On triangular matroids induced by n3-configurations

Abstract A triangular matroid is a rank-3 matroid whose ground set consists of the points of an n 3 {n}_{3} -configuration and whose bases are the point triples corresponding to non-triangles within the configuration. Raney previously enumerated the n 3 {n}_{3} -configurations which induce triangular matroids for 7 ≤ n ≤ 15 7\le n\le 15 . In this work, the enumeration is extended to configurations having up to 18 points. Several examples of such configurations and their symmetry groups are presented, as well as geometric representations of the triangular matroids induced by these configurations.


Introduction
A (finite) incidence structure is a triple ( ) , , , where is a set of points, is a set of lines (or blocks), and is an incidence relation on and , i.e., ⊆ × . A pair ( ) ∈ p L , is called a flag. In that case, we say that the point p and the line L are incident. We may also say that p lies on L or that L contains p.
A (combinatorial) configuration of type ( ) v n , r k is an incidence structure of v distinct points and n distinct lines. We further require that each point is incident with exactly r lines, each line is incident with exactly k points, and any pair of distinct points is contained in at most one line. Consequently, any pair of distinct lines has at most one common point.
If can be (realized) embedded in the real projective plane in the sense of points and straight lines, then we say that is geometric. Clearly, every geometric configuration is combinatorial, but the converse is not valid, see [1].
A configuration ( ) v n , r k with = v n (and by a simple counting argument also = r k) is called a symmetric configuration. In this case, we simply write n k -configuration.
A triangle in a configuration is a triple of non-collinear points { } p p p , , 1 2 3 in which every pair of distinct points is contained in a line. For the sake of simplicity, we write p p p 1 2 3 to denote the triangle { } p p p , , 1 2 3 . A configuration with no triangles is called a triangle-free configuration. In Figure 1, we see that each noncollinear triple of points in the Fano configuration (left) forms a triangle. On the other hand, the Cremona-Richmond configuration (right) has no triangles.
Two configurations = ( ) C , 1 1 1 and = ( ) , 2 2 2 are said to be isomorphic if there is a bijection → α : 1 2 which maps 1 onto 2 . Here, a line = { … } ∈ L p p , , k , then we simply say that has h-orbits. If ( ) Aut contains a cyclic subgroup acting transitively on the points, then is called cyclic.
Furthermore, a configuration is called k-cyclic if there exists an automorphism α of order k such that all orbits on points and lines under α are of the same size. In this setting, a cyclic n 3 -configuration is an ncyclic configuration. See [2] for more details.
To each ( ) v n , r k -configuration = ( ) , , we assign another configuration known as the dual configuration. It is the ( ) n v , k r -configuration = ( ) , ⁎ , with the roles of points and lines reversed, but with the same incidences. It is clear that and ⁎ have the same Levi graphs, except that the colors of classes (the classes of points and lines) are reversed. If furthermore is isomorphic to its dual ⁎ , we say that is self-dual and the corresponding isomorphism is called a duality. Moreover, a polarity is a duality of order 2. A configuration admitting a polarity is called self-polar.
We now turn our attention to the relation between n 3 -configurations and the class of rank-3 matroids. In particular, we consider the class of rank-3 matroids called triangular matroids, defined recently by Raney [3].
In the literature, a matroid is a structure that is related directly to the notion of linear independence in vector spaces. A matroid can be defined in many equivalent ways. Here, we define a matroid in terms of its bases.
of a finite set of elements , called the ground set of M, and a nonempty collection of subsets of , called bases (also called maximal independent sets), satisfying the following so-called "basis exchange property:" If B 1 and B 2 are two distinct bases in , then for every is a basis in . Applying the basis exchange property repeatedly, one can show that any two bases in share the same cardinality. A matroid with bases of cardinality r is said to have rank r.
Matroid theory has been extensively related and applied to different areas of mathematics including geometry, topology, group theory, and coding theory.
consists of all of the non-collinear point triples p p p i j k from . A simple counting argument shows that contains ( ) n 3 − n triples. In this case, we say that the linear matroid ( ) M is induced by . It is worth mentioning that ( ) M is of rank 3 since each basis in is formed by a triple of noncollinear points in . A fundamental example of a matroid is a uniform matroid. Let be a ground set of n elements and be the set of all possible k-subsets of . Then ( ) , defines a matroid called the uniform matroid k n , . In [3], Raney presented a special class of matroids, called triangular matroids, defined as follows: such that: (I) the ground set is the set of points of and (II) the collection of bases is the set of all non-triangular point triples in .
In this setting, the work in [3] intended to answer the question: if induces ( ) M tri , then what are the conditions on the triangles of that must be satisfied?
We note that if is any triangle-free n 3 -configuration, then induces ( ) M tri . In fact, ( ) M tri is isomorphic to n 3, . This is because each triple of points in forms a basis for ( ) M tri . Hence, the uniform matroid n 3, is constructed. See [3] for more details. It is possible that two or more non-isomorphic n 3 -configurations produce isomorphic copies of some induced triangular matroids. This is discussed in some detail in Section 2.
The search in [3] was established on n 3 -configurations for ≤ n 15. In this work, we extend that list to up to ≤ n 18. Furthermore, we present some structural properties for the constructed configurations. We also correct one miscalculated value which appeared in [3] (written in italics in Table 1). Table 1 presents the main results of the search for triangular matroids induced by n 3 -configurations. The new results are highlighted in bold. Here, # ( ) n c denotes the number of all non-isomorphic (combinatorial) n 3 -configurations, # ( ) n tri denotes the number of n 3 -configurations which induce triangular matroids, and # ( ) n mat denotes the number of non-isomorphic triangular matroids induced from these n 3 -configurations. Note that the entry for # ( ) 18 c , namely, 530,452,205, was faded to indicate that a special search was done in that case. That special search was used also for the cases = n 7 through = n 17 for further validation of the results. This search is explained in Section 3.

Properties
We now discuss some of the main properties of n 3 -configurations which induce triangular matroids. In order to keep the present work self-contained, we present some of the theoretical results related to triangular matroids. For proofs and further discussion, the reader can see [3].
A complete quadrangle in a configuration is a set of four points a, b, c, and d, no three collinear, for which all possible lines connecting each pair of distinct points exist. A near-complete quadrangle in a config- uration is a complete quadrangle missing exactly one line connecting one pair of points. A near-pencil in a configuration consists of the points of a line L and a point a not on L for which a is incident to all points on L. Figure 2 shows a near-complete quadrangle and a near-pencil. Theorem 2. Let be an n 3 -configuration with n triangles. If every point in is incident to exactly three triangles and no pair of points is incident to more than one triangle, then is isomorphic to tri , where tri is the configuration formed by the n triangles in .
Note that there exists no one-to-one correspondence between the triangular matroids themselves and the n 3 -configurations which induce them. For instance, there are several triangle-free n 3 -configurations (for ≥ n 18) inducing the same exact uniform matroid on n points, see [4,5].
But this same idea is also applicable for smaller n 3 -configurations. For that purpose, we present two 15 3 -configurations ( A 15 and B 15 ) in Figure 4. Each of these configurations consists of six triangles inducing two isomorphic triangular matroids. Figure 5 shows the geometric representations of the two isomorphic triangular matroids induced by configurations A 15 and B 15 . In the geometric representation, each collinear triple (that is, each non-basis element) corresponds to a triangle in the configuration it was induced by.    Assume that a list of all n 3 -configurations (for some ≥ n 7) is available at hand. One can test the configurations in the list for the tests stated in Theorem 1. This creates a (smaller or identical) list of all nonisomorphic n 3 -configurations inducing all triangular matroids ( ) M tri . We call this step "INDUCE." This list still can have isomorphic copies of triangular matroids (induced by several non-isomorphic configurations). Hence, we do one last test to clean up the isomorphic copies of these triangular matroids. We call this step "CLEAN UP." These two steps were implemented independently by the two authors for ≤ ≤ n 7 15, and by the first author for = n 16 and = n 17. The results were in full agreement after all. Another approach was to directly generate all n 3 -configurations inducing triangular matroids without having (storing) the whole list of non-isomorphic n 3 -configurations. In what follows, we describe the main steps of this (special) approach. This approach was initially applied for the case = n 18 as indicated in Table  1. The same approach was used again for the cases = n 7 through = n 17 for further validation. An n 3 -configuration can be represented by a { } 0, 1 -incidence matrix A with n rows and n columns corresponding to points and lines, respectively. The As A is representing an n 3 -configuration, it has exactly three ones in each row and exactly three ones in each column. Moreover, the dot product of two distinct rows (or two distinct columns) is at most one. Two incidence matrices (and hence their corresponding configurations) are said to be isomorphic if one can be obtained from the other by permuting the rows and the columns.
We now describe the search algorithm to classify the n 3 configurations inducing non-isomorphic triangular matroids. This algorithm can be considered as an example of the orderly generation method, see [6]. It has two main parts: the generation and the isomorphism test.
The search first starts with the generation procedure which carries out a row-by-row (or a point-bypoint) backtrack search to consider all possible incidence matrices of n 3 -configurations having neither a near-quadrangle nor a near-pencil. It starts initially with the empty matrix (all-zero entries), and it starts to build a row at each step.
Once a row is constructed by the generation procedure, the algorithm performs another test to check whether the created incidence matrix agrees with the canonical one. Here, canonical might have a different meaning depending on what our canonical matrix is defined to be. One example (which we chose) is to choose the lexicographically least form of the incidence matrix. If the incidence matrix is canonical, we proceed to the next row and continue the search. Otherwise, we reject it and backtrack.
The algorithm ends up with a list of non-isomorphic n 3 -configurations which induce triangular matroids. This list is similar (up to isomorphism) to the list created by the INDUCE step. At this point, we take that list again to the CLEAN UP step to produce all of the non-isomorphic triangular matroids induced by these non-isomorphic n 3 -configurations. This final CLEAN UP step is crucial since it is possible to produce isomorphic copies of induced triangular matroids from non-isomorphic copies of n 3 -configurations as discussed in Section 2.
We remark that we compute the lexicographically least representative of the isomorphism class of a matrix using our own algorithm. The complexity of this algorithm is exponential in the size of the input. No fast algorithm to solve this problem is known.
Moreover, the lexicographically least representative can be replaced by the canonical representative which can be computed using the idea of canonical augmentation due to McKay [7]. In almost all cases these representatives are different. We also tried this method using nauty [8] to compute the canonical representative. We found that orderly generation using the lexicographically least representative worked better for us. This may not be seen as a critique of "canonical augmentation." We did not try very hard to make it work, so a comparison is unfair. Again in either methods, no fast (i.e. polynomial) algorithm to solve this problem is known.

Results
The main results of the search described in Section 3 are presented in Table 1.
The search was done on a single Mac Laptop (with a Processor 2.2 GHz). The CPU needed for the search on = n 15, 16, 17, and 18 was 2 min, 26 min, 9 h, and 9.25 days, respectively. We note that all the presented figures in this paper were produced manually. The main purpose of these drawings is to emphasize some aspects such as symmetry of the groups and geometric realizations, if possible. Further study is advised here to study in more detail whether the objects can be realized geometrically. But this direction is not the purpose of the presented work.
We present some of the n 3 -configurations inducing triangular matroids that were constructed using the algorithm of Section 3 in what follows. But first, we present Table 2 which shows the triangle distribution of n 3 -configurations inducing triangular matroids. Here, #tri(n) and #mat(n) are defined in the same way as in Table 1. The two columns denoted by ( ) n Δ tri and ( ) n Δ mat present the number (and multiplicity) of the triangles of the related n 3 -configurations. That is, an entry a x means that there are x n 3 configurations inducing triangular matroids each of which have a triangles. For instance, the first entry written as 20 1 means that the 10 3 -configuration has 20 triangles and this type occurs once.
We might have (in the same table row) a x and a y in the third and fourth columns, respectively, with ≤ y x. This means that we have several ( − + y x 1) non-isomorphic n 3 -configurations, each of whose induced triangular matroids is shared by at least one other n 3 -configuration. For instance, in the row of = n 15 we have 6 4 (in ( ) n Δ tri ) and 6 3 (in ( ) n Δ mat ). This means that there exist two non-isomorphic 15 3 -configurations, each having six triangles, that produce isomorphic copies of the same induced triangular matroid. This fact has already been discussed in Section 2 ( Figure 4). Table 3 gives the distribution of non-trivial automorphism groups of n 3 -configurations which only produce non-isomorphic induced triangular matroids. The 'ago' column presents the automorphism group orders while the 'Aut' column presents the automorphism group type. An entry a x in the ago column means that we have x geometries with automorphism group order a. In the same corresponding position of the Aut column, the group type is shown. The group types listed in the last column of Table 3 can be described using the Gap-command "StructureDescription," see [9] for further details.
We use C n and D n to denote the cyclic group of order n and the dihedral group of order n 2 , respectively. For groups H and K, let H K : be a split extension of H by K (with normal subgroup H).

101,910
We now present some properties of some of the n 3 -configurations inducing non-isomorphic triangular matroids. In what follows, we write n x 3 -configuration to denote an n 3 -configuration whose automorphism group order is x.

14 3 -configurations inducing triangular matroids
For the 14 3 -configurations, there is a unique 14 3 -configuration which induces a triangular matroid with a transitive group. Its automorphism group is C 14 of order 14. Hence, it has one orbit on points (and one orbit on lines as well). It is the 14 3 14 -configuration presented in Figure 3. It is cyclic as it has a transitive automorphism group C 14 . Figure 6 shows one of the four 14 3 -configurations (namely, 14 3 8 with six triangles) which induces the triangular matroid described by its geometric representation. This configuration has 4-orbits with automorphism group D 4 of order 8. Points of the same color belong to the same point-orbit.        There are 220 15 3 -configurations inducing triangular matroids. Among these we found several examples of configurations inducing isomorphic triangular matroids. Thus, we cleaned up this list to get exactly 173 15 3 -configurations which induce non-isomorphic triangular matroids. There are only two configurations among these with transitive groups. Namely, these are the ones with automorphism groups S 6 and × C D 3 5 of orders 720 and 30, respectively. An example of a 15 3 -configuration which induces a triangular matroid is already presented in Figure 1. It is the Cremona-Richmond 15 3 720 -configuration. Its automorphism group (S 6 ) is transitive on its flags (hence on points and on lines). This configuration is the smallest triangle-free configuration. Hence, its induced triangular matroid is the uniform matroid on 15 points. Figure 7 presents the unique 15 3 5 -configuration. It induces a triangular matroid whose geometric representation is also presented. As it can be seen, it has ten triangles. It has five points incident with three triangles, five points incident with two triangles, and five points incident with only one triangle. This configuration has 3-orbits as it has three point-orbits (of the same size 5) under the action of C 5 . The three point-orbits are shown in the figure as different colors. Its automorphism group can be generated by ( )( )( ) 1 6 14 12 9 2 5 11 3 7 4 8 13 15 10 .
We next present the unique 15 3 10 -configuration which induces a triangular matroid. These two structures are shown in Figure 8. The automorphism group here is D 5 of ago 10. Some points in the diagram of the configuration were duplicated to show a better presentation of the structure of the group which can be generated by It induces two point-orbits of lengths 10 and 5. These two orbits are shown in two different colors in the diagram. Each point of the configuration is incident to two triangles where the number of triangles is 10. The geometric representation of the induced triangular matroid is also presented. As it can be seen, it has ten triangles. It is in fact an ( ) 15 , 10 2 3 -configuration. There is a unique 15 3 12 -configuration which induces a triangular matroid. It has 14 triangles. Figure 9 shows this configuration along with its induced triangular matroid. It has (4, 3)-orbits under its automorphism group D 6 of order 12. Again, we give different colors to points in different orbits. Note that in the geometric representation of the induced triangular matroid, four collinear points in the geometric representation means that each of the four possible point triples taken from these four points defines a triangle in the configuration. For instance, any point triples taken from the four collinear points (complete quadrangle) { } 3, 4, 12, 15 defines a triangle in the configuration. This configuration has 15 triangles. Each point is incident to three triangles, with no pair of points incident to a triangle more than once. Its triangles are blocks of another 15 3 configuration isomorphic to this 15 3 30 -configuration, see Theorem 2.
The 15 triangles of this configuration are as follows:   One of the four configurations with a transitive automorphism group is the cyclic 16 3 16 -configuration.
This configuration is presented in Figure 11. Its automorphism group is C 16 and it is flag-transitive. It has 16 triangles inducing an isomorphic 16 3 -configuration.
There are two 16 3 32 -configurations ( A 16 and B 16 ) inducing non-isomorphic triangular matroids. The automorphism groups for A 16 and B 16 are ( × ) C C C : 8 2 2 and SD 16 (the semidihedral group of order 32), respectively. Both configurations have 16 triangles in the setting of Theorem 2. These two groups are transitive (but not cyclic). Figure 12 presents these two configurations.
The 16 3 -configuration inducing a triangular matroid with the largest automorphism group order is presented in Figure 13. Its automorphism group (( × ) ) C C C C : :

17 3 -configurations inducing triangular matroids
In this case, we found 166,286 17 3 -configurations inducing triangular matroids. Applying the CLEAN UP procedure results in exactly 19,930 configurations inducing non-isomorphic triangular matroids. Among these, we found two 17 3 12 -configurations with the automorphism group D 6 of order 12. One is the unique triangular-free (self-polar) configuration on 17 points. The other one is a self-polar configuration with 14 triangles. Both have 4-orbits and the length of their orbits on points and lines is 6, 6, 3, and 2. Figure 14 shows their incidence matrices.     In this case, we see a considerable increase in the number of n 3 -configurations inducing triangular matroids. We found (up to isomorphism) 4,126,028 18 3 -configurations which induce triangular matroids, and 101,910 non-isomorphic triangular matroids in total. There are 23 triangle distributions in this case. We first present a 6-cyclic (in the terminology of [2]) 18 3 6 -configuration which induces a triangular matroid. Its automorphism group is C 6 of order 6. It has three orbits on points and three orbits on lines where each orbit has length 6. Figure 15 shows this configuration which happens to be a self-polar configuration. The point-orbits are presented as different colors. We also present the geometric representation of the induced triangular matroid. We found three flag-transitive configurations among the 4,126,028 18 3 -configurations. Their automorphism groups are C C , 18 18 , and ( × ) S S C : 3 3 2 with orders 18, 18, and 72, respectively. We also found one point-transitive and another line-transitive configuration. Both of these have × S S 3 4 as their automorphism group with order 144. Figure 16 shows a realization for the 18 3 72 -configuration which is flag transitive. It has six triangles where each point is incident to exactly one triangle.
The other two flag-transitive configurations have automorphism group C 18 . They are cyclic configurations having 18 triangles each, but their respective triangular matroids are non-isomorphic. We present these two configurations in Figure 17.  1,2 -orbits), respectively. Configuration L 18 has two point-orbits of length 12 and 6. On the other hand, configuration P 18 has two line-orbits of length 12 and 6. In fact, these two configurations are dual. By dual configuration, we mean that the roles of points and lines in the dual configuration are    Figure 19.
Another example of a dual pair of 18 3 -configurations inducing non-isomorphic triangular matroids is presented in Figure 20. This time we provide the incidence matrices associated with this dual pair of 18 3 48 -configurations.