Quellcode-Bibliothek partition.tex   Sprache: Latech

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%A  partition.tex        DESIGN documentation           Leonard Soicher
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\def\GRAPE{\sf GRAPE}
\def\DESIGN{\sf DESIGN}
\def\nauty{\it nauty}
\def\Aut{{\rm Aut}\,} 

\Chapter{Partitioning block designs}

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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Partitioning a block design into block designs}

\>PartitionsIntoBlockDesigns( <param> )

Let <D> equal `<param>.blockDesign'. This function returns a list
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 `.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 `<param>.tSubsetStructure.t', let be
`<param>.tSubsetStructure.partition', and let be
`<param>.tSubsetStructure.lambdas'. Then 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 -subsets of
`[1..<param>.v]', and 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 `[]' times, counted
over all blocks of the design.  For binary designs, this means that each
<t>-subset in `<partition>[<i>]' is contained in exactly `[]'
blocks.  The `partition' component is optional if has length 1.
We require that <t> is less than or equal to each element of
`<param>.blockSizes', and that each block of `.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 -th
element in this list specifies the number of blocks whose size is equal to
`<param>.blockSizes[<i>]' (in each design in each partition in ). The
length of `<param>.blockNumbers' must equal that of `.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 `.blockSizes'.

`<param>.isoGroup' must be a subgroup of the automorphism group of
`<param>.blockDesign'. We consider two elements of 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 `.blockDesign'.

`<param>.requiredAutSubgroup' must be a subgroup of `.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.

\beginexample
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])));
[  ]
\endexample

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\Section{Computing resolutions}

\>MakeResolutionsComponent( <D> )
\>MakeResolutionsComponent( <D>, <isolevel> )

This function computes resolutions of the block design <D>, and stores
the result in `<D>.resolutions'. If the component `.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 =0 or =2
then this component will be `true';

`allClassesRepresented': `true', `false' or `"unknown"', depending on the
resolutions in `list' and what is known. If =1 or =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.

\beginexample
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 ) )
\endexample