An incidence structure is specified by a list of points, blocks, or an incidence matrix ([1], [2]). IncidenceStructure instances have the following methods:
ground_set()  Return the ground set (i.e the list of points). 
num_points()  Return the size of the ground set. 
num_blocks()  Return the number of blocks. 
blocks()  Return the list of blocks. 
block_sizes()  Return the set of block sizes. 
degree()  Return the degree of a point p 
is_connected()  Test whether the design is connected. 
is_simple()  Test whether this design is simple (i.e. no repeated block). 
incidence_matrix()  Return the incidence matrix \(A\) of the design 
incidence_graph()  Return the incidence graph of the design 
packing()  Return a maximum packing 
is_t_design()  Test whether self is a \(t(v,k,l)\) design. 
dual()  Return the dual design. 
automorphism_group()  Return the automorphism group 
edge_coloring()  Return an optimal edge coloring` 
REFERENCES:
[1]  Block designs and incidence structures from wikipedia, Wikipedia article Block_design Wikipedia article Incidence_structure 
[2] 

AUTHORS:
Peter Dobcsanyi and David Joyner (20072008)
This is a significantly modified form of part of the module block_design.py (version 0.6) written by Peter Dobcsanyi peter@designtheory.org.
Vincent Delecroix (2014): major rewrite
Bases: object
A base class for incidence structures (i.e. hypergraphs, i.e. set systems)
An incidence structure (i.e. hypergraph, i.e. set system) can be defined from a collection of blocks (i.e. sets, i.e. edges), optionally with an explicit ground set (i.e. point set, i.e. vertex set). Alternatively they can be defined from a binary incidence matrix.
INPUT:
points – (i.e. ground set, i.e. vertex set) the underlying set. If points is an integer \(v\), then the set is considered to be \(\{0, ..., v1\}\).
Note
The following syntax, where points is ommitted, automatically defines the ground set as the union of the blocks:
sage: H = IncidenceStructure([['a','b','c'],['c','d','e']])
sage: H.ground_set()
['a', 'b', 'c', 'd', 'e']
blocks – (i.e. edges, i.e. sets) the blocks defining the incidence structure. Can be any iterable.
incidence_matrix – a binary incidence matrix. Each column represents a set.
name (a string, such as “Fano plane”).
check – whether to check the input
copy – (use with caution) if set to False then blocks must be a list of lists of integers. The list will not be copied but will be modified in place (each block is sorted, and the whole list is sorted). Your blocks object will become the IncidenceStructure instance’s internal data.
EXAMPLES:
An incidence structure can be constructed by giving the number of points and the list of blocks:
sage: designs.IncidenceStructure(7, [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
Incidence structure with 7 points and 7 blocks
Only providing the set of blocks is sufficient. In this case, the ground set is defined as the union of the blocks:
sage: IncidenceStructure([[1,2,3],[2,3,4]])
Incidence structure with 4 points and 2 blocks
Or by its adjacency matrix (a \(\{0,1\}\)matrix in which rows are indexed by points and columns by blocks):
sage: m = matrix([[0,1,0],[0,0,1],[1,0,1],[1,1,1]])
sage: designs.IncidenceStructure(m)
Incidence structure with 4 points and 3 blocks
The points can be any (hashable) object:
sage: V = [(0,'a'),(0,'b'),(1,'a'),(1,'b')]
sage: B = [(V[0],V[1],V[2]), (V[1],V[2]), (V[0],V[2])]
sage: I = designs.IncidenceStructure(V, B)
sage: I.ground_set()
[(0, 'a'), (0, 'b'), (1, 'a'), (1, 'b')]
sage: I.blocks()
[[(0, 'a'), (0, 'b'), (1, 'a')], [(0, 'a'), (1, 'a')], [(0, 'b'), (1, 'a')]]
The order of the points and blocks does not matter as they are sorted on input (see trac ticket #11333):
sage: A = designs.IncidenceStructure([0,1,2], [[0],[0,2]])
sage: B = designs.IncidenceStructure([1,0,2], [[0],[2,0]])
sage: B == A
True
sage: C = designs.BlockDesign(2, [[0], [1,0]])
sage: D = designs.BlockDesign(2, [[0,1], [0]])
sage: C == D
True
If you care for speed, you can set copy to False, but in that case, your input must be a list of lists and the ground set must be \({0, ..., v1}\):
sage: blocks = [[0,1],[2,0],[1,2]] # a list of lists of integers
sage: I = designs.IncidenceStructure(3, blocks, copy=False)
sage: I.blocks(copy=False) is blocks
True
Return the subgroup of the automorphism group of the incidence graph which respects the P B partition. It is (isomorphic to) the automorphism group of the block design, although the degrees differ.
EXAMPLES:
sage: P = designs.DesarguesianProjectivePlaneDesign(2); P
Incidence structure with 7 points and 7 blocks
sage: G = P.automorphism_group()
sage: G.is_isomorphic(PGL(3,2))
True
sage: G
Permutation Group with generators [(2,3)(4,5), (2,4)(3,5), (1,2)(4,6), (0,1)(4,5)]
A non selfdual example:
sage: IS = designs.IncidenceStructure(range(4), [[0,1,2,3],[1,2,3]])
sage: IS.automorphism_group().cardinality()
6
sage: IS.dual().automorphism_group().cardinality()
1
Examples with noninteger points:
sage: I = designs.IncidenceStructure('abc', ('ab','ac','bc'))
sage: I.automorphism_group()
Permutation Group with generators [('b','c'), ('a','b')]
sage: designs.IncidenceStructure([[(1,2),(3,4)]]).automorphism_group()
Permutation Group with generators [((1,2),(3,4))]
This method is deprecated and will soon be removed (see trac ticket #16553). You could use is_t_design() instead.
This is not a wrapper for GAP Design’s IsBlockDesign. The GAP Design function IsBlockDesign http://www.gapsystem.org/Manuals/pkg/design/htm/CHAP004.htm apparently simply checks the record structure and no mathematical properties. Instead, the function below checks some necessary (but not sufficient) “easy” identities arising from the identity.
INPUT:
WARNING: This is very fast but can return false positives.
EXAMPLES:
sage: BD = designs.IncidenceStructure(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.is_t_design(return_parameters=True)
(True, (2, 7, 3, 1))
sage: BD.block_design_checker(2, 7, 3, 1)
doctest:...: DeprecationWarning: .block_design_checker(v,t,k,lmbda) is deprecated; please use
.is_t_design(v,t,k,lmbda) instead
See http://trac.sagemath.org/16553 for details.
True
sage: BD.block_design_checker(2, 7, 3, 1,"binary")
doctest:...: DeprecationWarning: .block_design_checker(type='binary') is
deprecated; use .is_binary() instead
See http://trac.sagemath.org/16553 for details.
True
sage: BD.block_design_checker(2, 7, 3, 1,"connected")
doctest:...: DeprecationWarning: block_design_checker(type='connected') is
deprecated, please use .is_connected() instead
See http://trac.sagemath.org/16553 for details.
True
sage: BD.block_design_checker(2, 7, 3, 1,"simple")
doctest:...: DeprecationWarning: .block_design_checker(type='simple')
is deprecated; all designs here are simple!
See http://trac.sagemath.org/16553 for details.
True
Return the set of block sizes.
EXAMPLES:
sage: BD = designs.IncidenceStructure(8, [[0,1,3],[1,4,5,6],[1,2],[5,6,7]])
sage: BD.block_sizes()
[3, 2, 4, 3]
sage: BD = designs.IncidenceStructure(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.block_sizes()
[3, 3, 3, 3, 3, 3, 3]
Return the list of blocks.
INPUT:
EXAMPLES:
sage: BD = designs.IncidenceStructure(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.blocks()
[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]
What you should pay attention to:
sage: blocks = BD.blocks(copy=False)
sage: del blocks[0:6]
sage: BD
Incidence structure with 7 points and 1 blocks
Return the degree of a point p
The degree of a point \(p\) is the number of blocks that contain it.
INPUT:
EXAMPLES:
sage: designs.steiner_triple_system(9).degree(3)
4
sage: designs.steiner_triple_system(9).degree()
{0: 4, 1: 4, 2: 4, 3: 4, 4: 4, 5: 4, 6: 4, 7: 4, 8: 4}
Return the dual of the incidence structure.
INPUT:
algorithm – whether to use Sage’s implementation (algorithm=None, default) or use GAP’s (algorithm="gap").
Note
The algorithm="gap" option requires GAP’s Design package (included in the gap_packages Sage spkg).
EXAMPLES:
The dual of a projective plane is a projective plane:
sage: PP = designs.DesarguesianProjectivePlaneDesign(4)
sage: PP.dual().is_t_design(return_parameters=True)
(True, (2, 21, 5, 1))
TESTS:
sage: D = designs.IncidenceStructure(4, [[0,2],[1,2,3],[2,3]])
sage: D
Incidence structure with 4 points and 3 blocks
sage: D.dual()
Incidence structure with 3 points and 4 blocks
sage: print D.dual(algorithm="gap") # optional  gap_packages
IncidenceStructure<points=[0, 1, 2], blocks=[[0], [0, 1, 2], [1], [1, 2]]>
sage: blocks = [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]]
sage: BD = designs.IncidenceStructure(7, blocks, name="FanoPlane");
sage: BD
Incidence structure with 7 points and 7 blocks
sage: print BD.dual(algorithm="gap") # optional  gap_packages
IncidenceStructure<points=[0, 1, 2, 3, 4, 5, 6], blocks=[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]>
sage: BD.dual()
Incidence structure with 7 points and 7 blocks
REFERENCE:
Deprecated: Use dual() instead. See trac ticket #16553 for details.
Deprecated: Use dual() instead. See trac ticket #16553 for details.
Compute a proper edgecoloring.
A proper edgecoloring is an assignment of colors to the sets of the incidence structure such that two sets with nonempty intersection receive different colors. The coloring returned minimizes the number of colors.
OUTPUT:
A partition of the sets into color classes.
EXAMPLES:
sage: H = Hypergraph([{1,2,3},{2,3,4},{3,4,5},{4,5,6}]); H
Incidence structure with 6 points and 4 blocks
sage: C = H.edge_coloring()
sage: C # random
[[[3, 4, 5]], [[2, 3, 4]], [[4, 5, 6], [1, 2, 3]]]
sage: Set(map(Set,sum(C,[]))) == Set(map(Set,H.blocks()))
True
Return the ground set (i.e the list of points).
INPUT:
EXAMPLES:
sage: designs.IncidenceStructure(3, [[0,1],[0,2]]).ground_set()
[0, 1, 2]
Return the incidence graph of the design, where the incidence matrix of the design is the adjacency matrix of the graph.
EXAMPLE:
sage: BD = designs.IncidenceStructure(7, [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.incidence_graph()
Bipartite graph on 14 vertices
sage: A = BD.incidence_matrix()
sage: Graph(block_matrix([[A*0,A],[A.transpose(),A*0]])) == BD.incidence_graph()
True
REFERENCE:
Return the incidence matrix \(A\) of the design. A is a \((v \times b)\) matrix defined by: A[i,j] = 1 if i is in block B_j and 0 otherwise.
EXAMPLES:
sage: BD = designs.IncidenceStructure(7, [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.block_sizes()
[3, 3, 3, 3, 3, 3, 3]
sage: BD.incidence_matrix()
[1 1 1 0 0 0 0]
[1 0 0 1 1 0 0]
[1 0 0 0 0 1 1]
[0 1 0 1 0 1 0]
[0 1 0 0 1 0 1]
[0 0 1 1 0 0 1]
[0 0 1 0 1 1 0]
sage: I = designs.IncidenceStructure('abc', ('ab','abc','ac','c'))
sage: I.incidence_matrix()
[1 1 1 0]
[1 1 0 0]
[0 1 1 1]
Deprecated: Use is_t_design() instead. See trac ticket #16553 for details.
Test whether the design is connected.
EXAMPLES:
sage: designs.IncidenceStructure(3, [[0,1],[0,2]]).is_connected()
True
sage: designs.IncidenceStructure(4, [[0,1],[2,3]]).is_connected()
False
Test whether this design is simple (i.e. no repeated block).
EXAMPLES:
sage: designs.IncidenceStructure(3, [[0,1],[1,2],[0,2]]).is_simple()
True
sage: designs.IncidenceStructure(3, [[0],[0]]).is_simple()
False
sage: V = [(0,'a'),(0,'b'),(1,'a'),(1,'b')]
sage: B = [[V[0],V[1]], [V[1],V[2]]]
sage: I = designs.IncidenceStructure(V, B)
sage: I.is_simple()
True
sage: I2 = designs.IncidenceStructure(V, B*2)
sage: I2.is_simple()
False
Test whether self is a \(t(v,k,l)\) design.
A \(t(v,k,\lambda)\) (sometimes called \(t\)design for short) is a block design in which:
INPUT:
EXAMPLES:
sage: fano_blocks = [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]]
sage: BD = designs.IncidenceStructure(7, fano_blocks)
sage: BD.is_t_design()
True
sage: BD.is_t_design(return_parameters=True)
(True, (2, 7, 3, 1))
sage: BD.is_t_design(2, 7, 3, 1)
True
sage: BD.is_t_design(1, 7, 3, 3)
True
sage: BD.is_t_design(0, 7, 3, 7)
True
sage: BD.is_t_design(0,6,3,7) or BD.is_t_design(0,7,4,7) or BD.is_t_design(0,7,3,8)
False
sage: BD = designs.AffineGeometryDesign(3, 1, GF(2))
sage: BD.is_t_design(1)
True
sage: BD.is_t_design(2)
True
Steiner triple and quadruple systems are other names for \(2(v,3,1)\) and \(3(v,4,1)\) designs:
sage: S3_9 = designs.steiner_triple_system(9)
sage: S3_9.is_t_design(2,9,3,1)
True
sage: blocks = designs.steiner_quadruple_system(8)
sage: S4_8 = designs.IncidenceStructure(8, blocks)
sage: S4_8.is_t_design(3,8,4,1)
True
sage: blocks = designs.steiner_quadruple_system(14)
sage: S4_14 = designs.IncidenceStructure(14, blocks)
sage: S4_14.is_t_design(3,14,4,1)
True
Some examples of Witt designs that need the gap database:
sage: BD = designs.WittDesign(9) # optional  gap_packages
sage: BD.is_t_design(2,9,3,1) # optional  gap_packages
True
sage: W12 = designs.WittDesign(12) # optional  gap_packages
sage: W12.is_t_design(5,12,6,1) # optional  gap_packages
True
sage: W12.is_t_design(4) # optional  gap_packages
True
Further examples:
sage: D = designs.IncidenceStructure(4,[[],[]])
sage: D.is_t_design(return_parameters=True)
(True, (0, 4, 0, 2))
sage: D = designs.IncidenceStructure(4, [[0,1],[0,2],[0,3]])
sage: D.is_t_design(return_parameters=True)
(True, (0, 4, 2, 3))
sage: D = designs.IncidenceStructure(4, [[0],[1],[2],[3]])
sage: D.is_t_design(return_parameters=True)
(True, (1, 4, 1, 1))
sage: D = designs.IncidenceStructure(4,[[0,1],[2,3]])
sage: D.is_t_design(return_parameters=True)
(True, (1, 4, 2, 1))
sage: D = designs.IncidenceStructure(4, [range(4)])
sage: D.is_t_design(return_parameters=True)
(True, (4, 4, 4, 1))
TESTS:
sage: blocks = designs.steiner_quadruple_system(8)
sage: S4_8 = designs.IncidenceStructure(8, blocks)
sage: R = range(15)
sage: [(v,k,l) for v in R for k in R for l in R if S4_8.is_t_design(3,v,k,l)]
[(8, 4, 1)]
sage: [(v,k,l) for v in R for k in R for l in R if S4_8.is_t_design(2,v,k,l)]
[(8, 4, 3)]
sage: [(v,k,l) for v in R for k in R for l in R if S4_8.is_t_design(1,v,k,l)]
[(8, 4, 7)]
sage: [(v,k,l) for v in R for k in R for l in R if S4_8.is_t_design(0,v,k,l)]
[(8, 4, 14)]
sage: A = designs.AffineGeometryDesign(3, 1, GF(2))
sage: A.is_t_design(return_parameters=True)
(True, (2, 8, 2, 1))
sage: A = designs.AffineGeometryDesign(4, 2, GF(2))
sage: A.is_t_design(return_parameters=True)
(True, (3, 16, 4, 1))
sage: I = designs.IncidenceStructure(2, [])
sage: I.is_t_design(return_parameters=True)
(True, (0, 2, 0, 0))
sage: I = designs.IncidenceStructure(2, [[0],[0,1]])
sage: I.is_t_design(return_parameters=True)
(False, (0, 0, 0, 0))
Return the number of blocks.
EXAMPLES:
sage: designs.DesarguesianProjectivePlaneDesign(2).num_blocks()
7
sage: B = designs.IncidenceStructure(4, [[0,1],[0,2],[0,3],[1,2], [1,2,3]])
sage: B.num_blocks()
5
Return the size of the ground set.
EXAMPLES:
sage: designs.DesarguesianProjectivePlaneDesign(2).num_points()
7
sage: B = designs.IncidenceStructure(4, [[0,1],[0,2],[0,3],[1,2], [1,2,3]])
sage: B.num_points()
4
Return a maximum packing
A maximum packing in a hypergraph is collection of disjoint sets/blocks of maximal cardinality. This problem is NPcomplete in general, and in particular on 3uniform hypergraphs. It is solved here with an Integer Linear Program.
For more information, see the Wikipedia article Packing_in_a_hypergraph.
INPUT:
EXAMPLE:
sage; IncidenceStructure([[1,2],[3,"A"],[2,3]]).packing()
[[1, 2], [3, 'A']]
sage: len(designs.steiner_triple_system(9).packing())
3
Deprecated function. You should use is_t_design() instead.
EXAMPLES:
sage: I = designs.IncidenceStructure('abc', ['ab','ac','bc'])
sage: I.parameters()
doctest:...: DeprecationWarning: .parameters() is deprecated. Use
`is_t_design` instead
See http://trac.sagemath.org/16553 for details.
(2, 3, 2, 1)
Deprecated: Use ground_set() instead. See trac ticket #16553 for details.
Deprecated function that builds an incidence structure from a matrix.
You should now use designs.IncidenceStructure(incidence_matrix=M).
INPUT:
EXAMPLES:
sage: BD1 = designs.IncidenceStructure(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: M = BD1.incidence_matrix()
sage: BD2 = IncidenceStructureFromMatrix(M)
doctest:...: DeprecationWarning: IncidenceStructureFromMatrix is deprecated.
Please use designs.IncidenceStructure(incidence_matrix=M) instead.
See http://trac.sagemath.org/16553 for details.
sage: BD1 == BD2
True