resolution of a block design D is a partition of the blocks into subsets, each of which forms a partition of the point set. We say that two resolutions R and S of D are isomorphic if there is an element g in the automorphism group of D, such that the g-image of R is S. (Isomorphism defines an equivalence relation on the set of resolutions of D.)

  The parameter isolevel (default 2) determines how many resolutions are computed: isolevel=2 means to classify up to isomorphism, isolevel=1 means to determine at least one representative from each isomorphism class, and isolevel=0 means to determine whether or not D has a resolution.

  When this function is finished, ``D`.resolutions` will have the following three components:

  `list`: a list of distinct partitions into block designs forming resolutions of D;

  `pairwiseNonisomorphic`: `true`, `false` or `"unknown"`, depending on the resolutions in `list` and what is known. If isolevel=0 or isolevel=2 then this component will be `true`;

  `allClassesRepresented`: `true`, `false` or `"unknown"`, depending on the resolutions in `list` and what is known. If isolevel=1 or isolevel=2 then this component will be `true`.

  Note that ``D`.resolutions` may be changed to contain more information as a side-effect of other functions in the DESIGN package.

      gap> L:=BlockDesigns(rec(v:=9,blockSizes:=[3],
      >          tSubsetStructure:=rec(t:=2,lambdas:=[1])));;
      gap> D:=L[1];;
      gap> MakeResolutionsComponent(D);
      gap> D;
      rec( isBlockDesign := true, v := 9,
        blocks := [ [ 1, 2, 3 ], [ 1, 4, 5 ], [ 1, 6, 7 ], [ 1, 8, 9 ],
            [ 2, 4, 6 ], [ 2, 5, 8 ], [ 2, 7, 9 ], [ 3, 4, 9 ], [ 3, 5, 7 ],
            [ 3, 6, 8 ], [ 4, 7, 8 ], [ 5, 6, 9 ] ],
        tSubsetStructure := rec( t := 2, lambdas := [ 1 ] ), isBinary := true,
        isSimple := true, blockSizes := [ 3 ], blockNumbers := [ 12 ], r := 4,
        autGroup := Group([ (1,2)(5,6)(7,8), (1,3,2)(4,8,7)(5,6,9), (1,2)(4,7)(5,9),
            (1,2)(4,9)(5,7)(6,8), (1,4,8,6,9,2)(3,5,7) ]),
        resolutions := rec( list := [ rec( partition :=
                      [ rec( isBlockDesign := true, v := 9,
                            blocks := [ [ 1, 2, 3 ], [ 4, 7, 8 ], [ 5, 6, 9 ] ] ),
                        rec( isBlockDesign := true, v := 9,
                            blocks := [ [ 1, 4, 5 ], [ 2, 7, 9 ], [ 3, 6, 8 ] ] ),
                        rec( isBlockDesign := true, v := 9,
                            blocks := [ [ 1, 6, 7 ], [ 2, 5, 8 ], [ 3, 4, 9 ] ] ),
                        rec( isBlockDesign := true, v := 9,
                            blocks := [ [ 1, 8, 9 ], [ 2, 4, 6 ], [ 3, 5, 7 ] ] ) ],
                    autGroup := Group(
                      [ (2,3)(4,5)(6,7)(8,9), (1,3,2)(4,8,7)(5,6,9),
                        (1,8,9)(2,4,6)(3,7,5), (1,2)(5,6)(7,8), (1,2)(4,7)(5,9),
                        (1,2,9,6,8,4)(3,7,5) ]) ) ], pairwiseNonisomorphic := true,
            allClassesRepresented := true ) )

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