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.
The number of blocks r containing any point of the t-(v, k, lambda) design D, where t > 0.
The number of points in a block of the design D.
Given a t-(v, k, lambda) design D and an integer s such that 0 <= s <= t, return the value of lambda_s; i.e. the number of blocks that contain an arbitary s-subset of the points of D.
The order of the t-(v, k, lambda) design D. This is defined only for designs with t >= 2.
The block intersection number lambda_i^j; i.e. the number of blocks of D containing an i-set and disjoint from a j-set. The arguments i and j must satisfy i + j <= t.
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), ..., 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.
> 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