The Universality of the Resonance Arrangement and Its Betti Numbers

The resonance arrangement An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}_n$$\end{document} is the arrangement of hyperplanes which has all non-zero 0/1-vectors in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} as normal vectors. It is the adjoint of the Braid arrangement and is also called the all-subsets arrangement. The first result of this article shows that any rational hyperplane arrangement is the minor of some large enough resonance arrangement. Its chambers appear as regions of polynomiality in algebraic geometry, as generalized retarded functions in mathematical physics and as maximal unbalanced families that have applications in economics. One way to compute the number of chambers of any real arrangement is through the coefficients of its characteristic polynomial which are called Betti numbers. We show that the Betti numbers of the resonance arrangement are determined by a fixed combination of Stirling numbers of the second kind. Lastly, we develop exact formulas for the first two non-trivial Betti numbers of the resonance arrangement.

The term resonance arrangement was coined by Shadrin, Shapiro, and Vainshtein in their study of double Hurwitz numbers stemming from algebraic geometry [SSV08]. Billera, Billey, Rhoades, and Tewari proved that the product of the defining linear equations of A n is Schur positive via a so-called Chern phletysm from representation theory [BBT18,BRT19].
Recently, Gutekunst, Mészáros, and Petersen established a connection between the resonance arrangement and the type A root polytope [GMP19].
The arrangement A n is also the adjoint of the braid arrangement [AM17, Section 6.3.12]. It was studied under this name by Liu, Norledge, and Ocneanu in its relation to mathematical physics [LNO19]. The relevance of the resonance arrangement in physics was also demonstrated by Early in his work on so-called plates, cf. [Ear17].
In earlier work, the arrangement A n was called (restricted) all-subsets arrangement by Kamiya, Takemura, and Terao who established its relevance for applications in psychometrics and economics [KTT11,KTT12].
A first contribution of this article is a universality result of the resonance arrangement for rational hyperplane arrangements: Theorem 1.2. Let B be any hyperplane arrangement defined over Q. Then B is a minor of A n for some large enough n, that is B arises from A n after a suitable sequence of restriction and contraction steps. Equivalently, any matroid that is representable over Q is a minor of the matroid underlying A n for some large enough n.
The proof is constructive and the size of the required A n depends on the size of the entries in an integral representation of B.
1.2. Chambers of A n . The chambers of A n are the connected components of the complement of the hyperplanes in A n within R n . We denote by R n the number of chambers of the arrangement A n . The arrangement A 3 for instance has 32 chambers as shown in Figure 1.
These chambers appear in various contexts, such as quantum field theory where these regions correspond to generalized retarded functions [Eva95]. Cavalieri, Johnson, and Markwig proved that the chambers of A n are the domains of polynomiality of the double Hurwitz number [CJM11]. Subsequently, Gendron and Tahari demonstrated the significance of the chambers of the resonance arrangement in geometric topology [GT20].
Billera, Tatch Moore, Dufort Moraites, Wang, and Williams observed that the chambers of A n are also in bijection with maximal unbalanced families of order n + 1. These are systems of subsets of [n + 1] that are maximal under inclusion such that no convex combination of their characteristic functions is constant [BTD + 12]. Equivalently, the convex hull of their characteristic functions viewed in the n + 1-dimensional hypercube does not meet the main diagonal. Such families were independently studied by Björner as positive sum systems [Bjö15].
The values of R n are only known for n ≤ 8 and are given in Table 1, cf. also [Slo,A034997]. There is no exact formula known for R n . The work of Odlyzko and Zuev [Odl88,Zue92] together with the recent one by Gutekunst, Mészáros, and Petersen [GMP19] gives the bounds (1) n 2 − 10n 2 / ln(n) − n + log 2 (n + 1) < log 2 (R n ) < n 2 − 1, which in turn yields the asymptotic behavior log 2 (R n ) ∼ n 2 . Deza, Pournin, and Rakotonarivo obtained the improved upper bound of log 2 (R n ) < n 2 − 3n + 2 + log 2 (2n + 8) [DPR]. Due to a theorem of Zaslavsky the number of chambers of any arrangement over R equals the sum of all Betti numbers of the arrangement [Zas75]. The Betti numbers can be defined via the characteristic polynomial of an arrangement: Definition 1.3. For any arrangement of hyperplanes A in F n for any field F its characteristic polynomial χ(A; t) is defined to be where for any subset S ⊆ A we set r(S) := codim ∩ H∈S H. The absolute value of the coefficient of t n−i in the characteristic polynomial χ(A; t) is called i-th Betti number. One always has b 0 (A) = 1 and b 1 (A) = |A|.
In the case of a complex arrangement of hyperplanes, the Betti numbers coincide with the topological Betti numbers of the complement of the arrangement C n \ (∪ H∈A H) with coefficients in Q, cf. [OT92,Chapter 5] for an overview of the topological study of arrangement complements.
A formula for χ(A n ; t) would also yield a formula for R n . Unfortunately, there is also no such formula known for χ(A n ; t). In fact, the polynomial χ(A n ; t) itself is only known for n ≤ 7 as computed in [KTT11].
The next result of this article proves that the Betti numbers b i (A n ) for any fixed i > 0 can be computed for all n > 0 from a fixed finite combination of Stirling numbers of the second kind S(n, k) which count the number of partitions of n labeled objects into k non-empty blocks. The proof is based on Brylawski's broken circuit complex [Bry77].
Theorem 1.4. There exist some positive integers c i,k for all i ≥ 0 and i + 1 ≤ k ≤ 2 i such that for all n ≥ 1, The first two trivial cases of this theorem are b 0 (A n ) = S(n + 1, 1), b 1 (A n ) = S(n + 1, 2).
One can obtain exact formulas for the higher Betti numbers b i (A n ) from Theorem 1.4 if one knows b i (A n ) for all 1 ≤ n ≤ 2 i since the matrix of Stirling numbers (S(n, k)) n,k=1,...,2 i is invertible. Unfortunately, this already fails for b 3 (A n ) since χ(A n ; t) is only known for n ≤ 7.
Combining the upper bound on the constants c i,k given in Theorem 1.4 with the formula for the Stirling numbers given in (5) yields the upper bound b i (A n ) < 2 in i! for i, n ≥ 1. Summing up these bounds for i = 0, 1, . . . , n we obtain for n > 1 log 2 (R n ) < n 2 − n + 1.
Analyzing the triangles in the broken circuit in detail we obtain exact formulas for the first two non-trivial coefficients of χ(A n , t), namely b 2 (A n ) and b 3 (A n ), in terms of Stirling numbers of the second kind. That is, we determine the exact constants c 2,k and c 3,k for all relevant k. The resulting values of b 2 (A n ) and b 3 (A n ) are displayed in Table 1.
This article is organized as follows. After reviewing necessary definitions of matroids and their minors in Section 2 we will prove Theorem 1.2 in Section 3. Subsequently, we state the necessary facts on broken circuit complexes in Section 4 and prove Theorem 1.4 in Section 5. Lastly, we give the proof of Theorem 1.5 in Sections 6 and 7. ACKNOWLEDGMENTS I would like to thank Karim Adiprasito for his mentorship and for introducing me to the topic of resonance arrangements. Furthermore, I am grateful to Louis Billera, Michael Joswig, and José Alejandro Samper for helpful conversations and feedback on earlier version of this manuscript. Last but not least, I am indebted to the graphics department of the Max Planck Institute for Mathematics in the Sciences for helping me to create Figure 1.

MATROIDS AND THEIR MINORS
In this section we review some basics of matroids and their minors. Details can be found in [Oxl11].
Definition 2.1. A matroid M is a pair (E, I) where E is a finite ground set and I is a non-empty family of subsets of E, called independent sets such that Given some set finite set E and an r × E-matrix A with entries in some field F we obtain a matroid M (A) on the ground set E whose independent sets are the columns of A that are linear independent. A matroid M is called representable over a field F if there exists an r × E-matrix A such that M = M (A).
An arrangement of hyperplanes A also gives rise to a matroid by writing the coefficients of a linear equation for each H ∈ A as columns in a matrix and applying the above construction. Similarly, we also get a matroid M (A) underlying an arrangement A with ground set A whose independent set are precisely those whose hyperplanes intersect with codimension equal to the cardinality of the subset.  Minors play a central role in the theory of matroids. For instance, Geelen, Gerards and Whittle announced a proof of Rota's conjecture which asserts that matroid representability over a finite field can be characterized by a finite list of excluded minors [GGW14].
The restriction of a representable matroid to some subset S is again representable by the same matrix after removing the columns that are not in S. The following lemma establishes a similar connection for contractions of representable matroids. This also motivates the term minor of a matroid as it corresponds to a minor of a matrix in the representable case.

UNIVERSALITY OF THE RESONANCE ARRANGEMENT
Let M be a matroid of rank r and size n that is representable over Q. Thus after scaling, we can assume that there is a r × n matrix A with entries in Z that represents M . Let a 1 , . . . , a n ∈ Z r be the column vectors of the matrix A. Expressing each vector a i for 1 ≤ i ≤ n as a sum of positive and negative characteristic vectors yields We work in the extended vector space for some appropriate N ∈ N. Hence, the vectors a 1 , . . . , a n naturally live in the first factor Q r of Q N . We fix the standard basis of Q N as Now, we describe a construction which will be used in the proof in Theorem 1.2. To this end, we define 0/1-vectors v 1 , . . . , v n which will eventually represent the matroid M after contracting several other 0/1-vectors. We define for each 1 ≤ i ≤ n: Example 3.1. Consider the vectors a 1 := (1, −2, −1) T and a 2 := (−1, 0, −1) T in Z 3 . They can be expressed as Thus, m − 1 = 2, m + 1 = 1, m − 2 = 1, and m + 2 = 0. The above construction yields the following column vectors in Q 8 depicted in the left matrix below. The matrix on the right arises from the one on the left after suitable row operations as described below in the proof of Theorem 1.
All columns apart from v 1 , v 2 became standard basis vectors and removing those columns together with all rows apart from the first three yields the matrix with columns a 1 , a 2 .
Proof of Theorem 1.2. Assembling the vectors in R and V to a matrix yields: Now, we perform row operations on the matrix in (3) to ensure that all columns corresponding to vectors in R are standard basis vectors. To this end, we apply the following steps for all 1 ≤ i ≤ n: (a) We pivot on the entry in row e i,− k and column r i,− k for each 1 ≤ k ≤ m − i . (b) Lastly, we pivot on the entry in row e i,s j and column r i,s j for each 1 ≤ j ≤ m + i and each s ∈ {+, ++}. By construction and Equation (2), this procedure yields the following matrix: Therefore, we obtain the matrix A from the one given in Equation (4) by removing all columns corresponding to vectors in R and all rows apart from the first r ones. Hence, Lemma 2.3 implies that the matroid M equals the matroid of the resonance arrangement A N restricted to V ∪ R and contracted by R, that is M is a minor of the matroid of A N .

THE BROKEN CIRCUIT COMPLEX
The Stirling numbers of the second kind are denoted by S(n, k) and count the number of ways to partition n labeled objects into k nonempty unlabeled blocks. We will use the standard formula A tool to compute the Betti numbers of an arrangement is the broken circuit complex: Definition 4.1. Let A be any arrangement and fix any linear order < on its hyperplanes. A circuit of A is a minimally dependent subset. A broken circuit of A is a set C \ {H} where C is a circuit and H is its largest element (in the ordering <). The broken circuit complex BC(A) is defined by BC(A) := {T ⊂ A | T contains no broken circuit}.
Its significance lies in the following result: Theorem 4.2. [Bry77] Let A be any arrangement in a vector space F n for some field F with a fixed linear order < on its hyperplanes. Then for any 1 ≤ i ≤ n it holds that where f i is the f -vector of the broken circuit complex.
For the rest of the article we will study the broken circuit complex of the resonance arrangement A n . Each subset of I ⊆ [n] can be encoded as a binary number i∈I 2 i . This gives rise to a natural ordering of the hyperplanes in A n which we will use as to obtain its broken circuit complex. In the subsequent proofs we will identify a hyperplane H A with its defining subset A or its corresponding characteristic vector χ A if no confusion arises.

PROOF OF THEOREM 1.4
Throughout this section we use the following notation: Taking all possible intersections of the sets in an i-tuple (A 1 , . . . , A i ) of pairwise different non-empty subsets of [n] yields a partition π = {P 1 , . . . , P k } of [n+1] into k blocks with i+1 ≤ k ≤ 2 i (the block containing n + 1 exactly contains all elements of [n] which are not contained in any of the sets A j for 1 ≤ j ≤ i. We order the blocks in the partition π by their binary representation as detailed above; in particular we have n + 1 ∈ P k . We can recover the tuple (A 1 , . . . , A i ) from the partition π through a map Note that such a map is injective since the sets in the (A 1 , . . . , A i ) are assumed to be pairwise different. We call any injective map f : Conversely, given any partition π = {P 1 , . . . , P k } of [n + 1] and a (i, k)-prototype f we obtain an i-tuple (A 1 , . . . , A i ) which we denote by A f,π by setting for 1 ≤ j ≤ i where we define I f j := { ∈ [k −1] | j ∈ f ( )} for 1 ≤ j ≤ i and call these sets the building blocks of f .
In total, this construction gives a bijection between i-tuples of pairwise different nonempty subsets of [n] and pairs of (i, k)-prototypes together with partitions of [n + 1] into k blocks with i + 1 ≤ k ≤ 2 i . Now the main observation is the following. Whether an i-tuple A f,π is a broken circuit depends only on the prototype f but not on the partition π: Proposition 5.1. In the above notation, let f : [k − 1] → P([i]) \ {∅} be an (i, k)prototype. Assume there exists a partition π = {P 1 , . . . , P k } of [n + 1] such that the ituple A f,π = (A 1 , . . . , A i ) is a broken circuit of A n (in the order induced by the binary representation).
Let π = { P 1 , . . . , P k } be any partition of [ n + 1] for some n ≥ 1 into k non-empty parts. Then the i-tuple A f, π = ( A 1 , . . . , A i ) is also a broken circuit of A n .
Proof. By assumption, the tuple A f,π = (A 1 , . . . , A i ) is a broken circuit. Thus, there exists some C ⊆ [n] and λ 1 , . . . , λ i ∈ R * such that This implies that C is also a union of the first k − 1 parts of the partition π, that is there exists some I C ⊆ [k − 1] such that C = ∈I C P . Hence, we can rewrite Equation (6) as Subsequently, the fact A j < C yields I f j < I C for all 1 ≤ j ≤ i where I f j are the building blocks of the prototype f and the order is the one induced by the binary representation of subsets of [k − 1]. Now consider the partition π of [ n + 1]. Using the building block I C of C we can define a corresponding subset of [ n] by setting C := ∈I C P . Thus, Equation (7) Therefore, the tuple ( A 1 , . . . , A i , C) is a circuit of A n . Using the fact I f j < I C we obtain again A j < C for all 1 ≤ j ≤ i which completes the proof that A f, π is a broken circuit in A n .
In light of Proposition 5.1 we can subdivide prototypes into two sets. We call those which contain a broken circuit for some partition, and thus for all partitions, broken prototypes. Otherwise, we call a prototype functional.
Proof of Theorem 1.4. As explained above, any i-tuple of subsets of [n] can be obtained from an (i, k)-prototype and a partition π of [n + 1] into k blocks with i + 1 ≤ k ≤ 2 i . Theorem 4.2 then implies that we can compute the Betti number b i (A n ) for any i ≥ 0 through functional prototypes and partitions. We correct the fact that latter yields ordered tuples unlike the elements in the broken circuit complex by multiplying the Betti numbers b i (A n ) by i! in the following computation: This already proves that for each i ≥ 0 the Betti number b i (A n ) can be computed by a combination of Stirling numbers which is independent from n. This settles the first claim of the theorem.
For the second claim, note that the above argument shows for all i ≥ 1 and i + 1 ≤ k ≤ 2 i . Bounding the number of functional (i, k)-prototypes by the number of all (i, k)-prototypes which are merely injective functions f : Remark 5.2. The above upper bound on c 2,2 2 and c 3,2 3 actually agrees with the actual value of these constants given in Theorem 1.5 (3 and 840). It can be shown that the given bound on c i,2 i is attained for all i ≥ 1, that is all (i, k)-prototypes are functional. For c i,k with i ≥ 1 and k < 2 i the upper bound is not tight in general.

THE BETTI NUMBER b 2 (A n )
We compute b 2 (A n ) using Theorem 4.2.
Proposition 6.1. For all n ≥ 1 it holds that This case corresponds to a partition of [n + 1] into four nontrivial blocks P 1 , P 2 , P 3 , P 4 where we assume that n + 1 ∈ P 4 . Subsequently, we can choose any P i with 1 ≤ i ≤ 3 to be the intersection and set A := P j ∪ P i and B := P k ∪ P i where {j, k} := {1, 2, 3} \ {i}. Thus, there are 3S(n + 1, 4) many possibilities of that type. Now assume A ⊆ B. The subsets of the form {H A , H B } with A ⊆ B corresponds to a partition of [n + 1] into three nontrivial blocks P 1 , P 2 , P 3 where we again assume n+1 ∈ P 3 . In this situation we have the two families {H P 1 , H P 1 ∪P 2 } and {H P 2 , H P 1 ∪P 2 } which yields 2S(n + 1, 3) possibilities in total of that type.
Remark 6.2. In the language of the previous section, the above proof implies that all three (2, 4)-prototypes are functional whereas only two of the three (2, 3)-prototypes are functional.

THE BETTI NUMBER b 3 (A n )
To compute b 3 (A n ) we again use the broken circuit complex with the ordering induced by the encoding in binary numbers. Hence, we need to understand which families {H A , H B , H C } form a broken circuit of A n where A, B, C are subsets of [n] that are pairwise not disjoint. We use the following result due to Jovovic and Kilibarda: Theorem 7.1 ( [JK99]). For any n ≥ 1, the number of families {A, B, C} where A, B, C are subsets of [n] that are pairwise not disjoint is 1 3! (8 n − 3 · 6 n + 3 · 5 n − 4 · 4 n + 3 · 3 n + 2 · 2 n − 2).
We call such families pairwise intersecting.
As a first step we will classify the circuits of A n of cardinality four. To determine the broken circuits it suffices to consider circuits whose first three elements in the ordering < are pairwise intersecting. Otherwise, the edges between these elements are already broken circuits and therefore not part of BC(A n ).
Definition 7.2. We call a circuit in A n relevant if the corresponding subsets of [n] which are not maximal in the circuit are pairwise intersecting.
Proposition 7.3. For n ≥ 1, a four element family in A n is a relevant circuit if and only if it is one of the following types for subsets A 1 , A 3 , X ⊆ [n] such that In each case, we assume that the last element in each set is the largest with respect to the ordering <.
Before proving this proposition, we give examples for each such type of circuit of cardinality four. Proof of Proposition 7.3. Generalizing the relations given in Example 7.4 to arbitrary sets A 1 , A 3 , X satisfying the conditions in Equation ( ) shows that these given families are indeed families of four different subsets of [n] which form relevant circuits in A n . Conversely, let {A 1 , . . . , A 4 } be a family of subsets corresponding to a relevant circuit in A n with A i = A j for any i = j, A i ∩ A j = ∅ for 1 ≤ i, j ≤ 3 and A 4 is the maximal element in the ordering <. Since the hyperplanes form a circuit in A n there is a relation 4 i=1 λ i χ A i = 0 for some λ i ∈ Z for 1 ≤ i ≤ 4. The coefficients λ i need to be non-zero since the circuit would otherwise satisfy a dependency of cardinality less than four.
Case 1: λ 1 , λ 2 , λ 3 > 0 and λ 4 < 0: In this case, the relation implies A 1 ∪ A 2 ∪ A 3 = A 4 . Since the sets A 1 , A 2 , A 3 are by assumption pairwise intersecting every element in A 4 is contained in at least two of the sets A 1 , A 2 , A 3 . Not all elements of A 4 appear in all of the sets A 1 , A 2 , A 3 since otherwise these four sets would all be equal. Hence, the relation then implies that every element in A 4 is contained in exactly two of the sets A 1 , A 2 , A 3 which means that we can without loss of generality assume A 2 = A 1 A 3 . Therefore, the family is a circuit of type (i). Case 2: λ 1 , λ 3 > 0 and λ 2 , λ 4 < 0: Analogously to the first case, the relation now yields Hence, the maximality of A 4 yields A 1 ⊆ A 3 and A 1 ⊇ A 3 . Thus, the elements in A 1 ∪ A 3 are partitioned into the three blocks A 1 \ A 3 , A 3 \ A 1 and A 1 ∩ A 3 appearing with positive coefficients λ 1 , λ 3 and λ 1 + λ 3 respectively in the relation. Assume there is an element a ∈ (A 1 ∪ A 3 ) \ A 2 . Then, a ∈ A 4 which implies λ 4 = λ 1 + λ 3 since a ∈ A 2 . This yields A 4 ⊆ A 1 ∩ A 3 which contradicts the maximality of A 4 . Therefore, we must have A 1 ∩ A 3 ⊆ A 2 and it suffices to consider the following two subcases: Case 2.1: A 1 ∩ A 3 = A 2 : Then we obtain A 1 A 3 ⊆ A 4 . Since the positive coefficients in the relation are constant on the block A 1 ∩ A 3 we must have either The former case yields a circuit of type (ii) and the latter one of type (iii) as described in the statement of Proposition 7.3. Case 2.2: Since X ⊆ A 1 A 3 , the coefficient λ 2 can be at most λ 1 or λ 3 . However, the positive coefficient of the elements in A 1 ∩ A 3 is λ 1 + λ 3 . Hence, Since the positive coefficients of the elements in A 1 ∩ A 3 and A 1 A 3 are different we must have A 4 ∩ X = ∅. Therefore, A 4 = (A 1 ∪ A 3 ) \ X and the circuit is of type (iv).
Proposition 7.3 implies that all broken circuits of A n of cardinality three are of the form The former ones correspond to circuits of type (i) with the relation We call them tetrahedron circuits since they exhibit a tetrahedron if we regard the elements as vertices of the n-dimensional hypercube.
The latter broken circuits might not stem from a unique circuit of cardinality four. We can however fix a bijection between these broken circuits and the circuits of type (iii) and (iv) in Proposition 7.3. These all satisfy the relation The characteristic functions of these circuits viewed in the n-dimensional hypercube form rectangles which is why we call these circuit rectangle circuits in the following.
Using again Theorem 4.2 to determine b 3 (A n ) we will therefore start from Theorem 7.1 and subtract the number of tetrahedron and rectangle circuits which give broken circuits of cardinality three by removing the largest element in each circuit. Note that a broken circuit can not stem from a tetrahedron and rectangle circuit simultaneously since it can not satisfy a tetrahedron and a rectangle relation at the same time.
Proposition 7.5. For any n ≥ 1 there are S(n + 1, 4) tetrahedron circuits in A n .
We claim that the hyperplanes corresponding to A 1 , . . . , A 4 form a tetrahedron circuit in A n . By definition we have P k = A i ∩ A j for any possible ordering {k, i, j} = {1, 2, 3} and A i ⊂ A 4 for all 1 ≤ i ≤ 3 Hence, the family A 1 , . . . , A 4 is pairwise intersecting, i.e. A i ∩ A j = ∅ for all i = j. Next, consider l ∈ A 4 such that l ∈ P i for some 1 ≤ i ≤ 3 and set {j, k} := {1, 2, 3} \ {i}. Then, we conclude that l ∈ A j , l ∈ A k and l ∈ A i which implies that A 1 , . . . , A 4 corresponds to a tetrahedron circuit.
Conversely, given the subsets A 1 , . . . , A 4 of [n] corresponding to a tetrahedron circuit with largest subset A 4 we can define a partition of [n + 1] by setting P 4 := [n + 1] \ A 4 and P i := A 4 \ A i for 1 ≤ i ≤ 3. We claim this defines a partition of [n + 1]. By definition we have P i ∩ P 4 = ∅ for all 1 ≤ i ≤ 3. The assumption of A 1 , . . . , A 4 corresponding to a tetrahedron circuit implies that every l ∈ A 4 is contained in exactly two subsets A k , A j for some 1 ≤ k < j ≤ 3. This implies that every l ∈ A 4 is contained in exactly one block P i which proves that P 1 , . . . , P 4 is a partition of [n + 1].
Since these two constructions are inverse to each other the claim follows.
To count the rectangle circuits we construct corresponding tuples which will be easier to count. Throughout the subsequent discussion we regard the indices cyclically, i.e. given any family of sets X 1 . . . X n we set X 0 := X n and X n+1 := X 1 .
Proposition 7.6. Let (A 1 , . . . , A 4 ) be a family of distinct and non-empty subsets of [n] forming a relevant rectangle circuit, i.e. χ A 1 + χ A 3 = χ A 2 + χ A 4 and A i ∩ A j = ∅ for 1 ≤ i < j ≤ 3 with maximal element A 4 . Then ,we define its midpoint as M := 4 i=1 A i and the sides of the rectangle as S i : In this case, the tuple (S 1 , . . . , S 4 , M ) satisfies (S1) S i ∩ S j = ∅ for all i = j and in particular S i = S j for all i = j, (S2) M ∩ S i = ∅ for all 1 ≤ i ≤ 4, (S3) M = ∅, and (S4) at most one of two opposite sides are empty. We will call a tuple (S 1 , . . . , S 4 , M ) satisfying (S1) to (S4) a side-midpoint tuple.
Example 7.7. Figure 2 depicts the general case of a rectangle circuit together with its corresponding side-midpoint tuples as defined in Proposition 7.6 and two examples in A 5 . Proof of Proposition 7.6. To prove (S1) assume for a contradiction a ∈ S i ∩ S j . Without loss of generality we can assume a ∈ S 1 ∩ S 2 . By definition this yields a ∈ A 1 , A 2 , A 3 but a ∈ M . Thus a ∈ A 4 . This contradicts the relation χ {A 1 } + χ {A 3 } = χ {A 2 } + χ {A 4 } in the element a. Thus, S i ∩ S j = ∅ for all i = j.
The sides S i are defined as S i := (A i ∩ A i+1 ) \ M . This immediately implies property (S2) namely S i ∩ M = ∅.
Lastly, assume without loss of generality S 1 = S 3 = ∅. This implies A 1 = M ∪ S 4 ∪ A 1 for some A 1 ⊆ [n] disjoint from M and S 4 . This yields A 1 ∩ A 4 = ∅ since any intersection of these sets disjoint from M would be contained in S 4 . Hence using the fact A 1 ∪ A 3 = A 2 ∪ A 4 , we obtain A 1 ⊆ A 2 . Thus, Hence, A 1 = ∅ and A 1 = M ∪ S 4 . Analogously, we obtain A 4 = M ∪ S 4 which contradicts The next proposition shows that we can obtain a rectangle circuit from a side-midpoint tuple: Proposition 7.8. Let (S 1 , . . . , S 4 , M ) be a side-midpoint tuple. Set A i := M ∪ S i−1 ∪ S i . Then, the family (A 1 , . . . , A 4 ) corresponds to a relevant rectangular circuit which means it satisfies (C1) A i = A j for all i = j, (C2) A i = ∅ for all 1 ≤ i ≤ 4, (C3) A i ∩ A j = ∅ for all i = j and (C4) it forms a rectangle circuit, i.e. χ A 1 + χ A 3 = χ A 2 + χ A 4 .
Proof. Assume A 1 = A 2 . This implies M ∪ S 4 ∪ S 1 = M ∪ S 1 ∪ S 2 . Hence, S 4 = S 2 . By assumption (S1) these sets are disjoint which yields S 4 = S 2 = ∅. This contradicts the assumption (S 4 ) that at most one of two opposite sets is empty. Now assume A 1 = A 3 . This implies M ∪ S 4 ∪ S 1 = M ∪ S 2 ∪ S 3 . Thus, we have two partitions of the same set by pairwise disjoint sets which can not all be empty which is impossible. Thus we have without loss of generality proven (C1).
By assumption we have M = ∅. Our construction of the sets A i yields M ⊆ A i for all 1 ≤ i ≤ 4. This immediately implies A i = ∅ for all 1 ≤ i ≤ 4 and A i ∩ A j = ∅ for all i = j. Hence, properties (C2) and (C3) hold.
Lastly, we have by construction of the sets A i and due to the fact that the sets S 1 , . . . , S 4 , M are pairwise disjoint Proposition 7.9. The constructions defined in Propositions 7.6 and 7.8 are inverse to each other.
Proof. Let (A 1 , . . . , A 4 ) be the vertices of a relevant rectangle circuit satisfying (C1) to (C4). This yields by Proposition 7.6 the side-midpoint tuple with midpoint M A := 4 i=1 A i and sides (A i−1 ∩ A i ) \ M A . Fix some 1 ≤ i ≤ 4. The relation in property (C4) then implies A i ⊆ A i−1 ∪ A i+1 . Hence, we obtain A i = (A i−1 ∪ A i ) ∪ (A i ∪ A i+1 ). This yields, Thus, the vertices A i equal the resulting vertices from the construction in Proposition 7.8.
Conversely, let (S 1 , . . . , S 4 , M ) be a side-midpoint tuple. This yields by Proposition 7.8 the vertices of a rectangle circuit M ∪ S i−1 ∪ S i for 1 ≤ i ≤ 4. Since the sets S 1 , . . . , S 4 , M are pairwise disjoint the construction of Proposition 7.6 applied to these vertices yields the side-midpoint tuple (S 1 , . . . , S 4 , M ).