The Homogenized Linial Arrangement and Genocchi Numbers

We study the intersection lattice of a hyperplane arrangement recently introduced by Hetyei who showed that the number of regions of the arrangement is a median Genocchi number. Using a different method, we refine Hetyei's result by providing a combinatorial interpretation of the coefficients of the characteristic polynomial of the intersection lattice of this arrangement. We also show that the M\"obius invariant of the intersection lattice is a (nonmedian) Genocchi number. The Genocchi numbers count a class of permutations known as Dummont permutations and the median Genocchi numbers count the derangements in this class. We show that the signless coefficients of the characteristic polynomial count Dumont-like permutations with a given number of cycles. This enables us to derive formulas for the generating function of the characteristic polynomial, which reduce to known formulas for the generating functions of the Genocchi numbers and the median Genocchi numbers. As a byproduct of our work, we obtain new models for the Genocchi and median Genocchi numbers.


Introduction
The braid arrangement (or type A Coxeter arrangement) is the hyperplane arrangement in R n , n ≥ 1, defined by Note that the hyperplanes of A n−1 divide R n into open cones of the form where σ is a permutation in the symmetric group S n . Hence the braid arrangement A n−1 has | S n | = n! regions. The regions of the arrangement obtained by intersecting A 2 with the plane x + y + z = 0 are shown in the figure below. A classical formula of Zaslavsky [30] gives the number of regions of any real hyperplane arrangement A in terms of the Möbius function of its intersection (semi)lattice L(A). Indeed, given any finite, ranked poset P of length , with a minimum element0, the characteristic polynomial of P is defined to be (1. 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 (1.3) χ Πn (t) = n k=1 s(n, k)t k−1 , where s(n, k) is the Stirling number of the first kind, which is equal to (−1) n−k times the number of permutations in S n with exactly k cycles; see [26, Example 3. 10.4]. Hence χ Πn (−1) = (−1) n−1 | S n |. Therefore, from (1.2), we recover the result observed above that the number of regions of A n−1 is n!.
Deformations of the braid arrangement are often studied in the literature, and in many cases their number of regions is known. Among these deformations are the Linial arrangement, Shi arrangement, Catalan arrangement, semiorder arrangement, generic braid arrangement, and semigeneric braid arrangement; see [25] for a discussion of these arrangements and for a general introduction to the combinatorics of hyperplane arrangements. The Linial arrangement is the hyperplane arrangement in R n defined by, In [20] Postnikov and Stanley show that the number of regions of this arrangement is the number of alternating trees on node set [n + 1], where a tree is alternating if each node is either greater than all its neighbors or smaller than all its neighbors.
In [14], Hetyei proves that where h n is a median Genocchi number, 2 or equivalently by (1.2) that There are numerous characterizations of the median Genocchi numbers in the literature. One such characterization is given by the following formula for the generating function, which was obtained by Barsky and Dumont [3], h n x n = n≥1 n!(n + 1)!x n n k=1 (1 + k(k + 1)x) .
A combinatorial characterization in terms of a class of permutations now called Dumont derangements was also given by Barsky and Dumont in [3]. Another was given by Randrianarivony in [21] in terms of a class of objects called surjective staircases.
Hetyei's proof of (1.5) relies on the finite field method of Athanasiadis [2], which is a method for computing the characteristic polynomial by counting points in the complement of an associated arrangement over a finite field. Hetyei obtains a recurrence for a refinement of the associated point count polynomial and relates it to a recurrence of Andrews, Gawronsky, and Littlejohn [1] for Legandre-Stirling numbers. This yields a formula for χ L(H 2n−1 ) (−1) in terms of the Legandre-Stirling numbers, which is identical to a formula of Claesson, Kitaev, Ragnarsson, and Tenner [7] for the median Genocchi numbers.
In this paper we further study the intersection lattice L(H 2n−1 ) and its characteristic polynomial χ L(H 2n−1 ) (t) using an approach quite different from Hetyei's. The first few characteristic polynomials and their values at t = 1 and t = 0 are given in the table below.
We begin by showing that the intersection lattice L(H 2n−1 ) is isomorphic to the bond lattice of a certain bipartite graph, which belongs to the class of Ferrers graphs introduced by Ehrenborg and van Willigenburg [12]. This enables us to refine Hetyei's result by deriving a combinatorial formula for the Möbius invariant of the lower intervals of L(H 2n−1 ) in terms of a class of permutations that we call D-permutations, which are similar to the Dumont permutations. We obtain the following analog of (1.3): where (−1) k s D (2n, k) is equal to the number of D-permutations on [2n] with exactly k cycles. The D-permutations have a simple description. A permutation σ on [2n] is a D-permutation if i ≤ σ(i) whenever i is odd, and i ≥ σ(i) whenever i is even. A consequence of (1.7) and (1.5) is that the median Genocchi number h n is equal to the number of D-permutations on [2n]. A particularly interesting feature of this permutation model for the median Genocchi numbers is that the (nonmedian) Genocchi number 3 g n enumerates a subset of the set of D-permutations on [2n]; namely the set of D-cycles on [2n]. This follows from a simple bijection with a set of permutations on [2n − 1] shown by Dumont in [8] to have cardinality equal to g n . This feature is in contrast with the Dumont permutation model in which the Genocchi numbers enumerate the full set of Dumont permutations on [2n] and the median Genocchi numbers enumerate a subset of the these.
Our formula (1.7) therefore implies where and0 and1 are the minimum and maximum elements of L(H 2n−1 ), respectively. Hence the Genocchi numbers, as well as the median Genocchi numbers, play a fundamental role in the study of the homogenized Linial arrangement. We recover Hetyei's result (1.5), by constructing a bijection from the full set of D-permutations on [2n] to a set of surjective staircases shown by Randrianarivony in [21] to have cardinality equal to the median Genocchi number h n . Moreover, this bijection and the theory of surjective staircases enable us to prove two generating function formulas for the characteristic polynomials. One such formula is where (a) n denotes the falling factorial a(a − 1) · · · (a − n + 1). Note that (1.9) reduces to the Barsky-Dumont formula (1.6) for n≥1 h n x n when t is set equal to −1, and to a similar formula of Barsky and Dumont in [3] for n≥1 g n x n when t is set equal to 0.
In addition to the D-permutation model, our study of the characteristic polynomial leads to other interesting combinatorial models for the median Genocchi numbers and the Genocchi numbers. For instance, using our bond lattice result and Hetyei's formula (1.5), we show that a formula of Chung and Graham [6] for chromatic polynomials of incomparability graphs yields another nice permutation model for the median Genocchi numbers: h n is the number of permutations σ on [2n] such that i > σ(i) only if i is even and σ(i) is odd. This leads to the conjectural interpretation of −χ L(H 2n−1 ) (−t) as the enumerator of such permutations by their number of cycles, which would imply that the Genocchi number g n is equal to the number of such permutations that are cycles.
The paper is organized as follows. In Section 2 we review some basic material on hyperplane arrangements, bond lattices, and Genocchi numbers. Our result that the intersection lattice of the homogenized Linial arrangement is isomorphic to a bond lattice is proved in Section 3. 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 minimum and even maximum. Using Whitney's NBC theorem, we show that the signless Möbius invariant of each lower interval [0, π] of the bond lattice enumerates a certain class of alternating forests on [2n].
In Section 4, we prove (1.7) and a more general formula for chromatic polynomials of general Ferrers graphs by constructing a bijection between the alternating forests of Section 3 and the D-permutations on [2n]. We also give some alternative formulas for the characteristic polynomials in terms of D-permutations and we present some of their consequences. We observe that the characteristic polynomial χ L(H 2n−1 ) (t) is divisible by (t − 1) 3 and we give combinatorial and geometric interpretations of the polynomial (t − 1) −3 χ L(H 2n−1 ) (t).
Section 5 contains the bijection from the D-permutations on [2n] to the set of surjective staircases whose cardinality is equal to h n . This bijection, the formulas of Section 4 for the characteristic polynomial, and the theory of surjective staircases are used to prove (1.9) and a similar generating function formula.
In Section 6, we present the permutation models for the Genocchi numbers and median Genocchi numbers that arise from the Chung-Graham result on chromatic polynomials. Further work on a type B analog and a Dowling arrangement generalization, which will appear in a forthcoming paper, is discussed in Section 7.
Most of the results of this paper and of the forthcoming paper on the Dowling generalization were announced in the extended abstract [19].

Preliminaries
2.1. Hyperplane Arrangements. Let k be a field (here k is R or C). 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 H∈A H = ∅ then the intersection poset is a geometric lattice, otherwise it's a geometric semilattice.
If A is a real hyperplane arrangement in R n , then its complement R n \A is disconnected. By the number of regions r(A) 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.

2.2.
The Bond Lattice of a Graph. Let G = (V, E) be a graph. Given a subset B of V , let G| B denote the induced subgraph of G with vertex set B. Let Π V denote the lattice of partitions of the set V ordered in the usual way by reverse refinement. If V = [n] := {1, 2, . . . , n} we write Π n for Π V . The bond lattice of G is the induced subposet Π G of Π V consisting of partitions π = B 1 | · · · |B k such that G| B i is connected for all i. It is well known that the chromatic polynomial ch G (t) of G satisfies If V = [n], we can also associate to G the graphic hyperplane arrangement It is well known and easy to see that Note that Π n is the bond lattice of the complete graph K n and that the braid arrangement A n−1 is the graphical arrangement associated with K n . Another example is given below. Broken circuits provide a useful means of computing the Möbius function of the bond lattice of a graph G = (V, E) (or more generally, of a geometric lattice). Fix a total ordering of E and let S be a subset of E. Then S is called a broken circuit if it is the edge set of a cycle of G with its smallest edge (with respect to this ordering) removed. If S does not contain a broken circuit, we say that S is a non-broken circuit set or NBC set. Clearly the edge set of a cycle always contains a broken circuit. Hence if S is an NBC set, (V, S) is a forest. We call (V, S) for which S is an NBC set an NBC forest of G.
Given a tree T whose node set is a subset of Z >0 , we say the tree is increasing if when T is rooted at its smallest node, each nonroot vertex is larger than its parent. A forest on a subset of Z >0 is said to be increasing if it consists of increasing trees. Note that if G is K n then by ordering the edges lexicographically with the smallest element as the first component, the NBC forests of G are exactly the increasing forests on [n]. It therefore follows from (2.3) and (1.2) that r(A n−1 ) equals the number of increasing forests on [n], which can easily be shown to be equal to n!.
For any tree T (rooted or unrooted), let |T | denote the number of nodes of T . Recall that a plane tree is a rooted tree in which the children of each node are linearly ordered. Let T be a plane tree on a subset of Z >0 . When T is drawn in the plane, the children of each vertex v are drawn from left to right according to their linear order. Now let T be a plane tree on a subset of Z >0 . When drawn in the plane the order is depicted from left to right. For each vertex v, the order of its children determines a "left to right" order of the subtrees rooted at the children. If |T | > 1, let T 1 , T 2 , . . . , T k be the subtrees rooted at the children of the root r of T ordered from "left to right". The postorder word pw(T ) of T is defined recursively as the concatenation pw(T ) = pw(T 1 ) · pw(T 2 ) · · · · · pw(T k ) · r, if |T | > 1 and by pw(T ) = r if |T | = 1.
Every increasing tree T on node set V ⊂ Z >0 can be viewed as a plane tree on V by rooting it at its smallest node and then ordering the children of each node in increasing order. It is not difficult to prove that the map that sends an increasing tree T on V to the permutation whose cycle form is (pw(T )), is a bijection from the set of increasing trees to the set of cycles in S V . This bijection extends in the obvious way to a bijection from the set of increasing forests on V to the set of permutations in S V whose cycles correspond to the trees of the forest. Thus this bijection and (2.3) yield another way of proving (1.3).

2.3.
Genocchi and median Genocchi Numbers. The Genocchi numbers and median Genocchi numbers are classical sequences of numbers, which have been defined in many ways. Below we take a characterization of Dumont [8,Section 6] as our definition of the Genocchi numbers and a characterization of Barsky and Dumont [3] as our definition of the median Genocchi numbers.
n 0 1 2 3 4 5 6 g n 1 1 3 17 155 2073 h n 1 2 8 56 608 9440 198272 Our notation is nonstandard in that g n is usually denoted G 2n or |G 2n |, while h n is usually denoted H 2n+3 or |H 2n+3 |. The Genocchi numbers and median Genocchi numbers are also known as Genocchi numbers of the first and second kind, respectively.
Another permutation characterization of the Genocchi numbers obtained by Dumont in [8, Section 6] is given by We also mention the following exponential generating function formula for the Genocchi numbers (see [8, page 305 In [3], two formulas for the generating function of the Genocchi numbers and two formulas for the generating function of the median Genocchi numbers are given. We list these four generating function formulas here. Carlitz [5] and to Riordan and Stein [22], and the rest are due to Barsky and Dumont [3] .

The intersection lattice is a bond lattice
In this section we show that L(H 2n−1 ) is isomorphic to the bond lattice of a bipartite graph. This enables us to give a characterization of the intersection lattice L(H 2n−1 ) as an induced subposet of Π 2n and to compute its Möbius function by counting NBC sets.
Let Γ 2n be the bipartite graph on {1, 3, 5, . . . , 2n − 1} {2, 4, 6, . . . , 2n} with an edge {2i − 1, 2j} whenever 1 ≤ i ≤ j ≤ n. The graph Γ 6 is shown below. More generally for any finite subset V of Z >0 , let Γ V be the bipartite graph on V with edge set Remark 3.1. The graphs Γ V are easily seen to belong to the class of Ferrers graphs, which were introduced by Ehrenborg and van Willegenburg in [12]. In fact, it is not much more difficult to show that all Ferrers graphs are of this form. Ferrers graphs have been further studied in (among others) [7] and [24]. Proof. Let (e 1 , e 2 , . . . , e 2n+1 ) be the standard basis for R 2n+1 .
where G is the graph on [2n + 1] whose edge set is equal to E(Γ 2n ). In other words, G is Γ 2n with an appended isolated node 2n + 1. Since adding an isolated node to a graph yields an isomorphic bond lattice, by (2.2) we have We will prove the result by producing a vector space isomorphism ψ : R 2n+1 → R 2n+1 that takes K 2n to H 2n−1 . Indeed, such a map will induce an isomorphism from L(K 2n ) to L(H 2n−1 ).
First consider the linear operator φ : R 2n+1 → R 2n+1 defined on the standard basis by letting φ(e 2i−1 ) = e i − e n+1+i and φ(e 2i ) = e i+1 for all i ∈ [n] and letting φ(e 2n+1 ) = e 2n+1 . Let A be the matrix of φ with respect to the standard basis. One can easily check that | det A| = 1. Now let ψ : R 2n+1 → R 2n+1 be the linear operator whose matrix with respect to the standard basis is the transpose of A −1 . Clearly ψ is an isomorphism.
We claim that ψ takes the hyperplane K i,j to the hyperplane H i,j for all 1 ≤ i ≤ j ≤ n. To prove the claim, We also have which proves the claim. It follows from the claim that ψ takes K 2n to H 2n−1 as desired.
Hence, to study L(H 2n−1 ), it suffices to study Π Γ 2n . Proposition 3.3. For all finite subsets V of Z >0 , the bond lattice Π Γ V is the induced subposet of Π V consisting of those partitions π for which each nonsingleton block of π has an odd minimum and an even maximum.
Proof. Suppose that π ∈ Π Γ V . Let B be a nonsingleton block of π and let u = min B. Since Γ V | B is connected, there must be an element v ∈ B such that {u, v} is an edge of Γ V . By definition of Γ V , since u < v, the node u must be odd. A similar argument tells us that max B is even.
Conversely, suppose that we have a partition π = B 1 | · · · |B k ∈ Π V such that each nonsingleton B i has an odd minimum and an even maximum. We wish to show that Γ V | B i is connected. Indeed, let M i = max B i and m i = min B i . By the definition of Γ V , each odd vertex in B i shares an edge with M i and each even vertex of B i shares an edge with m i . Since, in particular, this means that m i and M i share an edge, We will use the Rota-Whitney formula (2.3) to evaluate the Möbius funtion on each lower interval [0, π] of Π Γ V . Let T be a tree whose nodes are in Z >0 . We say that T is increasing-decreasing (ID) if, when T is rooted at its largest node, each internal node v satisfies (1) if v is odd then v is less than all its descendants and all its children are even, (2) if v is even then v is greater than all its descendants and all its children are odd. An an increasing-decreasing (or ID) forest is a forest, each of whose components are increasing-decreasing trees. An example of an ID forest is given in the figure below.  (1) and (2) of the definition of ID tree when T is rooted at its smallest node.
Proof. Assume |T | > 1. Let T be an ID tree with odd smallest node s and even largest node t. Root T at t. Let T 1 , . . . , T k be the subtrees rooted at the children of t. Clearly, s must be the root of one of these subtrees say T 1 . Let S 1 , . . . , S j be the subtrees rooted at the children of s. Let T be the tree obtained by removing T 1 from T . When we root T at s we get the rooted tree in which the subtrees rooted at the children of s are S 1 , . . . , S j , T . Since T is an ID tree, these subtrees S 1 , . . . , S j , T must be ID trees rooted at their largest nodes, which are even. Hence the internal nodes of T rooted at s satisfy (1) and (2) of the definition of ID tree. The proof of the converse is analogous with the roles of even and odd exchanged, and the roles of larger and smaller exchanged.
The following result is implicit in the proof of Theorem 24 of [24]. We include a proof for the sake of completeness.
Theorem 3.5. For all π ∈ Π Γ V , we have that (−1) |π| µ(0, π) equals the number of ID forests on V whose trees have nodes sets equal to the blocks of π. Consequently, −µ Π Γ V (0,1) is equal to the number of ID trees on V and where F V,k is the set of ID forests on V with exactly k trees.
Proof. We prove this result by showing that under an appropriate ordering of the edges of Γ V , the NBC forests of Γ V are precisely the ID forests on V . The total order on the edges of Γ V we use is as follows: First order the odd vertices in increasing order and the even vertices in decreasing order, and then fix any linear extension ≺ of the product order of the two linear orders. By associating the edge {2i − 1, 2j} with the ordered pair (2i − 1, 2j), we get a total order on the edges of Since adding an edge between two different components of a forest can never complete a cycle, a forest is an NBC forest of Γ V if and only if its trees are NBCs trees of the subgraph of Γ V induced by the node set of the tree. By definition, a forest is an ID forest if and only if its trees are ID trees. Hence, it suffices to prove that the NBC trees of Γ V are precisely the ID trees on V . We can assume |V | > 1, min V is odd and max V is even.
Let T be an ID tree on V . Suppose that e = {2i − 1, 2j} is an edge of Γ V that is not in T . If we add e to T , we create a unique cycle ρ, which contains e. To show that T is NBC, it suffices to find an edge of ρ that strictly precedes e in the order ≺. Root T at its largest node and let h be the youngest common ancestor of 2i − 1 and 2j (that is, the furthest from the root). Clearly h is a node of ρ.
Suppose h is odd. Then h ≤ 2i − 1 since T is an ID tree. If h is the parent of 2j then h cannot be 2i − 1 since {2i − 1, 2j} is not an edge of T . Therefore h < 2i − 1, which implies that {h, 2j} ≺ {2i − 1, 2j} = e. Thus {h, 2j} is the edge of ρ that we seek. If h is not the parent of 2j then let 2k be the child of h that is a proper ancestor of 2j. We thus have 2k > 2j, which implies {h, 2k} ≺ {2i − 1, 2j} = e. Thus in this case, {h, 2k} is the edge of ρ that we seek.
The case that h is even is handled analogously with the role of even and odd exchanged and the role of < and > exchanged. We can now conclude that T is an NBC tree of Γ V .
Conversely, let T be an NBC tree of Γ V . Suppose that T is not an ID tree. Then T rooted at its largest node has a node x with a grandchild y satisfying x > y if x is odd and x < y if x is even. Choose x so that its distance in T from the root is minimal. We claim that x cannot be the root of T . Indeed, if x is even then it is not the largest node of T . Thus x is not the root. If x is odd then it is not the root since the largest node of T must be even. Let u be the parent of x and let v be the parent of y, which implies that x is that parent of v. Thus, u, x, v, y forms a path in T .
We claim that {u, y} is an edge of Γ V . Indeed, if u is even then x is odd. We thus have u > x > y. Also y is odd since it is a grandchild of x. It follows that {u, y} is an edge of Γ V . The case when u is odd is handled analogously.
Since {u, y} is an edge of Γ V we can use it to complete the cycle induced by the node set {u, x, v, y}. Clearly {u, y} ≺ {u, x}. By the minimality of the choice of x, since u is closer to the root than x is, we have u < v if u is odd and u > v if u is even. Hence {u, y} ≺ {u, x}, {v, x}, {v, y}. This implies that the subgraph of T induced by {u, x, v, y} is a broken circuit which contradicts the hypothesis that T is an NBC tree. Therefore T is indeed an ID tree.
By Theorem 3.2, we have the following consequence of Theorem 3.5.
Corollary 3.6. For all n ≥ 1, where F 2n,k is the set of ID forests on [2n] with exactly k trees.
Using Hetyei's formula (1.5) and Corollary 3.6, we obtain the following combinatorial interpretation of the median Genocchi numbers. In Section 5, we give an alternative proof that does not rely on these results. In the next section we prove the following analogous result. From this we see that the nonmedian Genocchi numbers also play an important role in the study of the intersection lattice of the homogenized Linial arrangement. Corollary 3.9. For all n ≥ 1,

From ID forests to Dumont-like permutations
Our next step is to introduce a class of permutations similar to the Dumont permutations discussed in Section 2.3 and then give a bijection between these permutations and the ID forests. This will enable us to express the characteristic polynomial in terms of D-permutations in several ways.

D-permutations. Let
A be a finite subset of Z >0 . We say σ ∈ S A is a Dpermutation on A if i ≤ σ(i) whenever i is odd and i ≥ σ(i) whenever i is even. We denote by D A the set of D-permutations on A and by DC A the set of D-cycles on A. If A = [n], we write D n and DC n . Note that all Dumont permutations are D-permutations, but not conversely. Indeed, a D-permutation can have both even and odd fixed points, while a Dumont permutation can only have odd fixed points. It follows immediately from the definitions that Recall that the two sets in the middle of this chain are enumerated by median Genocchi number h n−1 and Genocchi number g n+1 , respectively. It turns out that the sets on the ends of the chain are also enumerated by Genocchi and median Genocchi numbers, respectively. The count for DC 2n is an easy consequence of (2.4). We will see at the end of this section that the count for D 2n can be proved using Hetyei's formula (1.5). In Section 5 we will prove it without relying on Heyei's result.
Proof of (1). There is an easy bijection between the set of D-cycles on [2n] and the set S of permutations given in (2.4). Indeed, we send a D-cycle (a 1 , a 2 , . . . , a 2n ), where a 2n = 2n, to the permutation a 1 , a 2 , . . . , a 2n−1 (in one line notation), which is clearly in S. Hence the result follows from (2.4).

The bijection.
Let T be the set of ID trees. For T ∈ T , letT be the plane tree obtained by rooting T at its largest node (which is even when |T | > 1) and fixing the following total ordering on the children of each node v of T : • if v is even, place its children (all of which are odd) in increasing order from left to right • if v is odd, place its children (all of which are even) in decreasing order from left to right. Also letT be the plane tree obtained by rooting T at its smallest node (which is odd when |T | > 1) and fixing the above total ordering on the children of each node v of T .
Recall from Section 2.2, that pw(T ) denotes the postorder word of a plane tree T . For the ID tree T below, pw(T ) = 42156378 and pw(T ) = 56378421.  Proof. Suppose T ∈ T . We prove the result forT andT simultaneously by induction on |T |. The result is obviously true for |T | = 1, so assume |T | > 1. Let v 1 < · · · < v k be the children of the root r ofT . For each i, the subtree ofT rooted at v i is of the formT i for some ID tree T i by Proposition 3.4. We have where · denotes concatenation. Since the last letter of pw(T ) is the root ofT , it is even and is larger than all the other letters. By induction on |T |, each pw(T i ) is in W. We also know that the last letter of pw(T i ) is the root ofT i , which is odd. Hence to prove that pw(T ) ∈ W we need only show that the last letter v i of pw(T i ) is less than the first letter of pw(T i+1 ) for all i ∈ [k − 1]. (Note that the last letter of pw(T k ) is less than r). The first letter of pw(T i+1 ) is greater than the root v i+1 ofT i+1 . Since v i < v i+1 , we have that v i is less than the first letter of pw(T i+1 ). Thus pw(T ) ∈ W. An analogous argument yields pw(T ) ∈ W.
Given a word w = w 1 w 2 · · · w m over alphabet Z >0 , we say that w j is a right-to-left minimum if w j < w i for all i > j. Similarly we say that w j is a right-to-left maximum if w j > w i for all i > j. For example, the right-to-left minima of 2156437 are 1, 3, 7. We also let w denote w with its last letter removed, that is w = w 1 · · · w m−1 . Proof. Let v 1 < · · · < v k be the children of the root r ofT . Each v j is less than all its descendants inT , so for all i < j, we have that v i is less than all descendents of v j . Hence, v i , which is the last letter of the postorder word of the subtree rooted at v i , is less than all subsequent letters of pw(T ) . Conversely, if v is not a child of r then it has an odd anscestor u, which must be less than v. Since u appears after v in the postorder word, v is not a right-to-left minimum. The proof forT is completely analogous. We define γ recursively. Let w = w 1 · · · w m ∈ W. If m = 1, let γ(w) be the tree with the single node w 1 . Now suppose m > 1. To define γ, we need two cases: Case 1: Assume w m is even. Then it is the largest letter of w. Let j 1 < · · · < j k be such that w j 1 · · · , w j k are the right-to-left minima of the word w := w 1 w 2 . . . w m−1 . Note that j k = m − 1. For each i = 1, . . . , k, let α i (w) be the segment w j i−1 +1 w j i−1 +2 · · · w j i , where j 0 = 0. Hence w is the concatenation of the words α 1 (w), α 2 (w), . . . , α k (w).
Since for all i, α i (w) is a segment of w ∈ W, it satisfies conditions (1) and (2) of the definition of W. We claim that conditions (3) and (4) hold as well. Indeed, the last letter w j i of α i (w) is odd since it is a right-to-left minimum of w and is therefore is followed by a larger letter in w. The last letter of α i (w) is smaller than all the other letters of α i (w) since none of the other letters are right-to-left minima. Hence (3) holds and (4) holds vacuously. Thus α i (w) ∈ W. Now, for each i, we can recursively apply γ to α i (w) to obtain a plane tree γ(α i (w)) in {T : T ∈ T }. Let γ(w) be the plane tree constructed as follows: (1) The root is w m , which is even and largest.
(2) The children of w m are the right-to-left minima w j 1 < · · · < w j k , which are odd. (3) The subtree rooted at w j i is γ(α i (w)) for each i.

Clearly γ(w) ∈ {T : T ∈ T }.
Case 2: Assume w m is odd. An analogous argument allows us to decompose w = w 1 · · · w m−1 ∈ W into segments α i (w) ∈ W, whose last letter is the ith rightto-left maximum of w. We define γ(w) to be the plane tree constructed as follows: (1) The root is w m , which is odd.
(2) The children of w m are the right-to-left maxima w j 1 > · · · > w j k , which are even (3) The subtree rooted at w j i is γ(α i (w)) for each i.
We have γ(w) ∈ {T : T ∈ T }. Now that we have shown that γ is well-defined, it remains to check that the maps pw and γ are inverses of each other. We prove γ(pw(T )) =T and γ(pw(T )) =T , for all T ∈ T , by induction on |T |. If |T | = 1, this is clear, so suppose |T | > 1.
Case 1:T . By Lemma 4.4, the children v 1 < v 2 < · · · < v k of the root r ofT are exactly the right-to-left minima of the word pw(T ) = pw(T 1 ) · pw(T 2 ) · · · · · pw(T k ), whereT i is the subtree ofT rooted at v i . Since v i is the last letter of pw(T i ), we have α i (pw(T )) = pw(T i ) for all i. By induction, γ(α i (pw(T ))) = γ(pw(T i )) =T i . Now we have that γ(pw(T )) is the plane tree whose root is the last letter of pw(T ), which is r. The children of r are the right-to-left minima of pw(T ) which are v 1 < · · · < v k . The subtree rooted at v i is γ(α i (pw(T ))) which isT i . Hence γ(pw(T )) =T , as desired.
Hence γ • pw is the identity map on {T : T ∈ T } ∪ {T : T ∈ T }. A similar argument can be used to prove that the other composition pw •γ equals the identity map on W.
For any finite subset A of Z >0 , let T A = {T ∈ T : the node set of T is A} and let W A = {w ∈ W : the set of letters of w is A}. Recall that DC A is the set of D-cycles on A.
Theorem 4.6. For any finite suubset A of Z >0 , let ψ : T A → DC A be the map defined by letting ψ(T ) be the cycle (pw(T )) in S A . Then ψ is a well defined bijection.
Proof. Consider the following maps: (1) the map from T A to {T : T ∈ T A } defined by T →T (2) the map from {T : T ∈ T A } to {w ∈ W A : last letter of w is largest} defined byT → pw(T ) (3) the map from {w ∈ W A : last letter of w is largest} to DC A defined by w → (w). It is clear that the first map is a bijection. It follows from Lemma 4.5 that the second is also bijection. The third map is a well defined bijection since the last letter of w must be even. Since ψ is the composition of three bijections, it is also a bjection.
An analogous argument can be used to show that the map T A → DC A that takes T to the cycle (pw(T )) is also a well defined bijection. The following result shows that this map is, in fact, identical to ψ. Proof. Let s be the smallest element of A and t the largest element. Then s is the root ofT and t is its leftmost child. Let t 1 < . . . , < t k be be the children of t inT and let t > s 1 > · · · > s j be the children of s. For each i ≥ 1, let T i be the subtree ofT rooted at t i and let S i be the subtree ofT rooted at s i . Then pw(T ) is the concatenation pw(T 1 ) · · · pw(T k ) · t · pw(S 1 ) · · · pw(S j ) · s.

Now let us look at pw(T ).
Clearly t is the root ofT and s is its leftmost child. The children of t are s < t 1 < · · · < t k . The children of s are s 1 < · · · < s j . For each i ≥ 1, the subtree ofT rooted s i is S i and the subtree rooted at t i is T i . Then pw(T ) is the concatenation pw(S 1 ) · · · pw(S j ) · s · pw(T 1 ) · · · pw(T k ) · t.
From this we see that pw(T ) and pw(T ) are related by a cyclic shift. Hence the cycles are the same.
The bijection in Theorem 4.6 extends to a bijection between ID forests and Dpermutations. Under this bijection, the blocks of the cycle support of the image of an ID forest are the node sets of the trees of the ID forest. Hence we have the following consequence of Theorem 4.6.
Corollary 4.8. Let π be a partition of a finite subset of Z >0 . Then the ID forests whose trees have node sets equal to the blocks of π are in bijection with the Dpermutations whose cycle support is π.
As a consequence of Theorem 3.5 and Corollary 4.8, we have the following result. Theorem 4.9. Let A be a finite subset of Z >0 . For all π ∈ Π Γ A , Consequently, Note that it follows from Corollary 4.12 that Part (2) of Theorem 4.2 is equivalent to Hetyei's formula (1.5). So Part (2) is now proved. In the next section we will prove Part (2) without relying on (1.5), thereby recovering Hetyei's result.
Note also that Theorem 3.8 is now proved since it follows immediately from Theorems 4.2(1) and 4.6.

Alternative formulas.
We now give two alternative ways of expressing the characteristic polynomial in terms of D-permutations (equivalently ID forests), which will be used in Section 5.3. As a consequence of these formulas, we get a decomposition of the Genocchi numbers and the median Genocchi numbers into a sum of powers of 2.
Theorem 4.13. For all n ≥ 1, Proof. Let F V be the set of ID forests on node set V . For V = [2n], Theorem 3.5 can be restated as where m(F ) denotes the number of trees of the forest F . For each F ∈ F [2n] , let F be the forest obtained by removing node [2n] from F . Clearly F ∈ F [2n−1] . We thus have Fix G ∈ F [2n−1] and let T be a tree of the forest G. For all F ∈ F [2n] for which F = G, the tree T is either a tree of the forest F or it is attached to 2n in F . Hence to obtain a forest F ∈ F [2n] for which F = G, we have the option of not attaching T to 2n or attaching T at its smallest node, which can be done if and only if T is not an even single node. Hence if T is not an even single node then it contributes a factor of 1 − t to the inner sum, and if T is an even single node then it contributes a factor of −t to the inner sum. It follows that We now have Proof. Each F ∈ F [2n] can be obtained from a forest in F [2n]\{2} either by adding an isolated node 2 or by attaching the node 2 to 1. This implies that Similarly, For each F ∈ F [2n]\{2,2n−1} , letF be the forest obtained by removing the nodes 1 and 2n from F . ClearlyF ∈ F {3,...,2n−2} . It therefore follows from (4.4), (4.5), and (4.6) that Fix G ∈ F {3,...,2n−2} and let T be a tree of the forest G. For all F ∈ F [2n]\{2,2n−1} for whichF = G, the tree T is either a tree of the forest F or it is attached to either 1 or 2n in F . If T is a single node it can be attached to only one of the nodes in {1, 2n} depending on its parity. Hence singleton trees of G contribute a factor of 1 − t to the inner sum. If T has more than one node then there are exactly two ways to attach it; with an edge between its largest node and 1, or with an edge between its smallest node and 2n. Thus, nonsingleton trees of G contribute a factor 2 − t to the inner sum. It follows that We now have with the last equality following from (4.8).
Note that when t is set equal to 0, equation (4.2) reduces to (3.1). Indeed, the only terms to survive on right hand side of (4.2) are the ones corresponding to Dpermutations with no even fixed points. But these are the Dumont permutations, which were used in Section 2.3 to define the Genocchi numbers.
In the following corollary, we obtain the decomposition of the median Genocchi numbers by setting t = −1 in (4.2) and using (1.5). We obtain the decomposition of the Genocchi numbers by setting t = 0 in (4.3) and using (3.1). where h n,j is the number of D-permutations on [2n] with exactly j cycles that are not even fixed points and g n,j is the number of D-permutations on [2n] with exactly j cycles that are not fixed points.
We expect that the coefficients g n−1,j are equal to the coefficients in Sundaram's decomposition [27,Proposition 3.5] of the Genocchi numbers into powers of 2.
Corollary 4.16. For all n ≥ 3, Consequently, the number of bounded regions of P 2n is h n−3 .
It follows from Theorem 4.14 that χ L 2n (1) equals the number of D-permutations on [2n −4] with no fixed points. But these are precisely the Dumont derangements on [2n −4], which by the definition of the median Genocchi numbers given in Section 2.3, is equal to h n−3 . Hence (4.10) holds.
The consequence follows from another well known result of Zaslavsky [30], namely that the number of bounded regions of any affine arrangement A is equal to |χ L(A) (1)|.

From D-permutations to surjective staircases
In this section we prove (1.9) along with a similar formula for the generating function of the characteristic polynomial of the homogenized Linial arrangement. The proofs rely on the theory of surjective staircases introduced by Dumont [8]. We first construct a bijection from the D-permutations on [2n] to a certain class of surjective staircases known to be enumerated by h n , thereby proving Theorem 4.2 (2) and completing our path to Hetyei's formula (1.5).  It is not difficult to see that |X m | = m!. For our purposes, we will need a bijection from S m to X m constructed in [10]. An excedent function f ∈ X 2n is said to be a surjective staircase 4 if its image is {2, 4, . . . , 2n}. Since the odd labeled rows in the tableau T (f ) corresponding to a surjective staircase f are empty, we can delete these rows from the tableau. Let To prove the result we construct a bijection γ : G 2n → {f ∈ E 2n+2 : f has no even maxima}.
It is clear that f ∈ E 2n+2 . We claim that f has no even maxima. Indeed, the maxima of f are the j for which g(j) is odd. By the definition of G 2n , such a j must be a fixed point of g, so all the maxima of f are odd. We can now let It is helpful to visualize the map γ in terms of tableaux. First γ adds an empty row labeled 2n+2 of length 2n+2 to the top of the tableau T (g), where g ∈ G 2n . An X is placed in both of its two rightmost cells. An X in an odd labeled row of T (g) represents an isolated fixed point of g, so it is in the rightmost cell of its row. Next γ slides each such X to the top of its column. At this point, the odd labeled rows become empty. We remove these rows to get T (γ(g)). The figure below illustrates the map γ. 10 We claim γ is bijective. Indeed, given a surjective staircase f ∈ E 2n+2 with no even maxima, we can obtain an excedent function g : [2n] → [2n] by inserting empty odd labeled rows between the even labeled rows in T (f ) and then sliding the X's from the top row down to the bottom of their columns. After deleting the top row we obtain T (g). Clearly g ∈ G 2n . It is not difficult to see that the sliding down process and sliding up process are inverses of each other.
The sliding process also makes it easy to see that for all g ∈ G 2n and j ∈ We are now ready to give a proof of Theorem 4.2 (2) that does not rely on Hetyei's formula (1.5). This proof completes our path to (1.5).
But this is implicit in the work of Dumont and Randrianarivony [11] and stated explicitly in Corollary 10 (iii) of Randrianarivony [21].
5.3. The generating function formulas. In this subsection, we will use the bijections of the previous sections to derive generating function formulas for the characteristic polynomial of the homogenized Linial arrangement. To do so, we use a formula (due independently to Randrianarivony [21] and Zeng [31]) for the generating function of a multivariate polynomial that enumerates surjective staircases according to six statistics. • me(f ), the number of maxima j of f that are even (even maximum).
where c(σ) is the number of cycles of σ.
Proof. Let φ be the bijection of Lemma 5.2. By Lemma 5.2, We finally have the tools needed to prove (1.9), which is restated here.
It follows from (3.1) and (1.5) that equation (5.2) can be viewed as a unifying generalization of two of the four generating function formulas stated in Section 2.3. Indeed, when t is set equal to 0, equation (5.2) reduces to the Barsky-Dumont formula (2.5) for the Genocchi numbers. When t is set equal to 1, equation (5.2) reduces to the first Barsky-Dumont formula (2.6) for the median Genocchi numbers. We will see that the next result is a unifying generalization of the other two generating function formulas stated in Section 2.3.
Theorem 5.7. We have Equivalently, where L 2n is the geometric semilattice defined in Section 4.4.
Proof. By Theorem 4.14 and Lemma 5.2 we have that for all n ≥ 1, The result now follows from the Randrianarivony-Zeng formula given in Theorem 5.3.
Equation (5.4) can also be viewed as a unifying generalization of two of the generating function formulas given in Section 2.3. Indeed, by Corollary 4.16, when t is set equal to 0, equation (5.4) reduces to the Carlitz-Riordan-Stein formula (2.7) for the Genocchi numbers. When t is set equal to 1, equation (5.4) reduces to the second Barsky-Dumont formula (2.8) for the the median Genocchi numbers. It is interesting that the generating function formula (5.2) for the characteristic polynomial of Π Γ 2n reduces to one set of generating function formulas given in Section 2.3, while the generating function formula (5.4) for the characteristic polynomial of L 2n reduces to the other set.
When we set t = −1 in (5.3) we obtain a formula for the generating function of the median Genocchi numbers that is not listed in Section 2.3, namely . This formula might be known, but we have not seen it in the literature.
Remark 5.8. Many of the results in this section for V = [2n] are extended to general Ferrers graphs Γ V in a forthcoming paper [17].

Other permutation models
In this section, we show that a formula of Chung and Graham [6] for chromatic polynomials of incomparablity graphs yields an interpretation of the median Genocchi numbers in terms of yet another class of Dumont like permutations. This leads us to conjecture an analogous interpretation of the Genocchi numbers.
Before proving this theorem we state a few consequences. By setting t = −1 in (6.1) and using Hetyei's result (1.5), we obtain yet another combinatorial interpretation of the median Genocchi numbers. Corollary 6.2. For all n ≥ 1, h n = |{σ ∈ S 2n : σ has only even-odd drops}|.
By setting t = 0 in (6.1) and applying (3.1) we obtain yet another formula for the Genocchi numbers. A much nicer combinatorial formula for the Genocchi numbers analogous to that of Corollary 6.2 is given in the following conjecture. Conjecture 6.4. For all n ≥ 1, g n is equal to the number of cycles on [2n] with only even-odd drops.
We have verified this conjecture by computer for n ≤ 6. The following conjecture along with Theorem 4.2 implies both Corollary 6.2 and Conjecture 6.4. The first consequence follows by Theorem 4.9.
Conjecture 6.5. Let A be a finite subset of Z >0 . Then the number of cycles on A with only even-odd drops is equal to the number of D-cycles on A. Consequently, for all n ≥ 1, (1) if π ∈ Π Γ 2n then |µ Π Γ 2n (0, x)| is equal to the number of permutations on [2n] with only even-odd drops and cycle support π.
(2) for all j, the number of permutations on [2n] with j cycles and with only even-odd drops is equal to the number of D-permutations with j cycles.
We have verified (2) by computer for n ≤ 6.
Proof of Theorem 6.1. The incomparability graph inc(P ) of a finite poset P is the graph with vertex set P and edge set E = {{x, y} : x and y are incomparable in P }.
Given a poset P , we say that a permutation σ of the vertices of P has a P -drop (x, σ(x)) if x > P σ(x), and write d(P, k) for the number of permutations of P with exactly k P -drops. Next we observe that Γ 2n is the incomparability graph of a poset. Indeed, Γ 2n is the incomparability graph of the poset P 2n on [2n] with order relation given by x ≤ P 2n y if: • x ≤ y and x and y have the same parity • x < y, x is even and y is odd.
The Hasse diagram of P 6 and its incomparability graph Γ 6 are given below. Recall from Theorem 3.2 and Section 2.2 that where ch G (t) denotes the chromatic polynomial of a graph G. Chung and Graham [6,Corollary 7] prove that for any finite poset P , the chromatic polynomial ch inc(P ) has the following expansion: (6.2) ch inc(P ) (t) = |P |−1 k=0 d(P, k) t + k |P | .
Hence since Γ 2n = inc(P n ), Now note that (i, σ(i)) is a P 2n -drop of σ ∈ S 2n if and only if it is a drop of σ that is not even-odd. Hence d(P 2n , k) = d(2n, k) and the result holds.
We define an even-odd descent of a permutation σ to be a descent σ(i) > σ(i + 1) such that σ(i) is even and σ(i + 1) is odd. Since Foata's first fundamental transformation takes drops to descents, the word "drop" can be replaced by the word "descent" in the definition of d(2n, k) without changing the validity of Theorem 6.1 and Corollary 6.3. Similarly, Corollary 6.2 remains true if we replace the word "drop" with "descent".
There is a known permutation model for the Genocchi numbers that resembles the one we have been discussing. Kitaev and Remmel [16,15] conjectured and Burstein, Josuat-Vergès, and Stromquist [4] proved that the set of permutations in S 2n with only even-even descents has cardinality equal to g 2n−2 , where an even-even descent is defined in the obvious way.

Further results: Type B and Dowling arrangements
In a forthcoming paper [18], we will present a type B analog of our results and a Dowling arrangement generalization that unifies our results in types A and B. We define the type B homogenized Linial arrangement to be the hyperplane arrangement in R 2n , . . , n}. Our type B analog of (1.9) yields the following generating function formula for the number of regions r(H B 2n−1 ) of H B 2n−1 : (2n)!x n n k=1 (1 + 2k(2k + 1)x) .
For m ≥ 1, let ω m be the primitive mth root of unity e 2πi m . For m, n ≥ 1, we define the homogenized Linial-Dowling arrangement to be the hyperplane arrangement in C 2n given by H m 2n−1 := {x i − ω l m x j = y i : 1 ≤ i < j ≤ n, 0 ≤ l < m} ∪ {x i = y i : 1 ≤ i ≤ n}. By intersecting H m 2n−1 with the subspace y 1 = y 2 = · · · = y n = 0, one gets the Dowling arrangement A m n in C n . The intersection lattice of A m n is isomorphic to Π n+1 when m = 1 and to the type B partition lattice Π B n when m = 2. By introducing this Dowling analog of the homogenized Linial arrangement, we obtain unifying generalizations of the types A and B results. For instance, we obtain the m-analog of (1.9), , where (a) n,m = a(a − m)(a − 2m) · · · (a − (n − 1)m). Note that this reduces to (7.1) when we set m = 2 and t = −1.
We also obtain an m-analog of the formula g n = |µ L(H 2n−1 ) (0,1)| involving a well studied polynomial analog of the Genocchi numbers known as the Gandhi polynomials [13,11].
For a synopsis of some of our results on the homogenized Linial-Dowling arrangement, see the extended abstract [19, Section 4].