On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers

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1 Séminaire Lotharingien de Combinatoire XX (2019) Article #YY, 12 pp. Proceedings of the 31 st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana) On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers arxiv: v1 [math.co] 16 Nov 2018 Alexander Lazar and Michelle L. Wachs Department of Mathematics, University of Miami Abstract. Hetyei recently introduced the homogenized Linial arrangement and showed that its number of regions is equal to a median Genocchi number, which enumerates a class of permutations called Dumont derangements. Hetyei makes use of a a wellknown formula of Zaslavsky, which gives the number of regions as the characteristic polynomial of the intersection lattice of the arrangement evaluated at 1. Here, we refine Hetyei s result by deriving combinatorial formulas for the Möbius function and characteristic polynomial of the intersection lattice in terms of a class of permutations that are similar to the Dumont permutations. Our techniques also yield a type B analog and more generally a Dowling arrangement analog. This leads to a unifying generating function formula for the characteristic polynomial of the intersection lattice of the Dowling analog of the homogenized Linial arrangement evaluated at 1. 1 Introduction The braid arrangement is the hyperplane arrangement in R n defined by A n 1 := {x i x j = 0 : 1 i < j n}. Note that the hyperplanes of A n 1 divide R n into n! conical regions of the form R σ := {x R n : x σ(1) < x σ(2) < < x σ(n) }, where σ is a permutation in the symmetric group S n. A classical formula of Zaslavsky [18] gives the number of regions of any real hyperplane arrangement A in terms of Möbius function of the (semi)lattice of intersections L(A) of A (ordered by reverse containment). Indeed, given any finite, ranked poset P of length n, with a minimum element ˆ0, define the characteristic polynomial of P to be χ P (t) := µ P (ˆ0, x)t n rk(x), (1.1) x P alazar@math.miami.edu. Supported in part by NSF grant DMS ; wachs@math.miami.edu.

2 2 Lazar & Wachs where µ P (x, y) is the Möbius function of P and rk(x) is the rank of x. Zaslavsky s formula is #{regions of A} = χ L(A) ( 1). (1.2) It is well known and easy to see that the lattice of intersections of the braid arrangement A n 1 is isomorphic to the lattice Π n of partitions of the set [n] := {1, 2..., n}. It is also well known that the characteristic polynomial of Π n is given by χ Πn (t) = n s(n, k)t k 1, (1.3) k=1 where s(n, k) is the Stirling number of the first kind, which is equal to ( 1) k 1 times the number of permutations in S n with exactly k cycles; see [16, Example ]. Hence χ Πn ( 1) = S n. Therefore from (1.2), we recover the result observed above that the number of regions of A n is n!. In this extended abstract of [10], we obtain analogous results for a hyperplane arrangement introduced by Hetyei [9]. The homogenized Linial arrangement is the hyperplane arrangement in R 2n defined by H 2n 2 := {x i x j = y i 1 i < j n}. Note that by intersecting this arrangement with the subspace y 1 = y 2 = = y n = 0 one gets the braid arrangement A n 1. Similarly by intersecting H 2n 2 with the subspace y 1 = y 2 = = y n = 1 one gets the Linial arrangement in R n, {x i x j = 1 1 i < j n}. Postinikov and Stanley [12] show that the number of regions of the Linial arrangement is equal to the number of semiacyclic orientations of of the complete graph K n. (Note that the number of acyclic orientations of K n is n!, the number of the regions of the braid arrangement.) In [9] Hetyei uses the homogenized Linial arrangement to study certain orientations of K n that he calls alternation-acyclic. He shows that the regions of this arrangement are in bijection with the alternation-acyclic orientations of K n. Using the finite field method of Athanasiadis [1], Hetyei obtains a recurrence for χ L(H2n 2 )(t) and uses it to show that χ L(H2n 2 )( 1) = H 2n 1, (1.4) where H 2n 1 is a median Genocchi number. 3 Barsky and Dumont [2, Theorem 1] obtain the following generating function for the median Genocchi numbers 4, H 2n+1 x n = n 0 n 0 n k=1 3 Here we follow the standard indexing for the median Genocchi numbers. 4 This is stated in a slightly different form in [2]. n!(n + 1)!x n (1.5) (1 + k(k + 1)x).

3 Homogenized Linial 3 The median Genocchi numbers also have numerous combinatorial interpretations. One of these interpretations is given in terms of a class of permutations called Dumont permutations; see [5] and [6, Corollary 2.4]. Another is given in terms of surjective pistols in [6, Corollary 2.2]. Here, we further study the intersection lattice L n := L(H 2n ). We refine Hetyei s result (1.4) by deriving a combinatorial formula for the Möbius function of L n in terms of permutations in S 2n that are similar to Dumont permutations. A key step in our proof is to show that L n is isomorphic to the bond lattice of a certain bipartite graph. This bond lattice has a nice description as the induced subposet of the partition lattice Π 2n consisting of partitions all of whose nonsingleton blocks have odd smallest element and even largest element. Our Möbius function result yields a combinatorial formula for the characteristic polynomial of L n analogous to (1.3) with S n replaced by a class of permutations in S 2n that are similar to the Dumont permutations. By constructing a bijection between these permutations and the pistols mentioned above, we recover Hetyei s result that χ Ln ( 1) is a median Genocchi number. Moreover, we obtain the new result that the (nonmedian) Genocchi number G 2n is equal to µ Ln (ˆ0, ˆ1), where ˆ0 and ˆ1 are the minimum and maximum elements of L n, respectively. Our techniques also yield a type B analog of Hetyei s result and more generally a Dowling arrangement analog. We define the type B homogenized Linial arrangement to be the hyperplane arrangement in R 2n defined by H B 2n = {x i ± x j = y i : 1 i < j n} {x i = y i : i = 1..., n}. (1.6) We show that that the intersection lattice Ln B of H2n B is isomorphic to an induced subposet of the signed partition lattice Πn B and obtain a result for the Möbius function and characteristic polynomial analogous to that for L n. We use these results to prove the following generating function formula for the number of regions rn B of H2n B, rn B x n = n 0 n 0 n k=1 (2n)!x n (1.7) (1 + (2k)(2k + 1)x), thereby providing a type B analog of (1.5). Let ω be the primitive mth root of unity e 2πi m. For m, n 1, the Dowling arrangement Dn m is a hyperplane arrangement in C n defined by {x i ω l x j = 0 : 1 i < j n, 0 l < m} {x i = 0 : 1 i n}. (1.8) This is called a Dowling arrangement because L(D m n ) is isomorphic to the classical Dowling lattice Q n (C m ) (where C m is the cyclic group of order m), which is isomorphic to Π n+1 when m = 1, and to Π B n when m = 2. By introducing a Dowling analog of the homogenized Linial arrangement, we obtain unifying generalizations of the types A and B results discussed above.

4 4 Lazar & Wachs 2 Preliminaries 2.1 Hyperplane Arrangements Let k be a field (typically, R, C, or a finite field). A hyperplane arrangement A k n is a finite collection of affine codimension-1 subspaces of k n. The intersection poset of A is the poset L(A) of intersections of hyperplanes in A (viewed as affine subspaces of k n ), partially-ordered by reverse inclusion. If the hyperplanes have nonempty intersection then the intersection poset is a lattice. If A is a real hyperplane arrangement, then R n \A is disconnected. By the number of regions of A we mean the number of connected components of R n \A. This number can be detected solely from L(A) as Zaslavsky s formula (1.2) shows. If A is a complex hyperplane arrangement, its complement M A := C n \ H A H is a manifold whose Betti numbers can be detected solely from L(A). Indeed, this follows from the formula of Orlik and Solomon [11, Theorem 5.2], 2.2 The Bond Lattice of a Graph n β i (M A )t i = χ L(A) ( t). (2.1) i=0 Let G be a graph on vertex set [n]. The bond lattice of G is the subposet Π G of the partition lattice Π n consisting of partitions X = B 1 B k such that G Bi is connected for all i. Note that Π n is the bond lattice of the complete graph K n. Another example is given below G Π G Broken circuits provide a useful means of computing the Möbius function of the bond lattice of a graph (or more generally, of geometric lattices). We define them now. Let G = ([n], E) be a finite graph. Fix a total ordering of E and let X be a subset of E. Then X is called a broken circuit if it consists of a cycle in G with its least edge (with respect to this ordering) removed. If X does not contain a broken circuit, we say that X is a non-broken circuit or NBC set.

5 Homogenized Linial 5 Given any NBC set S, let π S be the partition of [n] whose blocks are the vertex sets of the connected components of the graph ([n], S). The following theorem is due to Whitney [17, Section 7] for graphs and Rota [14, Pg. 359] for general geometric lattices. Theorem 2.1 (Rota-Whitney). Let X Π G. Then µ(ˆ0, X) = #{NBC sets S of G : π S = X}. Given a rooted tree whose vertex set is a subset of Z +, we say the tree is increasing if each nonroot vertex is larger than its parent. A rooted forest on a subset of Z + is said to be increasing if it consists of increasing rooted trees. Note that if G is the complete graph on vertex set [n] then by ordering the edges lexicographically with the smallest element as the first component, the NBC sets of Π G are exactly the increasing forests on [n]. 2.3 Genocchi and median Genocchi Numbers The Genocchi numbers and median Genocchi numbers are classical sequences of numbers that have been extensively studied in combinatorics. There are many ways to define them. Here we define them in terms of Dumont permutations. A Dumont permutation is a permutation σ S 2n such that σ(2i) < 2i and σ(2i 1) 2i 1 for all i = 1,..., n. A Dumont derangement is a Dumont permutation without fixed points, i.e., σ(2i) < 2i and σ(2i 1) > 2i 1 for all i = 1,..., n. Example 2.2. When n = 3, the 8 Dumont derangements on [6] are: (1, 3, 5, 6, 4, 2) (1, 3, 4, 2)(5, 6) (1, 2)(3, 4)(5, 6) (1, 2)(3, 5, 6, 4) (1, 4, 3, 5, 6, 2) (1, 5, 6, 3, 4, 2) (1, 5, 6, 2)(3, 4) (1, 4, 2)(3, 5, 6) For each n 1, the Genocchi number G 2n+2 is defined to be the number of Dumont permutations on [2n] and the median Genocchi number H 2n 1 is defined to be the number of Dumont derangements on [2n]. 3 The (type A) homogenized Linial arrangement Recall that L n denotes the intersection lattice of the homogenized Linial arrangement H 2n := {x i x j = y i : 1 i < j n + 1}. In this section we give a characterization of L n as an induced subposet of Π 2n and compute its Möbius function. 3.1 The intersection lattice is a bond lattice We begin by showing that L n is the bond lattice of a very nice bipartite graph. Let Γ n be the bipartite graph on vertex set {1, 3,..., 2n 1} {2, 4,..., 2n} with an edge from 2i 1 to 2j iff i j.

6 6 Lazar & Wachs Our first step toward computing µ(ˆ0, X) for X L n is the following result: Theorem 3.1. The posets L n and Π Γn are isomorphic. In [10], we prove Theorem 3.1 by constructing an invertible Z-linear automorphism that sends H 2n to an arrangement whose intersection poset is Π Γn. Proposition 3.2. The bond lattice Π Γn is the subposet of Π 2n consisting of the partitions X = B 1 B k in which min(b i ) is odd and max(b i ) is even for all nonsingleton B i. We use the Rota-Whitney Theorem (Theorem 2.1) to compute the Möbius function of Π Γn. Our NBC sets have a nice description which we give now. By a tree (resp. forest) of a graph G = (V, E) we mean a tree (resp. forest) whose vertex set is a subset of V and whose edge set is a subset of E. Given a rooted tree T, any subset U of the vertex set of T induces a rooted forest T U on U defined as follows: for all u, v U, if u is an ancestor of v in T then u is an ancestor of v in T U. Let Γ be a bipartite graph on {1, 3, 5,..., 2n 1} {2, 4, 6,..., 2n}. A tree T of Γ is said to be doubly-increasing if, for some choice of root r, the rooted forests Γ odds and Γ evens are both increasing. A doubly-increasing forest is the union of doubly-increasing trees on disjoint vertex sets T T {1,3,5} T {2,6,8} The following lemma is a special case of a more general result about Ferrers graphs, which was obtained independently by Selig, Smith and Steingrimsson [15, Theorem 7.3] in a different context. Lemma 3.3. Order the edges of Γ n lexicographically with the odd vertex as the first component. With respect to this ordering, the NBC sets in Γ n are the edge sets of its doubly-increasing forests. Now by the Rota-Whitney Theorem (Theorem 2.1) we have the following result. Theorem 3.4. For all X Π Γn, we have that ( 1) X µ Ln (ˆ0, X) equals the number of doublyincreasing forests of Γ n whose trees have vertex sets equal to the blocks of X.

7 Homogenized Linial Dumont-like permutations Our next step is to introduce a class of permutations similar to the Dumont permutations and then give a bijection between these permutations in S 2n and the doubly-increasing forests of Γ n with vertex set [2n]. Let X be a finite subset of Z +. We say σ S X is a D-permutation on X if σ(i) i whenever i is even and σ(i) i whenever i is odd. We denote by D X the set of D- permutations on X and by DC X the set of D-cycles on X. If X = [n], we write D n and DC n. Note that all Dumont permutations are D-permutations, but not conversely. Indeed, the only difference between the two classes of permutations on S 2n is that fixed points can be even or odd in a D-permutation, while only odd fixed points are allowed in a Dumont permutation. It follows immediately from the definitions that DC 2n {Dumont derangements in S 2n } {Dumont permutations in S 2n } D 2n. Recall that the two sets in the middle of this chain are enumerated by median Genocchi numbers and Genocchi numbers, respectively. According to our next theorem the sets on the ends of the chain are also enumerated by these numbers. Theorem 3.5. For all n 1, (1) DC 2n = G 2n (2) D 2n = H 2n+1 The proofs appear in the full version of the paper [10]. The proof of (1) follows from an elementary bijection, while the proof of (2) is more difficult and relies on the theory of surjective pistols discussed in [13] and [6]. The above chain of containments and Theorem 3.5 yield G 2n H 2n 1 G 2n+2 H 2n+1. Next we define a bijection ψ X from the set T X of doubly-increasing spanning trees of Γ n X to DC X for all X [2n]. The map ψ X : T X DC X is defined recursively as follows. Clearly T {v} = {v} for all v [2n]. Let ψ {v} (v) be the cycle (v). Now, let X > 1 and let T be a doubly-increasing spanning tree of Γ n X. We root T at its least vertex r. Let T 1,..., T k be the subtrees of r, such that min(t 1 ) > min(t 2 ) > > min(t k ). Let X i be the set of vertices of T i for each i. Since each T i is doubly-increasing and has fewer verticies than T, we can apply ψ Xi. We write each cycle ψ Xi (T i ) in cycle form with the smallest element written first, and drop the parentheses to get the word w i. Next, we concatenate the words r, w 1,..., w k to get the word w. Finally, let ψ X (T) be the cycle (w).

8 8 Lazar & Wachs Example 3.6. In the tree T below, ψ X1 (T 1 ) = (7, 8), ψ X2 (T 2 ) = (5, 6), and ψ X3 (T 3 ) = (3, 4). Hence, T (1, 7, 8, 5, 6, 3, 4) (1,7,8,5,6,3,4) T Lemma 3.7. For all X [2n], the map ψ X : T X DC X is a well-defined bijection. Consequently T X = DC X. The proof appears in the full version of the paper [10]. In order to express the Möbius function of L n in terms of D-permutations, we need the notion of cycle support. The cycle support of σ S n is the partition cyc(σ) Π n whose blocks are the elements of the cycles of σ. For example, cyc((1, 7, 2, 4)(5)(6, 8, 9, 3)) = As a consequence of Theorem 3.4 and Lemma 3.7, we have the following theorem. Theorem 3.8. Let X Π Γn. Then ( 1) X 1 µ Ln (ˆ0, X) = #{σ D 2n cyc(σ) = X}. Now by Theorem 3.5, the following holds. Corollary 3.9. For all n 1, µ Ln (ˆ0, ˆ1) = DC 2n = G 2n. Finally, we have the analog of (1.3) promised in the introduction. Corollary For all n 1, χ Ln (t) = n s D (n, k)t k 1, (3.1) k=1 where ( 1) k 1 s D (n, k) is equal to the number of D-permutations in S 2n with exactly k cycles. By setting t = 1 in (3.1) and invoking Theorem 3.5 we recover Hetyei s result (1.4). We are also able to obtain the following characterization of the median Genocchi numbers by evaluating χ Ln ( 1) in another way. Theorem For all n 1, H 2n+1 is equal to the number of permutations σ on [2n] whose descents σ(i) > σ(i + 1) occur only when σ(i) is even and σ(i + 1) is odd.

9 Homogenized Linial 9 4 The homogenized Linial-Dowling arrangement In this section we extend the results of the previous section to the Dowling arrangements, which generalize the complexified types A and B braid arrangements. Let M be a finite group with identity e. An M-labeled partition X = B 0 B 1... B k is a partition of {0} [n] such that: 0 B 0 (B 0 is called the zero block). The elements of B i are labeled with elements of M for all i > 0, and min(b i ) is labeled with e. The Dowling lattice Q n (M) is defined to be the poset of M-labeled partitions of {0} [n], with the cover relation giving by a variant of the merging rule from Π n. Let X = B 0 B 1 B k. If B 0 and B i merge, erase all of the labels from B i and merge the blocks as in Π n to obtain a new zero block B 0. Suppose i, j = 0, and min(b i ) < min(b j ). When B i and B j merge, the labels of the elements of B i remain unchanged, while the labels of the elements of B j may all be multiplied by some element m M. Let C m be the cyclic group of order m generated by ω. Example 4.1. Suppose M = C 3. If W = 0 1 e 3 ω 2 e 4 ω, then we can merge the second and third blocks in three different ways to obtain X = 0 1 e 2 e 3 ω 4 ω, Y = 0 1 e 2 ω 3 ω 4 ω2 (note that the labels of 2 and 4 have both been multiplied by ω) or Z = 0 1 e 2 ω2 3 ω 4 e (note that the labels of 2 and 4 have both been multiplied by ω 2 ). It is not hard to see that for all m 1, Q n (C m ) is isomorphic to the intersection lattice of the Dowling arrangement defined in (1.8). When m = 1, Q n (C m ) is isomorphic to the partition lattice Π n+1 and when m = 2, Q n (C m ) is the type B partition lattice Π B n. See [4], [3, Section 5.3], and [8] for further information on Dowling lattices. Now we introduce a Dowling analog of the homogenized Linial arrangement. Let ω = e 2πi/m be a primitive mth root of unity. The homogenized Linial-Dowling arrangement is the complex hyperplane arrangement HQ n (m) = {x i ω l x j = y i 1 i < j n, 1 l m} {x i = y i 1 i n} C 2n. Note that when m = 2, the arrangement HQ n (m) is a complexified version of the type B homogenized Linial arrangement H2n B defined in the introduction. The proof of the following result is similar to that of the type A version except that we use a group-labeled graph in place of the graph Γ n.

10 10 Lazar & Wachs Theorem 4.2. For all n, m 1, the intersection lattice L m n := L(HQ n (m)) is isomorphic to the subposet of Q 2n 1 (C m ) consisting of all C m -labeled partitions such that for nonsingleton B 0, min(b 0 \ {0}) is odd, for all nonsingleton B i, with i > 0, min(b i ) is odd and max(b i ) is even. It follows from Theorems 3.1 and 4.2 and Proposition 3.2 that L m n is isomorphic to L n when m = 1. To compute the Möbius function of the intersection lattice L m n, we use the Rota- Whitney Theorem (Theorem 2.1) and then we construct a bijection from the NBC sets of the geometric lattice L m n to the class of C m -labeled D-permutations, which we now define. Assume m 2. Let h be some fixed nonidentity element of C m. A C m -labeled D-permutation is a D-permutation σ whose entries are labeled with elements of C m such that even fixed points of σ are labeled with e, cycle minima of σ other than the even fixed points are labeled with either e or h. We write D m n for the set of C m -labeled D-permutations on [n]. Let σ D m n. The cycle support of σ is the C m -labeled partition cyc(σ) = B 0 B k Q n (C m ) obtained from σ as follows: B 0 is the (unlabeled) union of the entries of the cycles of σ whose minima are labeled h, along with 0. Each cycle of σ whose minimum is labeled e gives rise to a labeled block B, with the labels of the entries of B being the same as the labels of the entries of the cycle in σ. Example 4.3. The C 3 -labeled D-permutation σ = (1 ω, 3 e, 5 ω2, 4 ω )(6 e )(7 e, 8 ω2 )(9 ω ) has cyc(σ) = e 7 e 8 ω2. Theorem 4.4. Let n 1 and m 2. If X L m n then ( 1) X 1 µ L m n (ˆ0, X) = #{σ D m 2n 1 cyc(σ) = X}. Corollary 4.5. For all n 1 and m 2, χ L m n (t) = n s D,m (n, k)t k 1, (4.1) k=1 where ( 1) k 1 s D,m (n, k) is equal to the number of C m -labeled D-permutations in S 2n 1 with exactly k cycles with minimum labeled e. Consequently, χ L m n ( 1) is the total number of C m - labeled D-permutations on [2n 1].

11 Homogenized Linial 11 In the full version of the paper [10], we use this corollary and the theory of surjective pistols in [13] to derive the following generating function formula for χ L m n ( 1) which reduces to (1.5) and (1.7) when m = 1 and m = 2, respectively. Theorem 4.6. For all n, m 1, χ L m n ( 1)x n = n 0 n 0 n k=0 where (a) n,m = a(a + m)(a + 2m) (a + (n 1)m). (1) n,m (2) n,m x n (1 + (mk)(mk + 1)x), Now by applying the Orlik-Solomon formula (2.1), we obtain a formula for the generating function of the Euler characteristic of the complement of the homogenized Linial- Dowling arrangement. 5 Further work The graph Γ n belongs to a class of graphs called Ferrers graphs, which were introduced by Ehrenborg and van Willegenburg [7]. We have been able to extend some of our results to more general Ferrers graphs and skew Ferrers graphs. We also have results on directed graph analogs of the homogenized Linial arrangement. Acknowledgements The authors thank José Samper for a valuable suggestion pertaining to bipartite graphs. MW thanks Gábor Hetyei for introducing her to his work on this topic when she visited him at the University of North Carolina, Charlotte during her Hurricane Irma evacuation, and for his hospitality at that time. References [1] C. A. Athanasiadis, Characteristic polynomials of subspace arrangements and finite fields, Adv. Math. 122 (1996), no. 2, [2] D. Barsky and D. Dumont, Congruences pour les nombres de Genocchi de 2e espèce, Study Group on Ultrametric Analysis. 7th 8th years: (Paris, 1979/1981) (French), Secrétariat Math., Paris, 1981, pp. Exp. No. 34, 13. [3] P. Doubilet, G-C. Rota, and R. P. Stanley, On the foundations of combinatorial theory. VI. The idea of generating function, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory (1972),

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