ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 14 (2018) 267-284 https://doi.org/10.26493/1855-3974.953.3e1 (Also available at http://amc-journal.eu) Trilateral matroids induced by n3-configurations Michael W. Raney * Georgetown University Washington, DC, U.S.A. Received 9 October 2015, accepted 9 June 2017, published online 4 September 2017 We define a new class of a rank-3 matroid called a trilateral matroid. When defined, the ground set of such a matroid consists of the points of an n3-configuration, and its bases are the point triples corresponding to non-trilaterals within the configuration. We characterize which n3-configurations induce trilateral matroids and provide several examples. Keywords: Configurations, trilaterals, matroids. Math. Subj. Class.: 05B30, 51E30, 05C38, 05B35 1 Introduction A (combinatorial) n3-configuration C is an incidence structure consisting of n distinct points and n distinct blocks for which each point is incident with three blocks, each block is incident with three points, and any two points are incident with at most one common block. If C may be depicted in the real projective plane using points and having (straight) lines as its blocks, then it is said to be geometric. As observed in [6] (pg. 17-18), it is evident that every geometric n3-configuration is combinatorial, but the converse of this statement does not hold. A trilateral in a configuration is a cyclically ordered set {p0, h0,p1, h1,p2, h2} of pair-wise distinct points pi and pairwise distinct blocks hi such that pi is incident with hi- 1 and hi for each i e Z3 [2]. We may without ambiguity shorten this notation by listing only the points of the trilateral as {p0,p1,p2}, or more simply as p0p1p2. A configuration is trilateral-free if no trilateral exists within the configuration. Unless stated otherwise, the n3-configurations we shall examine are point-line configurations, so that the blocks are lines. But we shall investigate an example of a point-plane configuration in Section 3. Following the terminology of [7], we define a matroid M to be an ordered pair (E, B) consisting of a finite ground set E and a nonempty collection B of subsets of E called bases which satisfy the basis exchange property: *The author wishes to acknowledge the anonymous referee for the suggestion to consider point-plane n3-configurations as potential sources for trilateral matroids. E-mail address: mwr23@georgetown.edu (Michael W. Raney) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 63 63 Ars Math. Contemp. 14 (2018) 267-284 Definition 1.1. If Bi, B2 G B and x G Bi — B2, then there exists y G B2 — Bi such that Bi — x U y G B. It is a consequence of this definition that any two bases of M share the same cardinality; this common cardinality is called the rank of the matroid. See [7], pg. 16-18 for the details. It is a standard result that any n3-configuration C defines a rank-3 linear matroid, or vector matroid, M (C) = (E, B) whose ground set E consists of the points {pi ,p2,...,pn} of C and whose set of bases B consists of the point triples {pa ,pb,pc} which are not collinear in C. Hence the cardinality of B is Q) — n for the linear matroid M (C) induced by C. In this work we pose the following associated question: under what conditions do the trilaterals of an n3-configuration C induce a rank-3 matroid Mtri(C) = (E, B) whose ground set E again consists of the points of C, but now whose bases are the point triples corresponding to non-trilaterals? This question, to our knowledge, has not previously been considered in the literature on configurations and matroids. Definition 1.2. A trilateral matroid Mtri (C) = (E, B), when it exists, is a matroid defined on the set E of points of an n3-configuration C whose set of bases B consists of all of the non-trilaterals of C. When Mtri(C) exists, we say that C induces Mtri(C). We shall see that, in contrast to the linear matroid setting, seldom is it the case that an n3-configuration C induces a trilateral matroid Mtri(C). But thankfully such matroids do exist; for instance, any trilateral-free configuration induces a trilateral matroid, since in this setting every point triple forms a base of the matroid. In other words, if C is a trilateral-free n3-configuration, then Mtri(C) exists and furthermore Mtri(C) = U3,n, the uniform matroid of rank 3 on n points. Thus our initial motivation to define this new class of matroids stems from the desire to enlarge the class of trilateral-free configurations. For purposes of instruction, we regard an example of a 153-configuration which induces a trilateral matroid on its points. Here is a combinatorial description of this configuration. li 12 h h I5 1& I7 1g I9 lio 1ll 1i2 1i3 114 115 1112233455 7 7 9 10 13 246468 11 68 9 8 9 11 11 14 3 5 7 14 10 12 13 12 10 13 14 15 12 15 15 This configuration has 10 trilaterals: ti ¿2 t3 t4 t5 ie ¿7 tg t9 tio 1 1 1 2 3 7 9 9 9 11 2 2 4 4 11 14 11 11 13 13 4 6 6 6 12 15 13 15 15 15 In Figure 1 we see both a diagram of this 153-configuration and a geometric representation of its trilateral matroid. In the geometric representation, each trilateral (that is, each nonbasis element) is collinear. Note that the configuration contains two complete quadrangles. The first complete quadrangle is determined by the point set {1, 2,4, 6}, and the second by {9,11,13,15}. This means, for example, that no three points in {1, 2,4,6} are collinear, and each pair of points is incident to a line of the configuration. So all four point triples present within {1, 2,4,6} give trilaterals, and hence are not bases of the matroid. Thus every 2-element subset of {1, 2,4,6} is independent, but no 3-element subset of {1, 2,4, 6} is. Therefore M. W. Raney: Trilateral matroids induced by us-configurations 269 11 9 13 15 Figure 1: A 153-configuration with 10 trilaterals, and a geometric representation of the matroid induced by these trilaterals. the four-point line that represents the level of dependency of {1,2,4, 6} in the geometric representation is appropriate. This minor is isomorphic to U2,4, which is the unique excluded minor for the class of binary matroids ([7], pg. 501). We must note that there is a fundamental difference between trilateral matroids and linear matroids. Admittedly a finite set of points and lines in the plane gives a (linear) matroid if and only if any pair of lines meet in at most one point. For suppose there exist two points a and b which are met by two lines, so that points a, b, c are collinear, points a, b, d are collinear, but a, b, c, d are not all on one line. Pick a new point e so that c, d, and e are not collinear, and so that a, b, and e are not collinear. Let B\ = abe and B2 = cde G B; both are bases of the linear matroid. We have B1 — B2 = ab and B2 — B1 = cd. Let x = e G B\ — B2, so B\ — x = cd. But if y G B2 — B\ = ab, then B\ — x U y equals either abc or abd, neither of which is a base. Hence a linear matroid cannot have two points common to more than one line. But a trilateral matroid can; if both abc and abd are trilaterals, then the configuration has a chance to induce a trilateral matroid if trilaterals acd and bcd are also present, meaning that points c and d are incident to a particular line of the configuration. In other words, points {a, b, c, d} form a complete quadrangle within the configuration. We shall explore this necessity further in Theorem 1.7. Any point of an n3-configuration is incident to three lines; these three lines are then incident to six points which are distinct from the original point and from each other. Consequently, the maximum number of trilaterals incident to a given point is (3) — 3 = 12, since lines are not trilaterals. This maximum is achieved by every point of the Fano 73-configuration (the smallest n3 -configuration) given in Figure 2. Proposition 1.3. Suppose an n3-configuration C induces a trilateral matroid Mtri(C) = (E, B). Then each point of the configuration is incident to at most six trilaterals. Proof. Let a be a point in C, and let abc, ade, and afg be the lines in C incident to a. Each of these lines belongs to B, and hence there are at most (3) — 3 = 12 trilaterals incident to a, namely abd, abe, abf, abg, acd, ace, acf, acg, adf, adg, aef, and aeg. 270 Ars Math. Contemp. 14 (2018) 267-284 Figure 2: The Fano 73-configuration. Since Bi = abc and B2 = ade are bases of Mtri(C), the basis exchange property applies to them. This means that if x G Bi — B2 = bc, there must exist some y G B2 — Bi = de such that Bi — x U y G B. Consequently, letting x = b, we find at least one of acd and ace must be a base, hence not a trilateral. Likewise, letting x = c, it follows that at least one of abd and abe is not a trilateral. Applying a similar analysis to the pair of bases Bi = abc, B2 = afg, we find that at least one of acf and acg is not a trilateral, and at least one of abf and abg is not a trilateral. Finally, given Bi = ade, B2 = afg, we find that at least one of aef and aeg is not a trilateral, and at least one of adf and adg is not a trilateral. Hence at least six of the 12 possible non-collinear triples are not trilaterals, so at most six are trilaterals. □ Corollary 1.4. Suppose an n3-configuration C induces a trilateral matroid Mtri(C) = (E, B). Then C contains at most 2n trilaterals. Although Corollary 1.4 admittedly serves as a crude necessary condition for an n3-configuration to induce a trilateral matroid, it does permit us to eliminate some of the smallest n3-configurations from consideration, such as the Fano 73-configuration (which contains 28 trilaterals) and also the Mobius-Kantor 83-configuration (which contains 24 trilaterals). Additionally, two of the three non-isomorphic 93-configurations may be dismissed from consideration by this criterion, although the Pappus 93-configuration, which contains 18 trilaterals, is still a possibility. We shall soon see, though, that the Pappus configuration does not induce a trilateral matroid on its points. The upper bound indicated by Proposition 1.3 is sharp, for it turns out that the Desargues 103-configuration induces a trilateral matroid. Each of the points of the Desargues configuration is incident to six trilaterals. M. W. Raney: Trilateral matroids induced by us-configurations 271 We now establish our main result. This will require the introduction of two types of geometric obstructions (near-complete quadrangles and near-pencils) that, when present within an n3-configuration C, individually preclude the existence of Mtri (C). Definition 1.5. A near-complete quadrangle [ab : cd] consists of four points a, b, c, and d of the configuration, no three of which are collinear, for which five of the six possible lines connecting each pair of points exist within the configuration, except for the pair cd. c X c d Figure 4: Near-complete quadrangle [ab : cd]. For example, we note the presence of the near-complete quadrangle [ab : cd] in the Pappus configuration in Figure 5. Figure 5: The Pappus 93-configuration. It is important to note that, by our conventions, a complete quadrangle determined by points {a, b, c, d} does not contain anear-complete quadrangle [ab: cd], since there exists a line in the configuration incident to both c and d. So the Desargues configuration, for example, possesses five complete quadrangles but no near-complete quadrangle. As we shall witness in greater detail, n3-configurations which induce trilateral matroids may contain complete quadrangles. Indeed, in a linear matroid, given any two points, at most one line passes between them. But, two trilaterals (call them abc and abd) may share the points a, b provided that acd and bcd are also trilaterals, that is, that line cd is also present within the configuration. Definition 1.6. A near-pencil [a : bcd] consists of four points a, b, c, and d of the configuration, with a incident to each of b, c, and d, and with bcd a line of the configuration. We regard the near-pencil [a : bcd] in the Mobius-Kantor 83-configuration given in Figure 7. The notations [ab : cd] and [a : bcd] for a near-complete quadrangle and a near-pencil, respectively, are similar in that the points incident to three of the lines which determine the object appear to the left of the colon, and those points incident to two lines appear to the right of the colon. 272 Ars Math. Contemp. 14 (2018) 267-284 a Figure 6: Near-pencil [a : bcd]. Figure 7: The Möbius-Kantor 83-configuration. Theorem 1.7. Let C be an n3-configuration, and let B be the set of the non-trilaterals of C. Then C induces a trilateral matroid Mtri(C) if and only if no four points of C determine either a near-complete quadrangle or a near-pencil. Proof. ( w) First suppose that C contains a near-complete quadrangle [ab: cd]. Let e be the third point on line ace. Case 1: bde is a line in C. Then the following subfiguration is present inside C. Let B\ = ace and B2 = bde; both Bi,B2 G B. Then B\— B2 = ac and B2 -B\ = bd. Let x = c G B\ — B2; then B\ — x = ae. But both abe and ade are trilaterals, so B1 — x U y G B for all y G B2 — B1. Hence B cannot be the set of bases of a matroid. Case 2: bde is not a line in C. Then inside of C we have b d ace Note that edge be cannot be present, for if so point b would have four lines incident to it, but every point in an n3-configuration is incident to three lines. Let B\ = abe,B2 = acd G B. Take e G B\ — B2; we have B\ — e = ab. But B2 — B\ = cd, and both abc and abd are trilaterals. Hence B cannot be the set of bases of a matroid. M. W. Raney: Trilateral matroids induced by us-configurations 273 Now suppose C contains a near-pencil [a: bcd] as indicated in the diagram. Let e be the third point on line ace. a em We have Bi = ace, B2 = bcd G B. Choose e G Bi — B2. Then Bi — e = ac. But B2 — B1 = bd, and both abc and acd are trilaterals. Hence B cannot be the set of bases of a matroid. Suppose that C does not induce a trilateral matroid Mtri (C). Since B cannot be the set of bases of a matroid, there must exist a pair Bi ,B2 in B for which the basis exchange property is violated. So there must exist x G Bi — B2 such that for all y G B2 — Bi, Bi — x U y is a trilateral. There are several cases to consider, some of which are vacuous. Case 1: Bi = B2. Then Bi — B2 = 0, so a violation of the basis exchange property cannot occur in this circumstance. Case 2: Bi = abc, B2 = abd (distinct letters label distinct points in C.) Then Bi — B2 = c and B2 — Bi = d. For a violation to occur, we require that Bi — cU d be a trilateral. But Bi — c U d = B2 G B. Hence no violation can occur in this case as well. Case 3: Bi = abc, B2 = ade. Then Bi — B2 = bc and B2 — Bi = de. Without loss of generality we assume that x = b. For a violation of the basis exchange property to occur, both acd and ace must be trilaterals. Subcase 3.1: ade is a non-collinear non-trilateral. Then [ac : de] is a near-complete quadrangle. a c P 13, the cyclic configuration C3(n, 1,4) induces a trilateral matroid on n trilaterals which equals the linear matroid on C3(n, 3,4). In other words, Mtri(C3(n, 1, 4)) = M(C3(n, 3,4)). Moreover, C3(n, 1,4) = C3(n, 3, 4). Proof. In order to determine the trilaterals of C3 (n, 1, 4), it suffices to ascertain the trilaterals which involve 0, and then extend from this via a cyclic pattern. The trilaterals involving 0 are: • 0 3 4 (using the lines {0,1,4}, {3,4, 7}, and {n — 1,0,3}) • n — 4 n — 30 (using the lines {n — 4, n — 3, 0}, {n — 1,0, 3}, and {n — 5, n — 4, n — 1}) • n — 3 0 1 (using the lines {n — 4, n — 3,0}, {n — 3, n — 2,1}, and {0,1, 4}) Since n > 13, no extra trilateral involving 0 is formed (for example, if n = 12, then 0 4 8 would be a trilateral.) Hence we see, after extending cyclically, that the trilaterals of C3(n, 1,4) form their own configuration, namely C3(n, 3,4), and thus Mtri(C3(n, 1,4)) is the linear matroid corresponding to C3(n, 3,4). Finally we may recognize that C3(n, 1,4) is isomorphic to C3(n, 3,4) either by utilizing Theorem 2.1 or by applying the correspondence t ^ (4 — t) mod n. □ It turns out that C3(16,1,4) and C3(16,1,7) are the smallest examples of non-isomorphic cyclic C3(n, k, m) configurations having n trilaterals each, and hence their corresponding trilateral matroids (which are isomorphic to the linear matroids associated with the respective original configurations) are non-isomorphic to each other as well. It is possible, however, for a non-cyclic n3-configuration to induce a trilateral matroid on its n trilaterals, with the trilaterals capable of determining an n3-configuration in their own right, without the original configuration needing to be cyclic. We have already seen 278 Ars Math. Contemp. 14 (2018) 267-284 Figure 13: A non-cyclic 163-configuration whose trilateral matroid is isomorphic to the linear matroid associated with the configuration. one example of this with the Coxeter 123-configuration given in Figure 9. Another example is the 163-configuration provided in Figure 13 whose automorphism group has order 32. It is additionally possible for an n3-configuration possessing n trilaterals to induce a trilateral matroid that is not isomorphic to the linear matroid associated with the original configuration. Figure 14 gives a diagram of such a configuration, a 203-configuration containing 20 trilaterals. It contains two points which are involved in six trilaterals and four points involved in four trilaterals. A geometric representation of the matroid is also provided. Figure 14: A 203-configuration with 20 trilaterals whose trilateral matroid is not isomor-phic to the linear matroid of the configuration, and a geometric representation of its trilateral matroid. We next offer an example of of an 183-configuration possessing 20 trilaterals which induces a trilateral matroid. In Figure 15 we provide a picture of this configuration (with several pseudolines) and the accompanying geometric representation of its trilateral matroid. This example presents another instance, in addition to the Desargues 103-configuration, of an n3-configuration containing more than n trilaterals which induces a trilateral matroid. M. W. Raney: Trilateral matroids induced by us-configurations 279 Figure 15: An 183-configuration with 20 trilaterals, and a geometric representation of its trilateral matroid. Note that this configuration contains four complete quadrangles. We now return to the enumeration of the smallest n3-configurations which induce trilateral matroids. There are four 143-configurations which do so. We label these configurations as 14A, 14B, 14C and 14D, and provide combinatorial depictions of them. 11 12 13 14 15 16 17 18 19 110 1ii 1i2 1i3 114 1 1 1 2 2 3 3 4 5 5 6 7 8 9 2 4 6 4 8 7 8 11 6 12 9 10 13 11 3 5 7 9 10 12 11 12 13 14 10 14 14 13 li 12 13 14 15 16 17 18 19 1i0 1ii 1i2 1i3 114 1 1 1 2 2 3 3 4 5 5 6 6 7 8 2 4 6 4 9 7 10 11 10 12 8 9 9 11 3 5 7 8 12 11 12 13 14 13 10 13 14 14 li 12 13 14 15 16 17 18 19 1i0 1ii 1i2 1i3 114 1 1 1 2 2 3 3 4 5 5 6 7 7 10 2 4 6 4 8 6 13 11 8 12 8 9 10 11 3 5 7 9 10 11 14 12 13 14 9 14 12 13 li 12 13 14 15 16 17 18 19 1i0 1ii 1i2 1i3 114 1 1 1 2 2 3 3 4 5 5 6 6 7 7 2 4 6 4 10 8 12 11 8 10 8 10 9 11 3 5 7 9 13 11 14 12 13 14 9 12 14 13 These configurations contain 14, 10, 10, and 6 trilaterals, respectively. Also, their automorphism groups have orders 14, 1, 4, and 8, respectively. Figure 16 gives a realization of Configuration 14A, which is isomorphic to the cyclic configuration C3(14,1,4). Hence we know its trilateral matroid is isomorphic to its linear matroid by Proposition 2.2. Configurations 14B and 14C both contain 10 trilaterals, so it is conceivable that their associated trilateral matroids could be isomorphic. But they are not, for 14B has three points which are each incident to three trilaterals and one point which is incident to only one trilateral, whereas Configuration 14C has two points each incident to three trilaterals 280 Ars Math. Contemp. 14 (2018) 267-284 11 Figure 16: Configuration 14A. and no point incident to only one trilateral. Figure 17 gives geometric representations of the trilateral matroids associated with Configurations 14B and 14C, respectively. Figure 17: Geometric representations for trilateral matroids for Configurations 14B and 14C. Figure 18 is a rendering for Configuration 14D with several pseudolines, along with a geometric representation of its associated trilateral matroid. Proceeding to the n = 15 setting, we encounter a substantial increase, to 220, of the number of 153-configurations which induce trilateral matroids. One such example is the Cremona-Richmond configuration provided in Figure 19. It is the smallest example of a trilateral-free n3-configuration. As it is trilateral-free, the trilateral matroid it induces is the uniform matroid on 15 points U3,15. Another example is the cyclic configuration C3(15,1,4), whose induced trilateral matroid (with 15 trilaterals) is isomorphic to the linear matroid on C3(15,1,4) by Proposition 2.2. Its automorphism group has order 30. Each of the other 153-configurations which induces a trilateral matroid contains k trilaterals, where k G {4,6,7,8, 9,10,11,12,13,14}. It is clearly not the case that for all n, there exists a one-to-one correspondence between the trilateral matroids themselves and the n3-configurations which induce them. We know this because there are four non-isomorphic trilateral-free 183-configurations [3], so each consequently must induce the same uniform matroid on 18 points. M. W. Raney: Trilateral matroids induced by us-configurations 281 Figure 18: Configuration 14D and its trilateral matroid. Figure 19: The Cremona-Richmond 153-configuration. It is of interest to contemplate whether smaller non-isomorphic n3-configurations exist that induce isomorphic trilateral matroids, and indeed this turns out to be true. In fact, this property is satisfied by the following pair of non-isomorphic 153-configurations given in Figure 20. Each contains 8 trilaterals and has a symmetry group of order 48. The Figure 20: Non-isomorphic 153-configurations which induce the same trilateral matroid on 15 points. set of points for both configurations consists of the eight vertices of a cube, the centers of the six faces of the cube, and the center of the cube itself. In the former configura- 282 Ars Math. Contemp. 14 (2018) 267-284 tion the diagonally-opposing points in each face of the cube are incident via a line which passes through the center of the same face, whereas in the latter configuration one pair of diagonally-opposing points in each face are incident via a "line" which passes through the center of the opposite face. The eight trilaterals involved in these respective configurations are identical, and thus their corresponding trilateral matroids are the same. Figure 21 gives this matroid, which is isomorphic to U2,4 © U2,4 © U3,7. Hence the number of Figure 21: The common trilateral matroid. trilateral matroids that are induced from 153-configurations is smaller than the number of 153-configurations which induce trilateral matroids. Our calculations indicate that there are 214 non-isomorphic trilateral matroids that may be found from the 220 153-configurations which induce trilateral matroids. We conclude this section with a table which summarizes the current state of affairs. Here #c(n) denotes the number of non-isomorphic n3-configurations, #tri(n) denotes the number of these configurations which induce trilateral matroids, and #mat(n) denotes the number of non-isomorphic trilateral matroids which arise from these configurations. n #c(n) #tri(n) #mat(n) 7 1 0 0 8 1 0 0 9 3 0 0 10 10 1 1 11 31 0 0 12 229 1 1 13 2036 1 1 14 21399 4 4 15 245342 220 214 3 A point-plane configuration A point-plane n3-configuration is an incidence structure consisting of n distinct points and n distinct planes for which each point is incident with three planes, each plane is incident with three points, and any two points are incident with at most one common plane. In such a configuration, we deem a trilateral to be a cyclically ordered set {p0, n0,p1,n1 ,p2,n2} of pairwise distinct points pi and pairwise distinct planes ni such that pi is incident with ni_1 and ni for each i e Z3. Once more we may without ambiguity shorten this notation by listing only the points of the trilateral as {po, p 1, p2}, or more simply as pop 1p2. In Figure 22 we offer an example of a point-plane 123-configuration which induces M. W. Raney: Trilateral matroids induced by us-configurations 283 a trilateral matroid on its points. The 12 points are selected from the 20 vertices of the regular dodecahedron so that each of the twelve pentagonal faces contains three points; note that each of the 12 points is the intersection of three faces, so a point-plane 123-configuration is achieved. We observe that each of the eight unlabeled red points in the 4 Figure 22: A 123 point-plane configuration which induces a trilateral matroid. diagram corresponds to a trilateral, and that this trilateral may be specified uniquely by cycling through the configuration points that are immediately adjacent to the red point. For example, the triple {1,3, 5} defines a trilateral. We start at 1, then pass through the plane containing both 1 and 3 to 3. We then pass through the plane containing both 3 and 5 to 5, and then finally pass through the plane containing both 5 and 1 back to 1 to complete the cycle. Here are the eight trilaterals. ¿1 ¿2 ¿3 ¿4 ¿5 ¿6 ¿7 ¿g 1 1 2 3 4 4 6 8 2 3 7 6 5 9 11 10 9 5 8 7 11 10 12 12 Figure 23 gives a geometric representation of the trilateral matroid. Figure 23: The trilateral matroid of the 123 point-plane configuration. 284 Ars Math. Contemp. 14 (2018) 267-284 After identifying each trilateral with its corresponding red point in Figure 22, we recognize that the trilateral matroid may also be represented as a point-plane configuration, namely an (83,122)-configuration. This means the configuration has eight points, with three planes incident to each point, and twelve planes, with two points incident to each plane. References [1] A. Al-Azemi and D. Betten, The configurations 123 revisited, J. Geom. 105 (2014), 391-417, doi:10.1007/s00022-014-0228-0. [2] M. Boben, B. Grunbaum and T. Pisanski, Multilaterals in configurations, Beitr. Algebra Geom. 54 (2013), 263-275, doi:10.1007/s13366-011-0065-3. [3] M. Boben, B. Grunbaum, T. Pisanski and A. Zitnik, Small triangle-free configurations of points and lines, Discrete Comput. Geom. 35 (2006), 405-427, doi:10.1007/s00454-005-1224-9. [4] R. Daublebsky von Sterneck, Die Configurationen 123, Monatsh. Math. Phys. 6 (1895), 223255, doi:10.1007/bf01696586. [5] H. Gropp, On the existence and nonexistence of configurations nk, J. Comb. Inf. Syst. Sci. 15 (1990), 34-48. [6] B. Grunbaum, Configurations of Points and Lines, volume 103 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 2009, doi:10.1090/gsm/103. [7] J. G. Oxley, Matroid Theory, volume 3 of Oxford Graduate Texts in Mathematics, Oxford University Press, New York, 1992.