Elementary Invariants of a Design

The following functions can be applied only to designs.

Parameters(D): Dsgn -> Record

The parameters \(t\)–\((v, b, r, k, \lambda)\) of the design \(D\) returned as a record.

ReplicationNumber(D): Dsgn -> RngIntElt

The number of blocks \(r\) containing any point of the \(t\)–\((v,k,\lambda)\) design \(D\), where \(t > 0\).

BlockDegree(D): Dsgn -> RngIntElt

BlockSize(D): Dsgn -> RngIntElt

The number of points in a block of the design \(D\).

Covalence(D, s): Dsgn, RngIntElt -> RngIntElt

Given a \(t\)–\((v, k, \lambda)\) design \(D\) and an integer \(s\) such that \(0 \le s \le t\), return the value of \(\lambda_s\); i.e., the number of blocks that contain an arbitrary \(s\)-subset of the points of \(D\).

Order(D): Dsgn -> RngIntElt

The order of the \(t\)–\((v,k,\lambda)\) design \(D\). This is defined only for designs with \(t \ge 2\).

IntersectionNumber(D, i, j): Dsgn, RngIntElt, RngIntElt -> RngIntElt

The block intersection number \(\lambda_i^j\); i.e., the number of blocks of the design \(D\) containing an \(i\)-set and disjoint from a \(j\)-set. The arguments \(i\) and \(j\) must satisfy \(i+j \le t\).

PascalTriangle(D): Dsgn -> SeqEnum

The “Pascal triangle” of the design \(D\), returned as a sequence; the \(i\)-th element of the sequence is a sequence representing the \(i\)-th row of the triangle. That is, the \(i\)-th element of the sequence is

\[[\lambda_0^{i-1}, \lambda_1^{i - 2}, \ldots, \lambda_{i-1}^0].\]

If \(D\) is a Steiner \(t\)–design, then the triangle returned has \(k + 1\) rows (where \(k\) is the size of a block of \(D\)); otherwise the triangle has \(t + 1\) rows.

Example: Design Invar (ex-3a71cb)

We illustrate some of the functions of the previous two sections with an example.

> F := Design< 2, 7 | {1,2,4}, {1,3,7}, {2,3,5}, {1,5,6}, {3,4,6}, {4,5,7},
>   {2,6,7} >;
> G := IncidenceStructure< 7 | Blocks(F), {1, 3, 7}, {1, 2, 4},
>   {3, 4, 5}, {2, 3, 6}, {2, 5, 7}, {1, 5, 6}, {4, 6, 7} >;
> F;
2-(7, 3, 1) Design with 7 blocks
> G;
Incidence Structure on 7 points with 14 blocks
> Points(G);
{@ 1, 2, 3, 4, 5, 6, 7 @}
> Blocks(F);
{@
    {1, 2, 4},
    {1, 3, 7},
    {2, 3, 5},
    {1, 5, 6},
    {3, 4, 6},
    {4, 5, 7},
    {2, 6, 7}
@}
> IncidenceMatrix(F);
[1 1 0 1 0 0 0]
[1 0 1 0 0 0 1]
[0 1 1 0 1 0 0]
[1 0 0 0 1 1 0]
[0 0 1 1 0 1 0]
[0 0 0 1 1 0 1]
[0 1 0 0 0 1 1]
> P := Points(F);
> P, Universe(P);
{@ 1, 2, 3, 4, 5, 6, 7 @}
Point-set of 2-(7, 3, 1) Design with 7 blocks
> S := Support(F);
> S, Universe(S);
{@ 1, 2, 3, 4, 5, 6, 7 @}
Integer Ring
> Covalence(G, {1, 2});
2
> Order(F);
2
> PascalTriangle(F);
         7
       4   3
     2   2   1
   0   2   0   1

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