ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 107-123 On some generalization of the Möbius configuration Krzysztof Petelczyc Institute of Mathematics, University of Bialystok, Ciolkowskiego 1 M, 15-245 Bialystok, Poland Received 13 November 2014, accepted 28 January 2017, published online 19 February 2017 The Möbius (84) configuration is generalized in a purely combinatorial approach. We consider (2nn) configurations M(„,v) depending on a permutation p in the symmetric group Sn. Classes of non-isomorphic configurations of this type are determined. The parametric characterization of M(„,v) is given. The uniqueness of the decomposition of into two mutually inscribed n-simplices is discussed. The automorphisms of M(„,v) are characterized for n > 3. Keywords: Möbius configuration, (84 ) configurations, Möbius pair, n-simplex. Math. Subj. Class.: 51D20, 05B30, 51E30 1 Introduction The Mobius (84) configuration is a certain configuration in a projective 3-dimensional space consisting of two mutually inscribed and circumscribed tetrahedra (cf. [7]). Each vertex of one tetrahedron lies on a face plane of the other tetrahedron and vice versa. Configurations with parameters (n4) were studied in detail in [4], but this is not the case, since the Mobius (84) configuration is not a point-line structure. An important role of the theorem connected with the Mobius configuration (which says, roughly speaking, that the Mobius configuration "closes") in a projective 3-dimensional space was presented in [12]: it is equivalent to the commutativity of the ground division ring. In this paper we deal with two n-simplices (simplices with n vertices, n > 3)1 instead of two tetrahedra (4-simplices). The way how an n-simplex is inscribed into another we define by a permutation p in the group Sn. The generalization of the Mobius configuration we obtain, is a (2nn)-configuration and it will be referred to as a Mobius pair of n-simplices, E-mail address: kryzpet@math.uwb.edu.pl (Krzysztof Petelczyc) 1 In geometry an n-simplex usually means a simplex having n +1 vertices. Our definition is slightly different. Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 108 Ars Math. Contemp. 13 (2017) 107-123 or shortly a Möbius n-pair. Only a combinatorial scheme (an abstract incidence structure, see e.g. [2, 10]) of a Möbius n-pair is investigated and we do not discuss problems regarding embeddability into projective (or other) spaces. Although these problems have been partially solved in [5] (the case with p = id), they are interesting and still open in general. As we know from [6], in a projective space, up to an isomorphism there are five (84) point-plane configurations with the property that at most two planes share two points, and dually at most two points are shared by two planes. These are precisely those configurations with two mutually circumscribed tetrahedra, and thus all of them are sometimes called the Mobius configurations. It is also known (cf. [10]), that these (84) configurations correspond to conjugacy classes of the permutation group S4. We shall prove, that two Mobius n-pairs are isomorphic if and only if the permutations, that determine them, are conjugate. Another important impact of the permutation on the geometry of the Mobius n-pair is that the cycle structure of p is associated with circuits in the incidence graph of the Mobius n-pair. As we shall see, the decomposition of the points of the generalized Mobius configuration into two complementary and mutually inscribed simplices is, generally, a unique one. Exceptions appear "near" the classical case n = 4. Three of five (84) Mobius configurations contain at least two distinct pairs of complementary 4-simplices. The next problem, which is considered in the paper, involves Mobius subpairs of a Mobius n-pair. We simply delete some number of points and blocks of one n-simplex and the same number of points and blocks of the second n-simplex with a hope to obtain a Mobius pair again. The conditions, under which we get a subpair in the Mobius n-pair, are determined. In the last part we use most of the established properties to characterize the automorphism group of the Mobius n-pair for n > 3. 2 Definitions, parameters and basic properties By a configuration we mean any point-block structure M = (S, L}, where the blocks are subsets of the set of points, i.e. L C 2S. The rank of a point is the number of blocks containing this point, and dually the size of a block is the number of points contained in this block. Let n be a natural number and X be a set. The family of all n-subsets of the set X will be denoted by P„(X). Let n > 3. We say that a configuration M is an n-simplex iff |L| = n, there is a subset V G Pn(S) such that for every V' G Pn-i(V) there is a unique block L G L containing V', and the rank of each point s G S \ V is less than n - 1. Elements of V will be called vertices of the simplex, and blocks of the simplex are said to be its faces. We say that two configurations M1 = (S1, L1}, M2 = (S2, L2} are isomorphic (and we write M1 = M2) iff there exists a bijective map f: S1 —> S2 such that conditions k G L1 and f (k) G L2 are equivalent. In case M1 = M2 = M the map f will be called an automorphism of M. Let us consider two sets A = ja1,..., a„} and B = {b1,..., bn} suchthat A n B = 0. Let p G Sn be a permutation of the set I = {1,..., n}. Now we introduce the following sets: La := {A' U {bj : A' G P„-1(A) and a G A'}, Lb := {B' U {avW} : B' G P„-1(B) and b G B'}. The configuration M(„jV) := (A U B, La U Lb }, K. Petelczyc: On some generalization of the Möbius configuration 109 will be called a Mobius n-pair. The Mobius configurations can be identified with the Mobius 4-pairs, which Levi graphs are Figures 1, 2, 3, 4, 5. All of them are also presented in [10]. In particular, M(4 id) is the classical (84) Mobius configuration. Let M be a Mobius n-pair. We write: A^ Bj for blocks of M not containing a.j, 6j, respectively; a-points, b-points, A-blocks, B-blocks for points in A, B, and blocks in LA, LB , respectively. The configuration M reflects the main abstract properties of the classical Mobius configuration. 1. The a-points yield a simplex in M: for any (n - 1)-subset A \ {a^ of the a-points there is a unique block of M, which contains this subset (Aj, a face of the simplex in question); the remaining points (b-points) yield another simplex. 2. The simplex with a-points and the simplex with b-points are mutually inscribed: on each face, Aj, of the first simplex there is a unique vertex (bj) of the second one; on each face, Bj, of the second simplex there is a unique vertex (av(j)) of the first simplex. Thus, we can decompose M into two complementary substructures SA(M) = (A, LA) and SB (M) = (B, LB), which we call simplices of M (although, formally, a block of each of them is not a subset of its points; there is one extra point on each of its faces). In the forthcoming part we will use the notion of the incidence graph (the Levi graph) Gm associated with M. Recall that a Levi graph is a bipartite graph with partition induced by points vs. blocks (cf. [9, 10]). Two of its vertices x, y are said to be adjacent (which is written x ~ y) if x is a point, y is a block (or vice versa) and x G y (or y G x). Otherwise x is not adjacent to y, which we write x ^ y. The rank of a vertex is the number of vertices adjacent to it. A vertex of Gm will be called point-vertex, block-vertex, a,b-vertex, A,B-vertex, or simply aj, bj, Aj, Bj as it corresponds to the point or to the block of M. The Levi graph associated with SA(M), SB (M) will be denoted by Gsa(m), Gsb(m), respectively. Remark 2.1. Let M be a Mobius n-pair. The Levi graph Gm has the following properties: (i) for X = A, B, every point-vertex from Gsx (m) is adjacent to all but one block-vertices from Gsx (m), and vice versa, (ii) for X, Y = A, B and X = Y, every point-vertex from Gsx (m) is adjacent to precisely one block-vertex from Gsy(m), and vice versa. Immediately from the definition of M(„iV), the number of its points coincides with the number of its blocks and equals 2n, and the rank of every point coincides with the size of every block and equals n. Thus the structures we investigate are (2nn)-configurations. A standard parametric question related to configurations is: what is the number of points that are contained in two distinct blocks, and dually: what is the number of blocks containing two distinct points. Proposition 2.2. Let k, l be two different blocks of the structure ffi(nif). Then |k n 1| G {0,1,2, n — 2}. If both k, l are A-blocks, or both k, l are B-blocks then |k n 1| = n — 2. Otherwise, k = Aj and l = Bj for some i, j G I, and the following equivalences hold (i) |Aj n Bj | =0 iff ^(j ) = i = j, (ii) |Aj n Bj | = 1 iff ^(j) = i = j or ^(j) = i = j, 110 Ars Math. Contemp. 13 (2017) 107-123 (iii) |A n Bj | = 2 iff fj ) = i = j. Proof. It is straightforward from the definition that if k, l are both A-blocks or B-blocks then k n l has n — 2 elements. Let k = Aj G and l = Bj G for some i, j G I. Let i = j. If f(j) = i then Aj n Bj = {b^a^j)}. Otherwise, for f(j) = i, we get Aj n Bj = {6j}. Let i = j. If f (i) = i we obtain Aj n Bj = {av(j}. In case f (i) = i it holds Aj n Bj = 0. □ Each conjugacy class of Sn corresponds to exactly one decomposition of a permutation f G Sn into cycles, up to a permutation of the elements of I. Now we describe how the cycle structure of f is reflected in block paths of M(„,v). Fact 2.3. A permutation f contains a cycle of length k < n iff there is a closed path of length 2k consisting of blocks of M(„,v) such that every two consecutive blocks intersect in precisely one point of M(„,v). Proof. Assume that f contains the cycle (iii2... ik). Then ajj+1 G Aj. n Bj. and bjj+1 G Bj, Ajj+1 for each j < k. Thus, the closed path in question is the following: Aj1, Bj1, Aj2 , Bj2 , . . . , Ajfc , Bjfc . Now assume that there exists a closed path 11, l1,..., lk, l'k of blocks of M(n,v) such that every two consecutive blocks intersect in a point. By Proposition 2.2(ii) every two consecutive blocks of the path are Aj g La, Bj g with f (j) = i = j or f (j) = i = j. Suppose f (j) = i = j holds for the first two blocks of our path, namely l1 = Aj, l[ = Bj and f (i) = i for some i G I .To obtain |l'x n l21 = 1 we must have l2 = Aj with f (i) = j. Thus the next two blocks are l2 = Av(j), l2 = Bv(j) and f (f (i)) = f (i). In general we obtain j = Avj-1(j), lj = Bvj-1 (j) and f j-1(i) = fj-2(i) for every j = 2,..., k. To close the path we need fk(i) = i. Let us put i = i0. Then the cycle (i0, i1,..., ik-1), where ij = fj (i) for j = 0,..., k — 1, is one of the cycles in the cycle decomposition of f. □ As the configuration M(n v) is symmetric, it makes sense to consider the dual configuration Fact 2.4. The configuration Mj?n is isomorphic to M(„,v). Proof. It is easy to note that Mj?n = M(n,v-1). Consider a G Sn such that a(1) = 1 and a(m) = n — m + 2 for m G I\ {1}. Let x G {a, b, A, B}, i G I. Then F : xj ^ xa(j) is an isomorphism mapping M(n,v-1) onto M(„,v). □ The problem of two isomorphic Mcibius n-pairs will be considered in general in the last section of the paper. Another parametric characterization is now a simple consequence of Proposition 2.2 and Fact 2.4. Proposition 2.5. Let x, y be two different points of M(„,v). There exist 0, 1, 2, or n — 2 blocks of M(„,v) containing x and y. 3 Hidden Mobius pairs The goal of this section is to characterize M = M(„,v) that can be transformed into Mobius pair with simplices distinct from SA(M), (M) by a decomposition of the points or by a deletion of some points and blocks. Informally, we say that these Mobius pairs are hidden in M. K. Petelczyc: On some generalization of the Möbius configuration 111 Figure 1: The Levi graph of M(4,id) (isomorphic to the hypercube graph Q4). Figure 2: The Levi graph of ffi(4,v) with p = (1234). Figure 3: The Levi graph of M(4jV) with p = (123)(4). Figure 4: The Levi graph of M(4,v) with p = (1)(2)(34). a\ ___a2 as ___04 bi ^-bi 63 64 Ai A2 A3 A4 B1 B 2 B3 B4 Figure 5: The Levi graph of M(4jV) with p = (12)(34). 112 Ars Math. Contemp. 13 (2017) 107-123 3.1 Möbius n-pairs with the special decompositions Let us start with the following combinatorial observation: Remark 3.1. The Möbius configuration M = M(4 id) can be presented in 3 distinct ways as two mutually circumscribed simplices such that each of them is distinct from Sa(M),Sb (M). One could say that there are four Mobius 4-pairs hidden in M(4,id). Let n > 4, M = M(„,v), and assume that it is possible to decompose the points of M into two complementary and mutually inscribed simplices Si(M), S2(M) suchthatSt(M) = SX(M) for each t = 1, 2, X = A,B. Such a decomposition will be called a special decomposition. Lemma 3.2. Let S1(M), S2(M) be two simplices, that arise from a special decomposition of M. (i) For each i G I, and each t =1,2, it is impossible to have both Bi,bi in St(M), or both Ai,ai in St(M). (ii) For each t = 1,2, the blocks of St(M) are two B-blocks and two A-blocks. Proof. The proof involves only S1(M), since the reasoning for S2 (M) will be the same. (i) Assume that S1(M) contains both of Bi,bi. Then also some aj is a point of S1(M) for j G I. Consider the graph Gm. The vertices Bi, bi are not adjacent, so from Remark 2.1(i) aj ~ Bi and j = ip(i). The unique block-vertex not adjacent to aj in Gsi(m) is Aj or Bs for some s = 4, there exists another block in Si(M), that is different from B^ Bs, Aj. We have two b-points in Si (M) so far, thus this block is a B-block. The B-vertex of Gm, that is associated with this block, must be adjacent to a^). So this block is Bj, which is already one of the blocks in S1(M) (comp. with the scheme presented in Figure 7), a contradiction. (ii) Let Bj be the unique B-block of S1 (M) for some i G I. Then the remaining blocks of S1(M) are A-blocks. In view of Lemma 3.2(i), there are n - 1 b-vertices in GSi(m): every A-vertex is associated with the b-vertex, which is not adjacent to it. For n > 4 a contradiction with Remark 2.1(i) arises: every b-vertex is adjacent to precisely one of A-vertices, and thus it is not adjacent to at least two A-vertices in GSi(m). Let S1(M) contain at least three B-blocks. Without loss of generality, assume B1, B2, B3 are blocks of S1 (M). From Lemma 3.2(i), b1, b2, b3 are not in S1(M). Thus, from Remark 2.1(i), S1(M) contains aj1, aj2, aj3 such that j = <(j) for j = 1,2,3. Every blockvertex Bj must be adjacent to at least two of the point-vertices ajj/ with j' = j. On the other hand, it is adjacent to at most one of them, what follows from Remark 2.1(ii) applied to Gm. This contradiction actually completes the proof as other cases run dually. □ By Lemma 3.2 we prove a generalization of Remark 3.1. Proposition 3.3. Let M = M(n,v). The following conditions are equivalent (i) there is a special decomposition of M, (ii) n = 4 and there is X c I such that |X | = 2 and <(X) = X. Proof. (i) ^ (ii): From Lemma 3.2(ii) we get n = 4, and two B-vertices and two A-vertices in GSi(m). Let (e.g.) B1,B2 be the B-vertices of GSi(m). In view of Remark 2.1(i), there are vertices x, y in GSi(m) such that x ~ B1, y ~ B2 and x ^ B2, y ^ B1. By Lemma 3.2(i), x = b2, y = b2, and thus x = aj, y = aj where <(1) = i, <(2) = j. Then two A-vertices in GSi(m) are As, At with s,t = i, j. The remaining two point-vertices must be of the form bs, bt> with s', t' = 1,2, since they must be adjacent to both of B1, B2. On the other hand, bs>, bt/ need to be adjacent to precisely one of As, At, so {s',t'} = {s,t}. Thus s,t = 1, 2, {1,2} = {i, j} = {<(1), <(2)}, and X = {1,2} is the required set. (ii) ^ (i): Assume, without loss of generality, X = {1, 2} and consider M = M(4,v) with <(X) = X. Take blocks B1, B2, A3, A4 and points av(1), av(2), b3, b4 of M, and consider Gm. We have B1 ^ av(2), B2 ^ av(1), and B1, B2 ~ b3, b4. Similarly A3 ^ b4, A4 ^ b3, and A3,A4 ~ av(1),av(2), since <(1),<(2) G {1, 2}. Thus the Levi graph we consider is a Levi graph of a 4-simplex. It is easy to verify that A1, A2, B3, B4 and b1, b2, a3, a4 form another 4-simplex. The two obtained simplices are mutually circumscribed. Indeed, B1, b2; B2, b1; A3, a4; A4, a3, and A1, av(1) (or A1, av(2)); A2, av(2) (or A2, av(1)); B3, b4; B4, b3 are all pairs of adjacent vertices representing blocks (points) of the first simplex and points (blocks) of the second simplex in each pair. In other words, we have found a special decomposition of M. □ Due to Proposition 3.3 there is a correspondence between the special decompositions of M(n,v) and 2-subsets of I preserved by <. The correspondence is established up to complements, since the special decompositions arise only for n = 4, and thus if < preserves a 2-subset of {1, 2,3,4} then it preserves its complement as well. So, directly from Proposition 3.3 we get: 114 Ars Math. Contemp. 13 (2017) 107-123 Corollary 3.4. All (up to an isomorphism) Möbius n-pairs with a special decomposition are the following: 1. M(4 ¡a) with 3 distinct special decompositions associated with X = {1, 2}, {1, 3}, {1, 4}, 2. M(4,(i3)(24)) with the special decomposition associated with X = {1,3}, 3. M(4,(i2)(3)(4)) with the special decomposition associated with X = {1,2}. 3.2 Subpairs of Möbius n-pairs Let M = M(„,v), n > 4, k > 3, k < n, and M' be a Möbius k-pair obtained from M by deleting 2(n - k) points and 2(n - k) blocks. We call M' a k-subpair of M. The blocks of M' are subblocks of M, that is every block of M' arises as a block of M with n - k points removed. The subblocks of the A-blocks, the B-blocks are called the A-subblocks, the B-subblocks, respectively. Let S1(M'), S2(M') be two simplices of M'. For any t =1, 2, X = A, B we write St(M') ^ Sx(M) if all the points and the blocks of St(M') are points and subblocks of SX(M). Otherwise we write Sj(M') ^ SX(M). For Y c I by ( \ Y we mean the restriction of ( to the set Y. In order to determine all Mobius n-pairs with k-subpairs we need to prove some auxiliary facts. Lemma 3.5. One of the following conditions holds (i) Si(M') ^ Sa(M) and S2(M') ^ Sb (M), (ii) S2(M') ^ Sa(M) and Si(M') ^ Sb (M), (iii) Si(M') ^ Sa(M),Sb (M) and S2(M') ^ SA(M),SB (M). Moreover, if M' satisfies (iii) then there is a special decomposition of M'. Proof. Let S1(M') ^ SA(M) and S2(M') ^ SB(M). So there is an a-point or A-subblock in S2(M'). We consider only the case with an a-point, as the case with an A-subblock is symmetric. From Remark 2.1(ii) applied to Gm, and Remark 2.1(i) applied to Gm', there are at most two B-subblocks in S2(M'). Since k > 3, there is at least one A-subblock in S2 (M'). Note that a unique A-subblock, which does not contain an a-point of S1(M'), is a block of S1(M'). Thus all the points of S1(M) are in an A-subblock of S2(M'). This yields a contradiction with Remark 2.1(ii). The proof for each of the remaining cases (i.e. S2(M') ^ Sa(M) and S1(M') ^ Sb(M), S1(M') ^ Sb(M) and S2(M') ^ SA(M),or S2(M') ^ Sb(M) and S1(M') ^ SA(M))is analogous. Let M' satisfy (iii). The steps of the proof of Lemma 3.2 can be repeated for simplices of M'. As a result we get k = 4, and two A-subblocks and two B-subblocks in each of simplices of M'. Let Y c I be the set of subscripts of A-subblocks and B-subblocks in one of these simplices. From the reasoning analogous to the first part of the proof of Proposition 3.3 we get that Y is the set of all the subscripts used for labelling the points and the blocks of M', and there is a two-element set X c Y such that ( \ Y (X) = X. Therefore, in view of Proposition 3.3, there is a special decomposition of M'. □ Lemma 3.6. If the number of deleted B-blocks and the number of deleted A-blocks coincide (and equals n — k), then there is X c I such that |X | = n — k and f(X) = X. K. Petelczyc: On some generalization of the Möbius configuration 115 Proof. Assume that Bil,..., Bin_k and Aj1,..., Ajn_k are removed blocks. Consider a vertex av(is ) with s = 1,... ,n — k of Gm' , and assume av(is ) is in GSl (M') (the case with a(¿1),... ,<(in-k)} = {¿1,..., in-k}, and X = {¿1,.. .,in-k} is the set from our claim. □ Let us present a condition, which is sufficient and necessary to find a k-subpair in M(n,V). Proposition 3.7. Let M = ffl(n,v). The following conditions are equivalent (i) there is M', which is a k-subpair of M, (ii) there is X c I such that |X| = n — k and <(X) = X. Furthermore, if (ii) holds then M' = M(k,v^ (AX)). Proof. (i) ^ (ii): By Lemma 3.5 M' satisfies one of (i) - (iii) of Lemma 3.5. In cases (i) and (ii) of Lemma 3.5 the numbers of A-blocks and B-blocks deleted from M coincide and are equal to n — k. The claim follows directly from Lemma 3.6. If case (iii) of Lemma 3.5 holds, then there is a special decomposition of M', and we get our claim by Proposition 3.3. (ii) ^ (i): Without any loss of generality, let X = {1,..., n — k}. Recall that the rank of every vertex in Gm is n. Observe the Levi graph obtained from Gm by removing the vertices ai, Ai and bi, Bi for every i G X, and all edges passing through these vertices. We denote this Levi graph by H. Note that av(i) is not a vertex of H, since <(i) G X. Let j G X and take Aj. Clearly Aj is a vertex of H. There are n — k edges joining Aj with all ai in Gm. Thus, the rank of Aj in H is n — (n — k) = k. Similarly we set ranks of the remaining vertices aj,bj, Bj of H. All these ranks are k. From this and the construction of H we get that H is the Levi graph of two mutually circumscribed k-simplices, where the way they are inscribed one into another is induced by the action of < on the set I \ X. Therefore H = Gm' for some M', which is a k-subpair of M. □ 4 Isomorphisms and automorphisms 4.1 Isomorphic Mobius n-pairs Recall that the Mobius (84) configurations (i.e. Mobius 4-pairs) correspond to conjugacy classes of the permutation group S4. In this section we generalize this property to all Mobius n-pairs. Let us start with a key lemma that gives an account on isomorphisms of configurations M(n,v) with the unique decomposition into two n-simplices. 116 Ars Math. Contemp. 13 (2017) 107-123 Lemma 4.1. Let f be an isomorphism mapping M(n,v) onto ffi(n,^). Assume that either n = 4 and both = id contain no cycle of length 2, or n > 5. There is a G Sn such that f (Bi) = Ba(i) for each i G I, or f (Bi) = A^) for each i G I. (i) If f (Bi) = Ba(i) then f (bi) = ba{i), f (Ai) = Aa{i), f (ai) = aa{i) for each i G I. (ii) If f (Bi) = Aa(i) then f (bi) = aa(i), f (Ai) = B^-i(a(i)), f (ai) = b^-i(a(i)) for each i G I. Furthermore, a<^> = ^a holds in both cases: (i) and (ii). Proof. Let M1 ■= ffl(n,tp) and M2 ■= Let i,j G I and Bi be an arbitrary B- block of Mi. Clearly, either f (Bi) = Bj for some B-block Bj of M2, or f (Bi) = Aj for some A-block Aj of M2. Assume that f (Bi) = Bj. In view of Corollary 3.4, both M1, M2 are Mobius n-pairs without the special decompositions. Thus all the B-blocks of M1 are mapped onto B-blocks of M2. We introduce a map a G Sn associated with f by the formula a : i ^ j iff f (Bi) = B ji for all i,j G I. Then f (Bi) = B^. Let us analyze graphs Gmi and Gm2 : f (bi) = ba(i) as bi, ba(i) are unique b-vertices not adjacent to Bi, Ba(i) respectively in graphs Gmi , Gm2; f (Ai) = Aa^i) as Ai, Aa^i) are unique A-vertices adjacent to bi, ba(i) respectively in G mi, Gm2 ; f (ai) = aa(i) as ai, aa(i) are unique a-vertices not adjacent to Ai, Aa(i) respectively in Gmi, Gm2. °n the other hand, f (av(i)) = a^(a(i)) as av(i), a^(a(i)) are unique a-vertices adjacent to Bi, Ba(i) respectively in Gm1 , Gm2 . So aa(v(i)) = a^(a(i)) and thus a 4 and G Sn. The following conditions are equivalent: (i) = (ii) M(n,v) = ffl(nrf). Proof. Let M1 = M(n,v) and M2 = (i) ^ (ii): Let i G I, ai, bi be points and A^ Bi be blocks of M1. Consider a map f associated to the permutation a given by the formula f (xi) = xa(i) for x G {a, b}, K. Petelczyc: On some generalization of the Möbius configuration 117 which maps the points of M 1 onto the points of M2. Then f (Aj) = Aa(i) and f (Bj) = Ba(i), as the conditions ai G Aj, bj G Bi uniquely determine blocks Ai; Bi; respectively. Clearly, conditions bi G Ai and ba(i) G Aa(i) are equivalent. Note that aa(v(i)) G Ba(i) is equivalent to a^,^)) G Ba(i) as well, since ay = ^a. Thus f is the required isomorphism. (ii) ^ (i): We restrict ourselves to n > 5 since for n = 4 this fact is well known, as it was mentioned at the beginning of this section. Let f be an isomorphism mapping M 1 onto M2. By Lemma 4.1, there is a G Sn associated with f such that ay = ^a. □ According to Theorem 4.2, the number of non-isomorphic configurations M(n,v) is equal to the number of partitions p(n) of a positive integer n. There is the generating function, recursive formula, asymptotic formula, and direct formula for p(n) (cf. [1]). The increase of n implies quick growth of p(n): p(5) = 7,p(6) = 11,... ,p(100) = 190569292,... ,p(1000) = 24061467864032622473692149727991. 4.2 The automorphism group structure of a Mobius n-pair For n = 3 the structure M(n,v) consists of two mutually inscribed triangles. From [8] the automorphism group of M(3,v) is isomorphic to S3 x C2. From the original paper of Mobius [7] the automorphism group of M(4,id) has order 192. The Mobius configuration is also a particular case of the Cox configuration. Recall the definition of the Cox configuration (comp. [3]). Let X be a set with n elements. The incidence structure (Cx)x = (Cx)n = (U {^2fc+i(X) : 0 < k < n}, [p2k (X): 0 < k < n}, CUd) is the (2n-1n) configuration, which is called the Cox configuration. Since the automorphism group of (Cx)n is established in [11] and M(4,id) = (Cx)4 (see Figure 8), we get the following: Fact 4.3. The automorphism group of M(4,id) is isomorphic to S4 x C3. It follows from Theorem 4.2 that the centralizer of y in Sn consists of automorphisms of M(n,v) for any n. Nevertheless, we will give a detailed characterization of automorphism group of M(4,v) with y = id, and of M(n,v) with n > 5. Let M = M(n,v) and 1 < v1 < ... < vr be the lengths of the cycles which are contained in the cycle decomposition of y G Sn. Assume that there are mt cycles of length vt, so n = J2r=1 mtvt. In other words Vl Vl Vl V2 V2 V2 Vr Vr V, y = y 11 y2 • • • y;yi2y22 ••.ym22...yiry2r -y;,, where yk4 is a cycle of length vt for k < mt, t < r. In view of Theorem 4.2 we can assume, that each cycle consists of consecutive natural numbers. If we set Mk := J2i-1 mi vi + (k -1)vt + 1 then yk4: Mfc ^ Mfc + 1 ^ Mfc + 2 ^ ... ^ Mfc + (V - 1) ^ Mfc, and the effective domain of yk4 is the set X^4 := {^k, Mk + 1, • • •, Mk + (v - 1)} C I. Taking all the domains of all the cycles we obtain the family of pairwise disjoint sets XV1,..., X;1, XV2,..., X;2,..., X Vr,..., X;r that yields a covering of I. Thus for any cycle yk4 we have y£ (X^4) = X^4 and yk4 fA X= id. 118 Ars Math. Contemp. 13 (2017) 107-123 {1,3,4} {2,3,4} Figure 8: The Möbius configuration as (Cx)4. The points and the blocks of M can be identified with the sequences (t, k, i, e) such that t < r, k < mt, i = 0,..., vt — 1, and e G {1,2, —1, -2} according to the formula: (t, k, i, e) = Ai+ßi for e = 1, for e = —1, for e = 2, Bi+Mt for e = —2. (4.1) Let vt = K ,...ymt) G Cm, at G Smt, and v = (vl,...,vr) G Xrt=1Cm, a = (ai,..., ar) G X r=1Smt. With the pair (v, a) we associate the map f(va) as follows: f(v,a)((t,k,i,e)) = (t,at(k),i + v\ mod vt,e). In like manner we define the map g(v,a) by: )((t, k, i, e)) (t, at(k), i + vjt — 1 mod vt, —e) for e = 1, 2, (t, at(k), i + vt mod vt, —e) for e = —1, —2. (4.2) (4.3) Lemma 4.4. The map f(v,a) is an automorphism of M, which preserves each ofsimplices Sa, SB . Proof. It follows directly from (4.2), that f(v,a) maps SA onto SA, and f(v,a) maps SB onto SB. Let i G and j G I. Assume that bj G Bj. By (4.1), Bj = (t, k, i0, —2) for some i0 G {0,..., vt — 1}, and bj = (t', k', j0, —1) for some t' < r, k' < mt,, j0 G {0,..., vv — 1}. Then f (Bj) = (t, at(k),io + v^t( t) mod vt, —2) and f (bj) = (t', a^(k'),jo + v Set i' (io + v at( k) mod vt) + k) and j' ( jo + vt at, (k') mod vt', —2). t' (k') mod vf) + (k')>so K. Petelczyc: On some generalization of the Möbius configuration 119 f (Bj) = Bv and f (bj) = Bj,. Recall that bj e Bi iff j = i. If j' = i' then: firstly t' = t, next at(k') = at(k) and thus k' = k, and finally jo = io. It means that j = i, which yields a contradiction. Hence f (bj) e f (Bi). Let aj e Bi. Then j = y(i). We have a^) = (t, k, i0 + 1 mod vt, 1), so f (av(i)) = (t, k, io + 1 + vk mod vt, 1) = a^,). Therefore f (avW) e f (Bj). The incidence (membership) relation is preserved by f(v,a) in case aj e Ai and in case bj e Ai as well, that can be easily proved by similar reasoning. □ Let vt = (v,..., v) for all t < r, and v = (vi,..., vr). Let us put g0 := g(0,id). t Lemma 4.5. The map g0 is an automorphism of M, which interchanges simplices SA, SB. Proof. Immediately from (4.2), g0 maps SA onto SB, and SB onto SA. We restrict our proof to the incidence relation involving the B-blocks, as the case with the A-blocks runs similarly. Let i e X^. From (4.1) Bi is represented by the sequence (t, k, i0, -2) for some i0 e {0,..., vt - 1}. The points that belongs to Bi are bj with j e I \ {i} and av(i). Clearly, g0(bj) = aj e Ai = g0(Bi). We have av(i) = (t, k, i0 + 1 mod vt, 1) and thus 00(av(j)) = (t, k, i0, -1) = bj. Then finally g0(av(j)) e g0(Bj). □ Since g(v,a) = g0f(v,a), from Lemma 4.4 and Lemma 4.5 we infer that: Corollary 4.6. The map g(v,a) is an automorphism of M, which interchanges simplices Sa, SB . We write Mk4 for the set of all the points and the blocks of M labelled by the elements of the set , and MVt = {Mk : k < mt}. Lemma 4.7. Let f be an automorphism of M, which (1) maps the B-blocks onto the B-blocks, or (2) maps the B-blocks onto the A-blocks. There is v e X^C^4 and a e Xr=1Smt such that (i) f = f(v,a) in case (1), or (ii) f = g(v,a) in case (2). In particular, for each k < mt there is k' < mt such that f (Mk4) = Mk,. Proof. (i): Let i e X^f. Assume that f (Bi) = Bj for some j e I. According to (4.1) there is i0 e {0,..., vt - 1} such that Bi = (t, k, i0, -2), and j0 e {0,..., vt, - 1} such that Bj = (t', k', j0, -2) for some t' < r, k' < mt,. Then, by Lemma 4.1(ii) we get f ((t, k, i0,e)) = (t', k',j0,e) for each value of e. The unique B-block containing ai = (t, k, i0,1) is Bv-i(i) = (t, k, i0 - 1 mod vt, -2), and a unique B-block containing aj is Bv-i(j) = (t', k, j0 - 1 mod vt,, -2). Hence, f maps (t, k, i0 - 1 mod vt, -2) onto (t', k', j0 - 1 mod vt,, -2), and f maps (t, k, i0 - 1 mod vt,e) onto (t',k', j0 -1 mod vt,, e) generally. By induction we get f : (t, k, ¿o — u mod vt, e) ^ (t', k', jo — u mod vt/, e) for all u = 0,..., vt — 1. 120 Ars Math. Contemp. 13 (2017) 107-123 This characterizes the action of f on M^, in particular, f (M^) C M^;'. Conversely, f-1 maps Bj onto Bj. By the reasoning, analogous to this, which has been already done, we come to f-i(Mukt') C M^. Consequently, f (M^) = M^', and therefore t' = t since f is a bijection. It provides that f preserves the set M Vt. We define the map a e Smt associated with f \ mvt by the formula a: k ^ k' iff f (Mkt ) = MJ£, for all k, k' < mt. Set = j0 — i0 mod vt. Finally the formula for f is the following: f: (t, k, i, e) ^ (t, a(k), i + v\ mod vt, e) for all i = 0,..., vt — 1. (ii): Based on Lemma 4.5, g0f is an automorphism of M, which maps the B-blocks onto the B-blocks. Then, from Lemma 4.7(i), g0f = f(v,a) for some v e X^C^ and a e Xr=1Smt, and thus f = g-1 f(v,a). Note that g-1 = g(1jid). Consequently, f = g(i,id)f(v,a) = gof(v+i,a) = g(v+i,a). What is more, f preserves the set MVt, that follows directly from (4.3). □ Now we characterize automorphisms of M(„iV), which can be uniquely decomposed into two mutually inscribed n-simplices. Theorem 4.8. Let M = M(n,v) and 1 < v1 < ... < vr be the lengths of the cycles in the cycle decomposition of p e Sn. Assume that either n = 4 and p = id contains no cycle of length 2, or n > 5. Then Aut(M) = 0^= (C^i x S„iz). Proof. Let F be an automorphism of M. By Proposition 3.3, there is no special decomposition of M. Thus, F either interchanges SA(M) with SB (M) or preserves each of them. According to Lemma 4.7 there is v0 e X^C^ and a0 e Xr=1Smt such that F = f(vo,ao) or F = g(vo,ao) = gof(vo,ao). Furthermore, every f(v,a), g(v,a) with v e X^C^ and a e Xr=1Smt is an automorphism of M by Lemma 4.4 and Corollary 4.6. Since, by Lemma 4.7, F preserves each of the sets MVt, we can restrict the proof to the one fixed set MVt. Thus, we assume that i = 0,..., vt — 1, k < mt. For the simplicity of the notation, we will write (a(k), i + va(k), e) instead of (t, at(k),i + v^t(k) mod vt,e). Moreover, we identify f(v,a) with f(v,a) f m^ , and g(v,a) with g(v,a) f mv , so we assume v e C^, a e Smt. Let w e C^, ft e Smt and note that t t t t f(w,j3)f(v,a)((k, i, e)) = f(w,^)((a(k), i + vk,e)) = (^a(k),i + vk + wa(k),e). Let 4>a : Smt —> Aut(Cm) be the map defined by ^a : (vi,..., vmt) ^ (v a(1), . . . , va Then the formula for the composition of /(v,a) and /(w„s) is It is not difficult to check that go and /(v,a) commute. Note also that i /(-f ,id) if z is even, I g(- ti ,id) if z isodd. K. Petelczyc: On some generalization of the Möbius configuration 121 Let k' < mt'. We introduce the family of maps g0k ((k',i,e)) = ffo((k',i,e)) if k' = k, (k', i,e) otherwise. Then the following equalities hold {got : z = 0,..., 2vt - 1 and z is even} = {g§, : z = 0,..., 2vt - 1 and z is odd} = {/(v,id) : vfc' =0 for k' = k}, {g(v,id) : vfc' = 0 for k' = k}. Therefore, for each v g Cm we have /(v,id) = go1 go2 .. .gcC^ ' where all numbers zk = 0,..., 2vt - 1 are even. Likewise g(v,id) = gOl gO2 ... gem*, where all numbers zk are odd. Hence, for each F g Aut(M) there is v g C™4 and a g Smt such that F = /(v,id)/(o,a) or F = g(v,id)/(o,a). To complete the proof it suffices to determine the remaining compositions: /(v,id)/( /(0,a)/(0,ß) /(v,id)/(0,a) g(v,id)f(0,a) = g0/(v,id)/(0,a) /(v,id)g(w,id) = /(v,id)g0/( g(v, id)g(w,id) = g0/(v,id)g0/( y"(w+v,id), y"(0,ßa), /"(^(v^a^ g0/(0a (v),a) = g(0a(v),a^ g0/(v,id)/(w,id) = g0/(w+v,id) = g( w+v,id), g0/(w+v,id) = /(-1,id)/(w+v,id) □ The Mobius n-pairs, which automorphism groups are not characterized by Theorem 4.8, admit a special decomposition. We say that an automorphism f of a Mobius n-pair M yields a special decomposition of M if f maps the pair {SA, SB} onto a distinct pair of mutually inscribed simplices. Theorem 4.9. The automorphism group of M(4jV) is isomorphic to (i) (C4 © S2) x C2 if ^ G S4 contains precisely one cycle of length 2, (ii) (C2 x S2) x C2 if ^ G S4 contains two cycles of length 2. Proof. In view of Theorem 4.2, without loss of generality we can consider M i = M(4,V1) with y>i = (1)(2)(34) in case (i), and M2 = M(4jV2) with ^2 = (12)(34) in case'(ii) (comp. Figures 4, 5). Let Fs G Aut(Ms) for s = 1,2. By Corollary 3.4, there is the special decomposition of each of Ms. Thus, Fs maps the pair {SA, SB } onto {SA,SB} or Fs yields the special decomposition of Ms. In case Fs maps the pair {SA,SB} onto {SA, SB}, by Lemma 4.7, there is v0 G {0} x {0} x C2, a0 G S2 x {id} for M1, or vo G C2 x C2, ao G S2 for M2 such that Fs = f(vo,ao) or Fs = g(vo,ao) = gof(vo,ao), respectively for s = 1, 2. By Lemma 4.4 and Corollary 4.6 all maps Fsf(v,a), Fsg(v,a), where v G {0} x {0} x C2 and a G S2 x {id} if s = 1, or v G C2 x C2, a G S2 if s = 2, are automorphisms of Ms preserving the pair {SA, SB}. Based on the proof of Theorem 4.8, these maps form the group C4 © S2 if s = 1, and the group x S2 if s = 2. 122 Ars Math. Contemp. 13 (2017) 107-123 Consider the following two transformations: x a 1 a2 a3 a4 b 1 b2 b3 b4 Z(x) bi b2 a4 a3 a a2 b3 b4 fx) a2 a b3 b4 b b2 a3 a4 The map f is an automorphism, which yields a special decomposition of Mi; and f is an automorphism, which yields a special decomposition of M2. Assume that Fs yields a special decomposition of Ms. Then F1 = /Fi and F2 = f F2', where Fs' is the automorphism of Ms given by (4.2) or (4.3). Let us set the commutativity rules in the automorphism group of Ms. By (4.1), the points of M1, M2 correspond to the sequences (t, i, k, e) with e = 1, -1. Using the convention introduced at the beginning of this paragraph we get t = 1,2, v1 = 1, v2 =2, m1 =2, m2 = 1 and X1 = {1}, X| = {2}, Xx2 = {3,4} for M1; t =1, v1 = 2, m1 = 2, and X2 = {1,2}, X2 = {3,4} for M2. To avoid any misunderstanding, in case M2 we will write Y2, Y22 instead of X2, X2 respectively. Then f maps the points of M1 by the formula: /((t,k,i,e)) = (t, k, i, —e) (t, k, i + 1 mod 2, e) (t, k, i, e) for i + pi G XJ^1, for e = 1, i + pi G X 2, for e = —1, i + pi G X 2. (4.4) The map f can be defined on points of M2 as: f ((t, k,i,e)) = ^ (t,k, i,e) (t, k, i, —e) (t, k, i +1 mod 2, e) for e = 1, i + pi G Y2, for e = —1, i + pi G Y2, for i + pi G Y2. (4.5) Note, that /2 = f2 = id. Hence the cyclic group generated by f and the cyclic group generated by f both coincide with C2. All the formulas for compositions of f with g0, and f with f = f(v,a) can be calculated using (4.4) and (4.5) (it is rather technical and thus omitted) and then we get ff = f/and Î90 = /g(0,id) = g(T(o,o,i)(0),id)L where T(0,0,i )(v) = T(0,0,i )((v J, vJ, v2)) = (vJ, vJ, v2 + 1). Analogous calculation can be done for f. 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