Some properties of the parking function poset

In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the framework of finite reflection groups. We present some enumerative and topological properties of this poset. In particular, we get a formula counting certain chains, that encompasses formulas for Whitney numbers (of both kinds). We prove shellability of the poset, and compute its homology as a representation of the symmetric group. We moreover link it with two well-known polytopes : the associahedron and the permutohedron.


Introduction
Parking functions are fundamental objects in algebraic combinatorics. It is well known that the set of parking functions of length n has cardinality (n + 1) n−1 , and the natural action of the symmetric group Sn on this set occurs in the deep work of Haiman [14] about diagonal coinvariants. Generalizations to other finite reflection groups lead to the parking space theory of Armstrong, Reiner, Rhoades [2,21].
The poset mentioned in the title was introduced by Edelman [13] in 1980, as a variant of the noncrossing partition lattice introduced by Kreweras [16] (hence the name noncrossing 2-partitions in [13]). One striking feature of Edelman's definition is that it really fits well in the noncrossing parking space theory mentionned above, so it seems that this overlooked poset can give a new perspective on recent results about parking functions.
Our goal is to obtain new enumerative and topological properties of Edelman's poset. Through various bijections, we will see that several variants of the same objects are relevant: • 2-noncrossing partitions (Section 1), • some pairs of a noncrossing partition together with a permutation (Section 1.3), • parking functions in the usual way (Section 1.4), • parking trees (Section 1.4).
The latter, which have the additional structure of a species, are defined on a set V as trees whose nodes are labelled by (possibly empty) parts of a weak partition of V and such that a node labelled by N as exactly N (possibly empty) children. They will be useful to write functional equations and get our enumerative results in Section 3. We draw below the different representations of the same element. What we get is the following formula counting chains of k elements whose top element has rank : ! kn S2(n, + 1).
A nice feature of this formula is that it encompasses a nice formula for Whitney numbers of the second kind at k = 1 (this one being obtained by Edelman), and one for Whitney numbers of the first kind at k = −1. We give a bijective proof of this formula in Section 3.2, introducing k-parking trees, which generalise parking trees and encode chains of k elements in the parking posets.

.1 Set partitions and noncrossing partitions
For integers i ≤ j, we will denote i; j := {i, i + 1, . . . , j}. Let Πn denote the lattice of set partitions of 1; n , endowed with refinement order. Note that we take the convention that the minimal element is { 1; n } (the set partition with one block, denoted 0n), and the maximal element is {{1}, {2}, . . . , {n}} (the set partition with n blocks, denoted 1n). A set partition π ∈ Πn is noncrossing if there exists no i < j < k < l such that i, k ∈ B1 and j, l ∈ B2 where B1 and B2 are two distinct blocks of π. Endowed with the refinement order, noncrossing partitions of 1; n form a lattice denoted N Cn, first defined by Kreweras [16]. It is also a (full) subposet of Πn. The cardinality of N Cn is the nth Catalan number, defined as Cn := 1 2n + 1 2n + 1 n .
More generally, the zeta polynomial of N Cn is given by the Fuß-Catalan number C n is also the number of k-trees with n internal vertices, which are by definition rooted plane trees where each of the n internal vertices has ki descendant for some i ∈ N. This can be proved by showing that the generating function T (z) of k-trees satisfy To end this section, we note the following facts about order ideals and filters in N Cn. It is easy to see that the order filter containing all elements above some π ∈ N Cn is isomorphic to the product N Ci 1 × N Ci 2 × · · · , where i1, i2, . . . are the block sizes of π. As the antiautomorphism K sends order ideals to order filters, the order ideal containing all elements below some π ∈ N Cn is isomorphic to the product N Ci 1 × N Ci 2 × · · · , where i1, i2, . . . are the block sizes of K(π).
Proof. Let B ∈ π , and write B = j i=1 Bi where Bi ∈ π. By the second condition in the definition of the order, the only possible choice is to define λ (B ) as j i=1 λ(Bi), and ρ as the set of all such λ (B ). This definition ensures that blocks of ρ are union of blocks of ρ (so ρ ≤ ρ in Πn), and (π , ρ , λ ) satisfies the required properties.
The following two propositions contains simple consequences of the definitions or of the previous lemma. Proofs are straightforward.
In particular, for each maximal element φ ∈ 2 Πn, the interval of 2 Πn bounded by the minimal element and φ is isomorphic to N Cn. Thus, 2 Πn can be seen as n! copies of N Cn, which blend in a rather nontrivial way.
Similarly, the following result is straightforward. This naturally extends the remark about the structure of order filters in N Cn. Proposition 6. The order filter of 2 Πn containing all elements above (π, ρ, λ) is isomorphic to a product 2 Πi 1 × 2 Πi 2 × · · · , where i1, i2, . . . are the block sizes π.
Next, we consider the poset2 Πn obtained from 2 Πn by adding a new maximal element1. It is thus a bounded poset (it has one minimal element and one maximal element).
Using the previous lemma, it is natural to consider the sequence (η φ (k) ∩ η ψ (k)) 1≤k≤n . Each element is either empty or a block of γ. We distinguish several cases.
In the second case, assume that there are b ∈ γ and indices k1, . . . , k such that η φ (ki) ∩ η ψ (ki) = b (for 1 ≤ i ≤ ), and > |b|. By way of contradiction, assume that there is a maximal element χ above φ and ψ. Then, the sets ηχ(ki) are pairwise distinct singletons (by maximality of χ), and they are included in b (by the previous lemma, since χ ≥ φ and χ ≥ ψ). This is not possible as > |b|, and it follows that1 is the least upper bound of φ and ψ.
In the remaining cases, we show that there exists χ ∈ 2 Πn such that ηχ(k) = η φ (k) ∩ η ψ (k) for 1 ≤ k ≤ n. Indeed, each b ∈ γ appears at most |b| times in the sequence (η φ (k) ∩ η ψ (k)) 1≤k≤n . Since the elements in the sequence are nonempty, a counting argument shows that each b ∈ π appears exactly |b| times, as needed. By the previous lemma, χ is a least upper bound of φ and ψ.

The parking space
The goal of this section is to describe a natural action of the symmetric group Sn on 2 Πn. It leads to a connection with parking spaces as defined by Armstrong, Reiner and Rhoades in [2], and to an alternative definition of 2-noncrossing partitions as some pairs (π, σ) ∈ N Cn × Sn.
The natural action of Sn on 1; n can be extended to various other combinatorial sets by the rule σ · X = {σ · x : x ∈ X}. This extended action automatically respects properties of sets such as inclusion, disjoint unions, etc. In particular, this gives a natural action of Sn on Πn which respects the poset structure. Proposition 9. There is an action of Sn on 2 Πn defined by where in σ • λ we identify σ with its action on set partitions. This action preserves the order relation of 2 Πn, so that it can be extended to an action on the chains of 2 Πn.
Proof. This is a direct application of the properties of the action of Sn on Πn.
Note that (π, ρ, λ) and (π , ρ , λ ) are in the same orbit if and only if π = π , so that the orbits are naturally indexed by N Cn.
For π ∈ N Cn, we denote by Sn(π) the set of σ ∈ Sn such that σ · b = b for any block b ∈ π. Then Sn(π) is a parabolic subgroup (it is conjugated to a Young subgroup). Via left multiplication, the quotient Sn/Sn(π) is acted on by Sn.

Proposition 10.
There is a Sn-equivariant bijection between 2 Πn and pairs (π, σ · Sn(π)) where π ∈ N Cn and σ ∈ Sn (where it is understood that Sn acts on the second element in the pair via left multiplication on the cosets).
By taking minimal length coset representatives in each coset, we immediately obtain the following.
We refer to [18] for characters of the symmetric group, symmetric functions, and the Frobenius characteristic map χ relating these two notions. The character of Sn/Sn(π) is Ind Sn Sn(π) (1), the trivial character of Sn(π) induced to Sn. It is such that where h λ is the homogeneous symmetric function, and λ is the integer partition obtained by sorting block sizes of π. (One can replace Ind Sn Sn(π) (1) with hπ = h λ to work with symmetric functions rather than characters in what follows, at the condition of being aware that the evaluation of a character Ψ at some permutation σ is χ(Ψ)| p λ z λ where the integer partition λ is the cycle type of σ.) Proposition 12. The character of the action of Sn on the orbit of (π, ρ, λ) ∈ 2 Πn is Ind Sn Sn(π) (1). Moreover, character of the action of Sn on 2 Πn is π∈N Cn Ind Sn Sn(π) (1).
Proof. From Proposition 10, we see that the orbit of (π, ρ, λ) ∈ 2 Πn is isomorphic (as a Sn-set) to Sn/Sn(π). The first statement follows. The second one is obtained by summing over the orbits in 2 Πn.
The character in (3) coincides with that of the noncrossing parking space from [2]. The evaluation of this character is given by σ → (n + 1) z(σ)−1 where z(σ) is the number of cycles of σ. Let us present a more general version of this character, defined by Rhoades [21]. He considered a character which comes from the action of Sn on chains of 2 Πn (in particular, the poset 2 Πn appears implicitly in [21]). Note that via Lemma 2, the orbit of a chain φ1 ≤ · · · ≤ φ k is isomorphic as a Sn-set to the orbit of φ k .  n denote the character of Sn acting on k-chains φ1 ≤ · · · ≤ φ k of 2 Πn, so that: Then, for any σ ∈ Sn, we have: This character Park (k) n will be called a zeta character, as it encompasses both the zeta polynomial of 2 Πn (see Section 3) and its character as a Sn-set. There is an alternative expression for Park (k) n , that will be useful below. Lemma 14. We have: Proof. Let (π1, ρ1, σ1) ≤ · · · ≤ (π k , ρ k , σ k ) be a k-element chain in 2 Πn. As noted above, the action of Sn on its orbit is isomorphic to the action on the orbit of (π k , ρ k , σ k ). This follows from Lemma 2 and the fact that the group only acts on the second and third elements of each triple. Therefore, the character of this action is Ind Sn Sn(π k ) (1). For a given π ∈ N Cn, the number of k-element chains having π as their top element is equal to indeed this follows from the result on the structure of order ideals in N Cn and knowing its zeta polynomial. Thus, by summing over π we get the desired formula for Park

Classical parking functions
In this section, we use the following terminology. A weak composition of an integer n ≥ 0 is a finite sequence of nonnegative integers, such that the sum is n. Similarly, a weak composition of a set X is a finite sequence of pairwise disjoint sets (that are possibly empty), such that the union is X.
We first need a lemma that gives an alternative encoding of noncrossing partitions.

Lemma 15.
There is a bijection between N Cn and weak compositions (a1, . . . , an) of n such that j i=1 ai ≥ j for any j ∈ 1; . . . , n . It is given by: Proof. We only give a quick description of the inverse bijection, and details are left as an exercise. First, there is a bijection between compositions as in the proposition and Lukasiewicz paths of length n (which are, by definition, lattice paths in N 2 from (0, 0) to (n, 0) with steps of the form (1, i) for i ∈ N ∪ {−1}). Explicitely, to (a1, . . . , an) we associate the lattice path with steps (1, a1 − 1), . . . , (1, an − 1). We build a noncrossing partition from a Lukasiewicz path as follows: • if the ith step is (1, 0) then there is a block {i}, • if the ith step is (1, j) with j > 0, and the j facing steps (1, −1) have indices i1, . . . , ij, then there is a block {i, i1, . . . , ij}.
The notion of facing steps in a path is illustrated by the horizontal arrows in Figure 2  The following is the classical definition of a parking function.
A parking function of length n can be rewritten as a weak set composition (A1, . . . , An) of 1; n satisfying k i=1 |Ai| ≥ k for any 1 ≤ k ≤ n. The correspondence is done by letting Ai be the set of positions of i in the parking function: Ai = {j | wj = i}. In this reformulation, the action of Sn on parking functions naturally extends the action of Sn on subsets of 1; n .
By seeing 2-noncrossing partitions as "enriched" noncrossing partitions, the previous lemma can be extended to give the following: There is a Sn-equivariant bijection between 2 Πn and parking functions of length n, defined by the following property: the image of (π, ρ, λ) ∈ 2 Πn is w1 . . . wn such that ∀B ∈ π, ∀i ∈ λ(B), wi = min B.
Equivalently (in terms of weak set compositions), this bijection sends (π, ρ, λ) to (A1, . . . , An) such that: Proof. It is straightforward to check that the two formulations are equivalent, so we only prove the second one. It is known that both sets have the same cardinality. To describe the inverse bijection, first note that the sequence (|A1|, . . . , |An|) is the weak composition of n corresponding to π via the bijection in Lemma 15, since |λ(B)| = |B| for B ∈ π. Second, note that ρ = {Ai | Ai = ∅}. Eventually, λ is such that λ(B) = A min(B) . The map in the proposition is thus injective, and bijective.
For example, this maps sends the 2-noncrossing partition in (2) to the parking function 41112712, with 1s in position 2347, etc.
It is worth making explicit what are the parking functions corresponding to (π, π, id), because these are orbit representatives. We will not use this result hereafter, and the proof is left as an exercise.
Proposition 18. The bijection from the previous proposition sends the elements (π, π, id) ∈ 2 Πn to parking functions w1 . . . wn such that: • wi ≤ i for all i ∈ 1; n , • w1 . . . wn is lexicographically maximal among parking functions in the same orbit and satisfying the previous condition.
Remark 19. It seems there is no clear and simple way to describe the poset structure of 2 Πn directly in terms of words w1, . . . , wn as in Definition 16. Indeed, the covering relation there are given by choosing a letter which appears more than twice and increasing it. The main difficulty lays in the value to which it is allowed to increase the letter.

Species and parking trees
We refer to [10] for the notion of combinatorial species and operations on them.
Definition 20. A parking tree on a set L is a rooted plane tree T such that: • internal vertices of T are labelled with nonempty subsets of L, which form a set partition of L, • leaves are labelled by empty sets, • each vertex has as many children as elements in its label.
The species of parking functions (or parking species), denoted P f , is the species which associates to any finite set L the set of parking trees on L as above.
Note that a parking tree on L has #L edges.  Proof. We construct a family of bijections φn : Pn → PT n by induction on n ≥ 0. When the index of φ k will be obvious, we will just omit it.
If n = 0, Pn contains only one pair ({}, id ∈ S0) and PT n contains only one tree, which is the empty parking tree. The map φ0 is defined by associating the elements in both sets.
Suppose that we have constructed bijections φ k for k < n. We now construct φn. Let us consider a pair p = (π, σ) in Pn. The noncrossing partition π admits the following decomposition, each "arch" of the non-crossing partition deliminating a non-crossing partition: • a part E (the first of π) of size • (possibly empty) non-crossing partitions π1, . . . , π of size strictly less than n By induction hypothesis, we can associate to each (π k , σ |π k ) a parking tree T k . We now construct a tree T , whose root is given by σ · E and has children given from left to right by T1, . . . , T k . Moreover it is a parking tree as the children of the root are parking trees and the root has a label of cardinality and exactly children. Proof. First, let us recall that parking functions can be represented as (weak) set compositions satisfying some properties, using the same bijection as in Proposition 17.
To a parking function f = w1 . . . wn of length n with k := max(wi, 1 ≤ i ≤ n) can be associated a weak set composition of {1, . . . , n}, ϕ(f ) = (E1, . . . , E k ) given by Ei = {j|wj = i}. The property of f being a parking function can immediately be translated into m i=1 |Ei| ≥ m, for 1 ≤ m ≤ k. Reciprocally, to a weak set composition of {1, . . . , n} c = (E1, . . . , E k ) satisfying m i=1 |Ei| ≥ m, for 1 ≤ m ≤ k, can be associated a word of length n, ψ(c) = w1 . . . wn, defined by wi = j for any i ∈ Ej. The conditions on the cardinality of the sets Ei ensure that the word ψ(c) is indeed a parking function.
To prove the above lemma, we will use the representation of parking functions in terms of weak set compositions.
From weak set compositions to parking trees : Let us consider a weak set composition of 1; n , c = (E1, . . . , En+1) satisfying m i=1 |Ei| ≥ m, for 1 ≤ m ≤ n (it is always possible to consider weak set compositions of 1; n of length n + 1 up to adding some empty sets at the end of the composition). We consider the parking tree T (c) obtained recursively as follows: • the vertices of T (c) are labelled by the sets Ei (hence T (c) has n + 1 vertices) • the root is labelled by E1 (which is non-empty thanks to the hypothesis) • if Ei is not empty, Ei+1 is the leftmost child of Ei • if Ei is empty, Ei+1 is grafted as the rightmost child of the closest ancestor Ej of Ei (j ≤ i) having strictly less than |Ej| children.
Here, we consider that a node is the last child of its parent if it is the pth child and the label of its parent as p elements.
We have to check that: 1. it is always possible to graft a vertex as mentioned above, 2. for each internal vertex, the cardinality of its label coincides with the number of its children.
For the first point, we have to check that such an ancestor Ej exists. By construction, the child of a node N is introduced only when the lineage of its older siblings is full. If every Ej ancestor of Ei is the last child of its parent, it means that every Ej, for j ≤ i has exactly |Ej| children. Hence the number of vertices of the constructed tree is: • i, since vertices are labelled by E1, . . . , Ei • i j=1 |Ej| + 1, since every vertex but the root is the child of another vertex. By hypothesis, i j=1 |Ej| + 1 > i: we then get a contradiction and the existence of Ej is proven. For the second point, the grafting algorithm ensures that an internal vertex does not have more children than the number of elements in its label. Denoting by ci the number of children of vertex Ei, we have ci ≤ |Ei|. Moreover, every Ei are grafted so the tree has n + 1 nodes, i.e. n + 1 = n+1 i=1 ci + 1 ≤ n+1 i=1 |Ei| + 1 ≤ n + 1. Hence every inequalities are equalities and every Ei has exactly |Ei| children. As it is clear that the obtained graph is a rooted planar tree, T (c) is then a parking tree.
From parking trees to weak set compositions : Reading nodes of a parking tree t in the prefix order gives a weak set composition C(t) = (E1, . . . , En+1). The node labelled by Em is the mth visited node, for 2 ≤ m ≤ n + 1. When we visit Em for the first time, all visited nodes (Em included) are E1 or descendants of some Ei, for 1 ≤ i < m. This leads to the inequality m−1 i=1 |Ei| + 1 ≥ m, hence the set composition satisfies the desired hypothesis.
Finally, the map C is the inverse bijection of T . Indeed, the grafting order of the map T corresponds to the prefix order. See Figure 3 for an example of these bijections.
Remark 25. Suppose that L (and consequently, any of its subsets) is endowed with a total order. Since parking trees are planar, we can for each vertex canonically associate each outgoing edge to an element in the vertex. In this case we can associate each internal vertex that is not the root to an element of L from which it descends. This defines a function f : L → L, that is not defined on the elements of the root, constant on each node of the tree, and which is nilpotent. Conversely, for each nilpotent partial function f on L, we can recursively build a parking tree: the root is the set of elements on which f is not defined, and for each i ∈ L, its descendant is labelled by f −1 (i). Note that if | Im(f )| = r, then the resulting tree has r + 1 vertices. This bijection falls in line with Laradji and Umar's in [17], in which they enumerate partial nilpotent functions with a fixed image set of size r.
The covering relations on parking trees corresponding to those of the 2-partition poset are then given as follows. From a parking tree T , another one U such that T U is obtained from T by a sequence of operations, represented on Figure 4: • choose a vertex A and partition it into two (non empty) sets A1 and A2, • deconcatenate the list of its (possibly empty) subtrees into three lists L1, L2 and L3, such that L1 is non empty and L2 and A2 have the same cardinality, • remove from the tree the elements of A2 and L2 • add the elements of A2 to the rightmost leaf of A1 in L1 • add L2 as the list of children of A2.
For the leftmost tree in Figure

Face poset of the permutahedron
We study in this subsection the restriction of parking poset to elements of the poset such that the associated non-crossing partitions are interval partitions. In terms of parking trees, it corresponds to parking trees which are right combs. We detail below this bijection: Lemma 26. The following objects are in bijection: • 2-partitions whose associated non-crossing partition is an interval partition in k parts, • parking trees which are right combs with k inner vertices, • composition of k non-empty sets.
Proof. We first consider 2-partitions whose associated non-crossing partitions are interval partition. Let us consider such a 2-partition (π, ρ, λ). If B1, . . . , B k are the blocks of π, ordered according to their minimal elements and bi the size of Bi,we then have max Bi < min Bi+1, for any i between 1 and k − 1. Moreover, the associated parking function has its value in Let us show by induction that the associated parking tree is a right comb. If k = 1, the 2-partition is trivial and has only one part. The associated parking tree has only one node containing all the labels {1, . . . , n} : it is a right comb. Let us now suppose the property true up to k−1 and consider a 2-partition with k parts (π, ρ, λ), π being an interval non-crossing partition. Considering the non-crossing partitioñ π = {B1, . . . , B k−1 }, we can associate to it a 2-partition (π, {λ(B1), . . . , λ(B k−1 )}, λ) on which we can apply the induction hypothesis: the associated parking tree is a right comb. To build the parking tree associated with π, we have to add the vertex associated with the value of B k and containing labels of λ(B k ). As the non-crossing partition is an interval partition, the value associated with B k is k−1 i=1 bi. The only way for the node associated with B k to be associated with this value is to graft it on the rightmost leaf of the tree: we then get a right comb. Let us denote by φ the map sending 2-partitions whose associated non-crossing partition is an interval partition to parking trees which are right combs.  When reading nodes on the right branch of a parking tree which are a right comb, we get a composition of sets which are known to be equivalent to surjection. We denote this map by ψ. This map is clearly a bijection. If we denote by C = (C1, . . . , C k ) the composition, the ith set of the composition is the Cith child in the parking tree of the i − 1th set of the composition, the other children being empty.
To finish the proof, we now have to describe the map µ which associates to a composition of sets C = (C1, . . . , C k ) a 2-partition whose associated non-crossing partition is an interval partition. The set partition is obtained by forgetting the order of the composition. The non-crossing partition is {{1, . . . , The map λ is given by associated to the part which is the ith of C the ith part of the non-crossing partition.
The map φ • µ • ψ sends a parking tree which is a right comb to a parking tree which is a right comb. Moreover, the kth node of the tree is send by ψ to the kth set of the composition which is send by µ to the kth part of the noncrossing partition, itself send by φ to the kth node of the parking tree. Hence, φ • µ • ψ is the identity and φ and µ are also bijections.
Compositions of sets are known to label the faces of the permutahedron. The link between parking trees and the permutahedron goes deeper as the order on parking tree studied in this article is exactly the inclusion order of the permutahedron.
Proposition 27. The subposet of 2 Πn of parking trees which are right combs is isomorphic to the face poset of the permutahedron.
Proof. The allowed covering relations are when a vertex A = A1 A2 is split into A1 and A2, with A2 grafted on the rightmost leaf of the |A1|th child of A1 and have children of A of indices between |A1| + 1 and |A1| + |A2| as children. As all children but the last one are empty, the covering relations can be simplified in choosing a node A with rightmost child C and replacing the right comb A − C by a right comb A1 − A2 − C, where P − C is a shortcut for "C is the rightmost child of P in the tree". This covering relation translating in term of composition of set is exactly the one of the face poset of the permutahedron. In this order, the composition (C1, . . . , C l−1 , C l ∪ C l+1 , C l+2 . . . , C k ) is covered by the composition (C1, . . . , C l−1 C l , C l+1 , C l+2 , . . . , C k )

Shellability of the parking functions poset
Recall that2 Πn is the bounded poset obtained by adding a new maximal element1 on top of 2 Πn. The goal of this section is to build a shelling of this poset. This shellability property of 2 Πn has geometric consequences, in particular it will be used to study its homology in Section 4. We refer to [8,22] for general notions of combinatorial topology (shellability, EL-labelings, etc.)

Construction of the shelling
Let P be a ranked poset of length n (this means that all maximal chains of P contains n + 1 elements). A maximal chain of P will be denoted p = (pi) 0≤i≤n , where it is understood that p0 is the minimal element, pn is the maximal element, and ∀i ∈ 0; n − 1 , pi pi+1.
Definition 28. A shelling of P is a total order < on its maximal chains, such that if p < p, there exists a maximal chain p such that (seeing chains of the poset as subsets): Let us mention that more generally, shellings can be defined for simplicial complexes. The definition above correspond to shellings of the order complex Ω(P ) (see Section 4).
For x ∈ P , its set of upper covers is: Consider a family (≺x)x∈P where each ≺x is a total order on Up(x). This data gives rise to a total order < lex on maximal chains of P , using lexicographic comparison: a chain p = (p i ) 0≤i≤n precedes another chain p = (pi) 0≤i≤n if p j+1 ≺p j pj+1 where j is the minimal index such that p j+1 = pj+1. This kind of structure is natural in the context of lexicographic shellability. For example, via the notion of recursive atom ordering [22] there is a criterion on (≺x)x∈P that ensures that < lex is a shelling of P . However we will use another method. Suppose that the poset P is endowed with an edge-labeling, i.e., a function λ(x, y) defined for each cover relation x y and taking values in a totally ordered set. This gives rise to total orders ≺x as above via the rule y ≺x y iff λ(x, y) ≤ λ(x, y ) (we do not discuss here the possibility that y = y and λ(x, y) = λ(x, y )). An interesting class of labelings whose associated lexicographic order are shellings are EL-labelings. They are defined as labelings such that each interval contains a unique strictly increasing chain, and it is the lexicographically minimal one (see [22] for details). Björner and Edelman showed that N Cn admit such an EL-labeling (see [8]). Moreover, there exists an EL-labeling λ having the additional property that ∀y, y ∈ Up(x), we have y = y ⇒ λ(x, y) = λ(x, y). Explicitly, we can take λ(x, y) =x −1ȳ (note that x y in N Cn implies thatx −1ȳ ∈ Sn is a transposition, and we order them with the lexicographic order on pairs (i, j) such that i < j). We will use such an EL-labeling λ of N Cn as an ingredient to show that 2 Πn is shellable.
Concretely, ci(σ) is the number of integers smaller than i on its right in σ. For instance, γ(15324) = 30100. In the sequel, it will be convenient to see elements of 2 Πn as pairs in N Cn × Sn, as explained in Section 1.3. Whenever an element of 2 Πn is written as a couple rather than a triple, it is understood that we take this convention.
Πn with rk(φ) < n − 1, we define a total order ≺ φ on Up(φ) by: where we use the lexicographic order to compare γ(τ ) and γ(τ ). If rk(φ) = n − 1, it has a unique cover in2 Πn, namely the maximal element1. In this case, the total order on Up(φ) is the obvious one.
The proof relies on the following lemma, which will be proved in the next sections and is illustrated on Figure 7.
Πn such that x y z, x y , and y ≺x y. Then: • either there exists y ∈ 2 Πn such that x y z and y ≺x y, • or there exists z ∈ 2 Πn such that y z ≤ y ∨ z and z ≺y z.
Proof of Theorem 31. Let p = (pi) 0≤i≤n and p = (p i ) 0≤i≤n be two distinct maximal chains of2 Πn such that p < lex p. Our goal is to find p = (p i ) 0≤i≤n as in Definition 28. Let x = pi = p i where i is the minimal index such that pi+1 = p i+1 . Let y = p i+1 , y = pi+1, and note that p < lex p means that y ≺x y. Let z = pi+2. If i = n − 2, the maximal chains p and p only differ in rank n − 1. So we can take p = p and it satisfies the requirements in the definition of a shelling. Otherwise, we have i < n − 2, so z <1 and we are in situation to use Lemma 32. We distinguish two cases.
• If there exists y ∈ 2 Πn such that x y z and y ≺x y, we define a maximal chain p by replacing y with y in p. By construction, it satisfies the requirements in the definition of a shelling.
• Otherwise, there exists z ∈ 2 Πn such that y z ≤ y ∨ z and z ≺y z. Let j be the minimal integer such that p j = pj and j > i. We have j ≥ i + 3 (it cannot be equal to i + 2 because that would mean y z, and this situation was already ruled out in the previous case). Note that y and z are both below pj, so y ∨ z ≤ pj, so z ≤ pj. Define p = (p k ) 0≤k≤n by: the elements strictly between p i+2 and p j are defined arbitrarily with the constraint that p is a maximal chain.
By construction, we can check: p < lex p (this holds because z ≺y z), p ∩ p p ∩ p (the inclusion holds because p k = p k if i < k < j , it is strict since y is in the second intersection but not in the first one). By iterating the argument, we can eventually construct a maximal chain satisfying the requirements in the definition of a shelling. Now, it remains only to prove Lemma 32.
Remark 33. It would be interesting consider the following problem. Given a finite poset P and a family (≺x)x∈P as above, we have showed that P is shellable if it satisfies a criterion as in Lemma 32 (if P is not a lattice, the condition z ≤ y ∨ z should be replaced with: ∀u ∈ P , u ≤ y and u ≤ z implies u ≤ z ). It is natural to ask whether this new criterion for shellability can be compared with other ones such as EL-shellability or CL-shellability. From definitions, it is clear that an order on atoms of any intervals in the poset which is a recursive atom ordering satisfies Lemma 32 (more precisely the second case). This criterion is then weaker or equivalent to CL-shellability.
Remark 34. We wrongly asserted in [12] that the orders (≺x) form a recursive atom ordering, a technical condition that characterizes CL-shellability. To see that it is not the case, consider x the minimal element together with: It is easily checked that y is the ≺x-minimal element among the 2 lower covers of z. However, there exists z ∈ Up(y) such that z ≺y z , and z covers an element y such that y ≺x y. Take

A few lemmas
Lemma 35. The statement in Lemma 32 holds when 2 Πn is replaced with N Cn.
Proof. Recall that λ is an EL-labeling for N Cn (the one in [8] for instance). We only use the existence of λ, and the fact that for distinct elements y, y ∈ Up(x) we have λ(x, y) = λ(x, y ). Let x, y, y , z ∈ N Cn as in the statement of Lemma 32. If λ(x, y) ≥ λ(y, z), it means that (x, y, z) is a decreasing chain of [x, z]. By properties of EL-labelings, it is not the lexicographically minimal chain of [x, z], so there exists y with x y z and λ(x, y ) < λ(x, y). Now, assume λ(x, y) < λ(y, z). Consider the maximal chain (p0, p1, . . . ) of [y, y ∨ z] which is strictly increasing. If p1 = z, by adding x at the beginning we obtain a strictly increasing chain from x to y ∨ z going through y and z. So this chain is lexicographically minimal. But this is a contradiction: there is a maximal chain from x to y ∨ z going through y , and it is lexicographically smaller than any chain going through y. We thus have p1 = z. Now, let z = p1. We have λ(y, z ) < λ(y, z) and z ≤ y ∨ z. This completes the proof.
Reciprocally, assume ∀i ∈ n − k + 1; n , i ∈ η φ (i). By induction on j ∈ 0; k , we prove that for all i ∈ n − j + 1; n , σ(i) = i. The case j = 0 is clear, and the details of the induction are left to the reader.
• N (φ, φ ) is the block of π that is split to obtain π .
Proof. This is clear upon inspection.
Lemma 43. Let x, y, y ∈ 2 Πn such that x y, x y , and N (x, y) = N (x, y ). Then for any u, v such that x ≤ u v ≤ y ∨ y , we have m(u, v) ≤ max(m(x, y), m(x, y )).
Proof. We first prove the case where x =0. We clearly have m(u, v) ≤ n − p0(v). On the other side, p0(v) ≥ p0(y ∨ y) = min(p0(y), p0(y )) by Lemma 40. So m(u, v) ≤ max(n − p0(y), n − p0(y )). Using x =0, we easily get m(x, y) = n − p0(y) and similarly for y . We thus get the desired inequality. Now, consider the general case (x =0). Let b = N (x, y). All elements in the interval [x, y ∨ y ] are obtained from x by splitting the block b. We can discard other blocks of x to identify the interval [x, y ∨ y] with an initial interval in 2 Π n with n = #b (initial means that the bottom element of this interval is the bottom element of the poset). Via this identification, it is straightforward to see that the quantities m(x, y), m(x, y ), etc., are changed via a relabelling which preserves their relative order. We thus get the result from the case x =0.
By a way of contradiction, assume m(φ, χ) > m(χ, ψ), so that m(χ, ψ) < n − p0(ψ). It remains to show that there exists χ such that φ χ ψ and m(χ , ψ) = n − p0(ψ), indeed it easily follows that χ ≺ φ χ. To build χ , denote n − p0(ψ) = l. It means that in ψ, the label l has a smaller label on its right, and labels l + 1, . . . , n are on the final part of the permutation. By merging in ψ the block containing the label l with the block containing the label right to l, we obtain χ which has the desired properties by construction.

Proof of Lemma 32
From hypotheses of Lemma 32, it follows that y, y and z satisfy γ(y ) ≤ γ(y) ≤ γ(z). We decompose the proof into several steps: the first three steps deals with equality cases in the previous relation. From Case 4, we will presume that γ(x) ≤ γ(y ) < γ(y) < γ(z)

Case 1: γ(y ) = γ(y) = γ(z)
By Lemma 37, γ(y ) = γ(z) is also the code of y ∨ z. So all element in the [x, y ∨ z] have the same code, except possibly x. It follows that the natural projection 2 Πn → N Cn sends [x, y ∨ z] to an interval in N Cn, in a way which is compatible with the orders ≺x, ≺y, etc. So this case follows from Lemma 35.
If there exists y ∈ 2 Πn such that x y z and y ≺x y, this case is settled. Assume otherwise, so that we have m(x, y) ≤ m(y, z) by Lemma 44.
We thus have m(x, y ) < m(x, y) ≤ m(y, z), so that m(x, y ) < m(y, z). Besides, m(x, y ) = m(y, y ∨ y ) by Lemma 42. We thus have m(y, y ∨ y ) < m(y, z). So, with z = y ∨ y we have γ(z ) < γ(z), and z ≺x z. It is straightforward to see that the non-inclusion N (y, z) ⊂ N (x, y) means: z = y ∨ȳ where x ȳ and N (x, y) = N (x,ȳ). We are thus in the situation described in Lemma 42. We can assume γ(x) < γ(y) (see the previous case), and it follows that m(x, y) = m(x,ȳ).
If γ(ȳ) < γ(y), we can take y =ȳ and we have x y z, y ≺x y so that this case is settled. Assume otherwise, so that m(x, y) < m(x,ȳ). By Lemma 42, we have m(x,ȳ) = m(y, z). So we have m(x, y) < m(y, z).
Let z be such that y z ≤ y ∨ y. By Lemma 43, we have m(y, z ) ≤ m(x, y). We thus have m(y, z ) ≤ m(x, y) < m(y, z), and m(y, z ) < m(y, z). It follows that γ(z ) < γ(z) and z ≺y z.

Case 8 (last case)
As Cases 6 and 7 are ruled out, we assume N (x, y) = N (x, y ) and N (y, z) ⊂ N (x, y). It follows that only one block of x is involved, i.e., all elements in [x, y ∨ z] are obtained from x by splitting this block. We assume x =0, as we can focus on this case by ignoring the other blocks (it is easily seen that discarding the other blocks is compatible with the orders ≺ φ ).
As Case 5 is ruled out, we assume p0(y) = p0(z), and denote this quantity n − l with l ∈ 1; n (l > 0 since we can assume γ(x) < γ(y), as was done in Case 6). This means c l (y) (resp. c l (z)) is the first non-zero element of γ(y) (resp. γ(z)). In particular, the label l has a strictly smaller label to its right in y (resp. z).
If c l (y) = c l (z), we define y from z by merging the block containing the label l with the block containing the label to the right of l. We have γ(y ) < γ(y) as p0(y ) ≥ p0(z) = p0(y) = n − l and c l (y ) < c l (z) = c l (y), so y ≺x y and this case is settled. We thus assume c l (y) = c l (z), so that c l (y) < c l (z). Now, let b1 = N (y, z), and let b2 denote the other block of y (since y has rank 1, it has only two blocks). We claim that b2 is not a block of y ∨ z. Indeed, otherwise it would also be a block of y , so that y and y have the same underlying noncrossing partition, and only their codes differ. This is impossible, because it would imply y ∨ y =1 (and this is already ruled out by Case 4).
Since b2 is not a block of y ∨ z, there exists z such that y z ≤ y ∨ z and N (y, z ) = b2. Note that p0(z ) ≥ p0(y ∨ z) = p0(z) = n − l.
Besides, the position of l in z is the same as in y, since l is not the label in y of an element of N (y, z ) = b2. Indeed, since c l (y) < c l (z), l is the label in y of some element in N (y, z) = b1. We thus get c l (z ) = c l (y).
In particular, setting k = 1 we see that noncrossing 2-partitions and parking functions are equienumerous. Another result from [13] is that for 0 ≤ k ≤ n − 1, the number of elements of rank in 2 Πn, called the th Whitney number of the second kind, is where S2(n, k) are the Stirling numbers of the second kind. Our main motivation for counting chains in the parking function poset comes from the relation with the homology of the poset, as explained Section 4. We will get in particular a nice formula for Whitney number of the first kind. This will follow from Corollary 50.

Species and generating functions
Proposition 45. The species P f of (non-empty) parking trees satisfies: where E k (V ) = δ |V |=k K (with K the ground field) and the species of non-empty sets is E + := This is obtained from the tree structure, and accordingly we can write an equation in terms of symmetric functions for the Frobenius image of the characters of P f .

Remark 46. This equation cannot be simplified, as for generating series, in
where E + is the species of non-empty sets and X is the singleton species. Indeed, this equation is not true when considering the action of the symmetric group usually defined on parking function. The action of the symmetric group associated with this equation would not only exchange letter in the associated word but also move values. For instance, the parking function 112 would be send by the transposition (12) on the parking function 113.
The set of weak k-chains of parking functions on I is the set PF I k of k-tuples (a1, . . . , a k ) where ai are parking functions on I and ai ≤ ai+1. The species which associates to any set I the set PF I k is denoted by C l k,t . Theorem 47. We have: where C l,p k−1,t (V ) = δ |V |=p C l k−1,t (V ) on any set V of size p. In terms of generating functions, this translates to: Proof. This decomposition is obtained by separating the root, of size p, and elements in the chain obtained from its splitting on one side and the subtree attached to its root in the minimal element of the chain and elements obtained from them on the other side. This is made possible by the fact that the splittings of the root and of its descendants do not mix. The chain obtained by restricting to the root and its splitting is equivalent to a chain of length k − 1 as the minimal element of the chain can easily be recovered by merging all the vertices.
Note that from the functional equation in terms of species, it is theoretically possible to find a formula for the character of Sn acting on the chains as above. Here we only consider the enumerative result.
Remark 48. In terms of usual generating series, the computations in terms of generating series are the same as if we considered chains in a poset of forests of rooted non planar trees. In such a poset, the corresponding species would satisfy the following equation, denoting by F l k,t the species of large chains: Finding such an order is however still an open question.
From Theorem 47, we show by induction the following formula, for any 1 ≤ i ≤ k, which leads to Corollary 49: Corollary 49. The generating function of weak k-chains in the 2-partition posets satisfies: From (12), C l k,t is the compositional inverse of ln(1 + x) (1 + tx) −k . By using Lagrange inversion, it is possible to extract the coefficients and we get: Corollary 50. The number of chains φ1 ≤ · · · ≤ φ k in 2 Πn where rk(φ k ) = is: ! kn S2(n, + 1).

Bijective proof of Corollary 50
We give here a bijective proof of this corollary, relying on the notion of k-parking trees.
Definition 51. A k-parking tree on a set L is a rooted plane tree T such that: • internal vertices of T are labelled with nonempty subsets of L, which form a set partition of L, • leaves are labelled by empty sets, • each vertex has as many children as k times the number of elements in its label.
We will group edges from a vertex into uplets of k edges, that will be called broods, as drawn on Figure 8. The ith element of this uplet will be called the child of index i. On the example on top of Figure 8, 7 is then the child of index 3 of the first brood of 12, whereas 6 and 4 are children of index 2, respectively of the first and second broods.
Yan [23] introduced a variant of parking functions, such that there are (kn + 1) n−1 of them. They can also be related with k-element chains in 2 Πn.
There is between k-parking trees and k-parking functions of the same kind as the bijection between parking trees and parking functions stated in Lemma 24.
Lemma 53. There is a bijection between k-parking trees on 1; n and k-parking functions of length n, which preserves the action of the symmetric group.
Proof. The proof of Lemma 24 can be adapted to the k case. The condition for a vertex labelled by Ei to have k|Ei| children is equivalent to the condition for a weak set composition to correspond to a k-parking function. This condition is given by Note that the set compositions considered here is a weak set composition of 1; n of length kn + 1. It is direct that a set composition read from a k-parking tree by reading the labels in a prefix manner satisfies Equation (14). Moreover, all the reasoning of the converse can be adapted to this case. The most tricky point of it is perhaps ensuring that for all inner vertices, the cardinality of its label coincides with the number of its children. Once again, the grafting algorithm ensures that an internal vertex does not have more children than k times the number of elements in its label. Denoting by ci the number of children of vertex Ei, we have ci ≤ k|Ei|. Moreover, every Ei are grafted so the tree has kn + 1 nodes, i.e. kn + 1 = n+1 i=1 ci + 1 ≤ n+1 i=1 k|Ei| + 1 ≤ kn + 1. Hence every inequalities are equalities and every Ei has exactly k|Ei| children.
We can now prove Corollary 50, which is illustrated on Figure 8. The proof of Corollary 50 will be in two steps : first proving that such chains are in bijection with k-parking trees having + 1 non-empty nodes and then that they are enumerated by the formula. The following corollary immediately follows from the bijection between k-chains in the poset and k-parking trees.
Corollary 54. Relations in the poset are given by 2-parking trees. In more details, for any a smaller than b in the poset, there exists a 2-parking tree T such that In other words, a parking tree a is smaller than a parking tree b in the poset if and only if: • the nodes of a are obtained as unions of some nodes of b • a is obtained from b by choosing a set of rightmost edges in b and for each edge e in it, between a parent p and a child c, by 1. deleting e 2. merging c with p or any of its ancestor for which e is on the rightmost branch of one of its child.
Proof of Corollary 50, step 1. Let us first explain the bijection Φ between chains φ1 ≤ · · · ≤ φ k in 2 Πn where rk(φ k ) = and k-parking trees having + 1 non-empty nodes. Consider a k-parking tree having + 1 non-empty nodes. The ith parking tree of the associated chain is obtained by merging every children of indices strictly more than i with their parent. In a same brood, the subtree whose root is the child of index j, for 2 ≤ j ≤ i is then grafted on: • the rightmost leaf of the subtree whose root is the child of index j − h if this subtree is not empty and the children of indices j − h + 1, ..., j − 1 are empty, • its parent otherwise.
We denote by Φ(T )[ ] the th tree of the chain Φ(T ). We do not create any cycle and transform every brood into a unique child: we then get a parking tree. Moreover, from a tree to another, the only differences are labels split with half of it brought to the rightmost leaf of the other: this exactly corresponds to covering relations in the poset. The obtained uplet of trees is then a chain of length k in the parking poset.
To exhibit the inverse bijection Ψ, we start from a chain φ1 ≤ · · · ≤ φ k in 2 Πn. Let us construct a k-parking tree T from it by induction on the number of vertices in φ k . If φ k has only one vertex R, the chain is constant (φi = φ k , for every i). The associated k-parking tree is the k-parking tree with only one node labelled by R. Otherwise, let us assume that φ k has N + 1 vertices. The vertices of T are the N + 1 vertices of φ k . Starting from the root, we then construct inductively T . The root R of T is exactly the root of φ k . The jth subtree Sj of this root in φ k gives the jth brood of R. Indeed, the root of Sj is the child of index i of the brood if it splits from R at time i. Then, the position of indices strictly smaller than i in the brood are empty (otherwise, they would be occupied by a descendant of the root who would also be a parent of R).
Let us split the rightmost branch of Sj into subtrees S i j , . . . , S k j . To do so, we run from R to the rightmost leaf of Sj. We define iteratively the forest and a strictly increasing function f : N * → N * . The first step is to cut the edge between R and Sj, then f (1) = i (the index of the root of Sj). At step , we cut the next edge such that the child is in the same vertex as R in φ f ( )−1 , but not in φ f ( ) , with k ≥ f ( ) > f ( − 1). We end the procedure when either we reach the rightmost leaf of Sj or at step m where f (m) = k. We then define S f ( ) j to be the subtree whose root is the child of the edge cut at step . The other S p j are empty. We call, in what follows, this procedure the cutting procedure.
Lemma 55. To each S j can be associated a chain of parking tree, i.e. φ1 |V (S j ) < . . . < φ k |V (S j ) is a chain of parking trees.
Proof of Lemma 55. The only thing to prove is that φi |V (S j ) is connected for every i. Let us consider a node D in S j , and C the root of S j . We want to prove that in every φi, it is either in the root of φi and i < or in a node which is a union of some nodes in S j (excluding any other node).
Suppose first that D is in the root of φ1. There is one t such that D is in the root of φt−1 but no more in the root of φt as D is not in the root of φ k . As D is in S j , t is smaller or equal to , otherwise it would have been split from the subtree S j by the cutting procedure.
Let us proceed now by the absurd and suppose that there is B in S k j (k < ) such that B and D are in the same vertex in φt, this vertex being different from the root. If t was smaller than , the root of S j could not be an ancestor of D because one cannot insert the splitting of a node on a path between two nodes. Then t is greater than or equal to . At step t − 1, the vertex D can only split to a position in one of the descendant subtree of the node bd containing both B and D. Hence, the only way for D to be in S j is that C is a descendant of bd and D splits at step t to the subtree rooted in C. This splitting is only allowed if C is not on the rightmost branch of this subtree. However, as C splitted from the root after k, it can only be on the rightmost branch of this subtree: we get a contradiction. Hence, every S k j evolves independently.
This proof is illustrated on Figure 9.
The root of the subtree S j , if it exists, is then defined to be the child of index of the jth brood of the root of T . Otherwise, this index of the brood correspond to an empty tree. Using Lemma 55, we can then apply the induction on each subtrees S i j , . . . , S k j , for every j, to get the full k-parking tree T .
Proof of Corollary 50, step 2. We now enumerate k-parking trees. Consider the set of kparking trees on 1; n having + 1 non-empty nodes. We can associate to it a kind of Prüfer code. First, we consider that a set A is smaller than a set B if the minimal element of A is smaller than the minimal element of B. Let us consider a k-parking tree T and number the half edges on the parent side from 1 to . Starting from T0 = T at step 0, we iteratively delete the smallest leaf l k of the remaining tree T k , obtaine dafter the kth iteration of the loop, until no vertex is left in T . To each l k can be associated the number of its associated edge : we get a permutation l0 . . . l −1 of {1, . . . , }. We describe this algorithm in pseudo-code below, with denoting the empty word: Algorithm 1: Construction of the code associated to a k-parking tree Result: permutation w of {1, . . . , } T initial k-parking tree ; w ← ; for i ← 0 to − 1 do l ← smallest leaf of T ; w ← w+number of the half-edges attached to l ; T ← tree obtained by deleting l in T ; end return w ; Conversely, from a choice of vertex set (S2(n, + 1)) V (with k|v| half-edges attached to each vertex v), a choice of used half-edges( kn ), and a permutation σ of {1, . . . , } ( !), one can build back a k-parking tree by iterating the following algorithm. We initialize the set L with all vertices whose half-edges are not numbered by an integer between 1 and and the word w by σ. As long as w is not empty, we pop (choose and remove) the element of L with the smallest root and attach it to the half-edge corresponding to the first letter in σ. We get a new tree t. We delete the first letter of σ and add t in L if no halfedge of it is labelled by an element of σ. We describe this algorithm in pseudo-code below: Algorithm 2: Construction of a k-parking tree Result: k-parking tree L ← vertices of V with no half-edges labelled; if t has no half-edge in w then L ← t ; end if length(w) = 0 then return t ; end end The termination of the algorithm is given by the strict decrease of the length of w. L is never empty because there are + 1 − k trees at the kth iteration and − k numbered half-edges. The two algorithms are reciprocal. This shows that a k-parking tree is equivalent to a partition, with elements of the partition having k half-edges, among which we choose half-edges and use a permutation to encode the grafting on this half-edges. Hence they are counted by ! kn S2(n, + 1).
Clearly, the formula in (13) specializes to (6), by letting k = 1. Also, using a general fact linking the zeta polynomial of a poset with its Möbius function, at k = −1 the formula above specializes to the Whitney numbers of the first kind of 2 Πn, defined by: Note that the number µ(0, φ) is a product of Catalan numbers. Indeed, this interval is isomorphic to an interval in N Cn, so it follows from the result on the Möbius function of N Cn [16]. By letting k = −1 in (13), we get w ( 2 Πn) = (−1) ! n + − 1 n S2(n, + 1).
In general, Whitney numbers of the first kind are the dimensions of the Whitney modules, which are useful to compute the homology of a poset (see the definition in the next section).

Homology of the parking function poset
We now study the homology associated to the parking function poset. The reader may read Wachs' article [22] as a general reference on this subject (in particular for Philip Hall's theorem, the Hopf trace formula, Whitney homology), and Munkres' book [20] for more details on simplicial homology.

A first derivation using the zeta character
Let2 Πn denote the proper part of 2 Πn, i.e., 2 Πn with its bottom element removed (the topology associated to 2 Πn is trivial, so2 Πn is the poset to consider here). We denote by Ω(2 Πn) the order complex of2 Πn, i.e., the simplicial complex having strict chains in2 Πn as simplices. We are interested in the reduced simplicial homology of Ω(2 Πn), but let us be more explicit.

{0}.)
Note that the action of Sn on chains in2 Πn permits us to view Cm as a Sn-module. It is clear that the maps ∂m are module maps, so thatHm(2 Πn) is also a Sn-module.
As a consequence of the shellability property obtained in Theorem 31, Ω(2 Πn) has the homotopy type of a bouquet of n − 2-dimensional spheres, so dimHm(2 Πn) = 0 for m = n − 2.
Theorem 57. The character ofHn−2(2 Πn) as a representation of Sn is given by: Proof. We can use the result in [11,Proposition 1.7], and it follows that the desired character is (−1) n−1 times the specialization at k = −1 of (4). This gives the desired formula. It's worth writing that more explicitely. First, the Hopf trace formula gives the equality in the representation ring of Sn. Since only one term is nonzero in the right-hand side, this is also equal to (−1) nH n−2(2 Πn). Let D k be the vector space freely generated by large chains φ1 ≤ · · · ≤ φ k in 2 Πn. It is a Sn-module in a natural way, and its character is Park (k) n (see (4)). A large chain in 2 Πn can be obtained from a strict chain in2 Πn by choosing some multiplicities for each element in the chain, and adding the minimal element of 2 Πn with some multiplicity. Omitting details, for any k ≥ 0 this gives the relation in the representation ring of Sn. By a polynomiality argument, we can set k = −1 in (17). What we get on the right hand side is the alternating sum in the left-hand side of (16), up to a sign. Thus, we have D−1 = (−1) n−1H n−2(2 Πn). So the character of Hn−2(2 Πn) is (−1) n−1 Park (−1) . This gives σ → (−1) n−1 (1 − n) z(σ)−1 , and we get (15).
Lemma 59. For π ∈ N Cn \ {0n} of rank , we have: Proof. By Philipp Hall's theorem, this dimension is the Möbius number of the interval [0n, π] in N Cn, up to a sign. Using the Kreweras complement, this interval is isomorphic to [K(π), 1n], thus isomorphic to N Ci 1 × N Ci 2 × · · · where i1, i2, . . . are the block sizes of K(π). The result follows from the fact that the Möbius number of N Cn is (−1) n−1 Cn−1.
We now give a combinatorial interpretation of Lemma 59.
Definition 60. Let T be a parking tree and x be a vertex of T . The right branch of x is the set of vertices v in the subtree of T rooted in x such that the unique path between v and x only contains x and vertices which are the rightmost child of their parent. We denote it by RB(T ) Πn and T the associated parking tree, the size of a block b of the Kreweras complement K(π) decreased by one is the number of vertices on the right branch of parking tree T b encoding the noncrossing partition under b. |b| = |RB(T b )| + 1 (19) Proof. This lemma is a direct consequence of the bijection described in Lemma 22. Indeed, the size of a block minus 1 is the number of "arches" of it. The bijection sends the non-crossing partition under the first arch to the root of the tree T b and the non-crossing partition left to the right subtree of T b .

Prime parking functions
In the context of the parking space theory, there is a character closely connected to the one in (15), combinatorially related to the notion of prime parking functions.
Definition 62. A noncrossing partition π ∈ N Cn is prime if 1 and n are in the same block of π. An element (π, ρ, λ) ∈ 2 Πn is prime if π is a prime noncrossing partition. Denote by N C n ⊂ N Cn the subset of prime noncrossing partitions, and 2 Π n ⊂ 2 Πn the subset of prime noncrossing 2-partitions.
Algebraically, note that π ∈ N Cn is prime iffπ does not belong to a proper Young subgroup of Sn. As a word w1 . . . wn, a parking function is prime iff #{i | wi ≤ k} > k for k ∈ 1; n − 1 . On parking trees, it corresponds for the root to have a leaf as its rightmost child. Following Section 1.3, the character of Sn acting on 2 Π n is: Proposition 63. For σ ∈ Sn, we have: Proof. This can be proved using a connection with rational parking functions of Armstrong, Loehr and Warrington [3]. For two coprime positive integers a and b, these authors define (a, b)-parking functions as a lattice path that stays above the diagonal in a a × b-rectangle, with some labels on the up steps. They show that parking functions (in the usual sense) correspond to the case (a, b) = (n, n + 1). A similar argument shows that prime parking functions correspond to the case (a, b) = (n, n − 1). As the character Sa acting on (a, b)parking functions is σ → b z(σ)−1 , we get the result. We omit details.
It would be very interesting to have a direct proof of this equality, without an explicit computation of both sides. This could be done by finding a basis (e φ ) φ∈ 2 Π n ofHn−2(2 Πn), such that σ · e φ = Sign(σ)e σ·φ .
Let's do that explicitly for n = 3. The Hasse diagram of2 Π3 is represented in Figure 10, in a way that respects the symmetry of the graph rather than the order. For each cycle of the underlying undirected graph, the alternating sum of its edges gives an element inH1(2 Π3). Note that each transposition (i, j) acts on this graph as a reflection in the plane.
• The 12-cycle at the boundary of the picture is fixed by S3. This cycle can be matched with 111, the element of 2 Π 3 fixed by S3.
• Choose a length 6 cycle going through 211 (there are two of them). The cyclic permutations of coordinates gives two other cycles, going through 121 and 112, respectively. These three cycles can be matched with 211, 121, and 112, the three remaining elements of 2 Π 3 .
The four cycles obtained in this way define four elements inH1(2 Π3). It is straightforward to identify the action of S3 on these elements.

Associahedra and parking functions
The initial goal of this section was to give a combinatorial interpretation to the numbers w ( 2 Πn). This leads us to define a simplicial complex ∆ Πn whose elements involve both faces of the associahedron and noncrossing 2-partitions. We call them cluster parking function, because the associahedron is related with the cluster complex coming from the theory of cluster algebras. This simplicial complex ∆ Πn might be useful in understanding the topology of the parking function poset. Indeed, we will show that it has the same topology as 2 Πn.

The complex of noncrossing alternating forests
The nth associahedron Kn is a simple n-dimensional polytope with a long history. For 0 ≤ i ≤ n, its i-dimensional faces are indexed by valid bracketings of n + 2 factors with n − i pairs of parentheses, moreover the incidence relations between faces are obtained by removing or adding pairs of parentheses. See [19] as a general reference. Here, we use slightly different objects, as indicated in the title of this section.
Definition 65. We denote by ∆n the set of noncrossing alternating forests on 1; n , i.e., forests (acyclic undirected graphs) that contains no pair of edges {i, j} and {k, } such that i < k ≤ j < . Moreover, let ∂∆n ⊂ ∆n denote the subset of forests not containing the edge {1, n}. (It will be explained later that ∂∆n is the boundary of ∆n in a precise sense.) Note that "noncrossing" refers to the forbidden relation i < k < j < , which means that edges can be drawn in a noncrossing way (see example below). And "alternating" refers to the forbidden relation i < k = j < , which means that the neighbours of a vertex i are all smaller or all bigger than i.
We can identify a forest with its edge set (we always understand that n, hence the vertex set of the forests, is fixed once for all). This way, ∆n is stable under taking subsets and can be seen as a simplicial complexes such that: • its vertices are pairs {i, j} with 1 ≤ i < j ≤ n, and can be identified with transpositions in Sn or coatoms of N Cn, • its facets (maximal faces) are noncrossing alternating trees.
In particular, ∆n is purely n − 2-dimensional. By taking the face poset of this simplicial complex (the set of faces ordered by inclusion), we also think of ∆n as a poset.
Proposition 66. The simplicial complex ∆n is a cone over ∂∆n. In particular, ∆n is topologically trivial.
Proof. It is easily checked that the forbidden relation i < k ≤ j < cannot hold if {i, j} = {1, n} or {k, } = {1, n}. So, for each face f ∈ ∂∆n, we have f ∪ {{1, n}} ∈ ∆n. This means that ∆n can be seen as a cone over its full subcomplex with vertices different from {1, n}, i.e., over ∂∆n.
There is a convenient way to represent f ∈ ∂∆n as valid bracketings of n factors, such as ((• • •)•)•, by writing a pair enclosing the ith and jth factor (and others inbetween) if {i, j} ∈ f . This way, we can identify ∂∆n with the poset of nonempty faces of the associahedron Kn−2 ordered by reverse inclusion. Using the dual polytope K * n−2 , we can identify ∂∆n with the poset of non-maximal faces of the simplicial polytope K * n−2 ordered by inclusion. The geometrical realization of ∂∆n can thus be identified with the boundary of K * n−2 , i.e., a n − 3-dimensional sphere. It follows that the geometric realization of ∆n is homeomorphic to a n − 2-dimensional ball, and ∂∆n is indeed its boundary.
Definition 67. For f ∈ ∆n, its set of connected components form a noncrossing partition that will be denoted f ∈ N Cn.

Proof.
A bijection between noncrossing alternating trees and complete binary trees can be given pictorially by drawing each edge {i, j} as two line segments from (i, 0) to ( i+j 2 , j−i 2 ) and from ( i+j 2 , j−i 2 ) to (j, 0), see Figure 11. This proves the case = n − 1, as we get Cn−1 on both sides.
This bijection can be extended componentwise to get the number of noncrossing alternating forests associated to a given π ∈ N Cn: This product of Catalan numbers is also (−1) n−1− µNC n (π, 1n), see Section 1.1. By summing over π of rank n − 1 − in N Cn, we get where in the last equality we used the self-duality of N Cn. The poset ∆n can be used to find what is the topology of N Cn. We briefly explain this, following the results of Athanasiadis and Tzanaki [5]. LetN Cn (resp.∆n) denote the poset N Cn (resp. ∆n) with its minimal element and maximal element(s) removed. (Note that this notation with a bar is not uniform for the posets considered in this article.) As ∆n is topologically a n − 2-dimensional ball, removing the Cn−1 top-dimensional simplices results in a wedge of Cn−1 many n − 3-dimensional spheres, which is thus the topology of∆n. The geometric realizations of∆n and Ω(∆n) are homeomorphic, since the latter is the barycentric subdivision of the former. Eventually, Athanasiadis and Tzanaki [5] proved that the map Ω(∆n) → Ω(N Cn) induced by f → f is a homotopy equivalence, using Quillen's fiber lemma. It follows that Ω(N Cn) is homotopy equivalent to a wedge of Cn−1 many n − 3-dimensional spheres, just like∆n.

Cluster parking functions
By analogy with our discussion about ∆n in the previous section, we introduce a simplicial complex ∆ Πn. In some sense, it is related to 2 Πn in the same way as ∆n is related to N Cn.
Note that ∆ Πn is a subposet of the product poset ∆n × 2 Πn, and it contains pair of elements having the same rank. It is easily seen that the projection on each factor is a rank-preserving poset map. Moreover the action of Sn respects the order of ∆ Πn.
Remark 70. The poset ∆ Πn can be seen as the fiber product of ∆n and 2 Πn over N Cn, along the two poset maps ∆n → N Cn, f → K(f ) and 2 Πn → N Cn, (π, ρ, λ) → π. This point of view will be useful to relate the topology of the two posets ∆ Πn and 2 Πn, as we will use a fiber poset theorem.
Proposition 72. ∆ Πn is the face poset of a simplicial complex.
Proof. If (f, φ) ∈ 2 Πn and f ⊂ f , it follows from Lemma 2 that there exist unique ρ and λ such that (K(f ), ρ, λ) ≤ φ in 2 Πn, so there exists a unique φ ∈ 2 Πn such that (f , φ ) ≤ (f, φ) in ∆ Πn. It follows that each order ideal in ∆ Πn is a boolean lattice. Let V denote the set of rank 1 element in ∆ Πn. It remains only to show that the map is injective to identify ∆ Πn with a simplicial complex having V as its vertex set. So, let (f, (π, ρ, λ)) ∈ ∆ Πn, and let (fi, (πi, ρi, λi)) 1≤i≤k be the rank 1 elements below it. Here k is the rank of (f, (π, ρ, λ)), since the order ideal of elements below it is boolean. First, note that f is the union of the singletons fi, as if f ⊂ f is a singleton there exists (f , φ ) ≤ (f, φ). It follows that f = f1 ∧· · ·∧f k in N Cn. By taking the Kreweras complement, we get π = π1 ∨ · · · ∨ π k . Eventually, we show that (π, ρ, λ) = ∨ 1≤i≤k (πi, ρi, λi). Otherwise, the join would be of rank < k, and taking the projection on N Cn give a contradiction since π = π1 ∨ · · · ∨ π k . This shows that (f, (π, ρ, λ)) is the join of rank 1 elements below it, so that the map in (22)  Proof. This is a direct application of Quillen's fiber poset theorem (see [22,Theorem 5.2.1]), similar to the argument in [5] (as described at the end of Section 5). The statement about homology follows from the equivariant version (see [22,Theorem 5.2.2]). Consider the pro-jection∆ Πn →2 Πn defined by (f, φ) → φ. To apply the fiber poset theorem, we need to check that the fibers are topologically trivial for any φ ∈ 2 Πn. Let (f , φ ) be in the set (23). Using Lemma 2, we see that φ ∈ 2 Πn is uniquely determined from f and φ. So the fiber in Equation (23) is isomorphic to its projection to the first factor ∆n (we have seen in the proof of the previous proposition that this projection respects the order). If φ = (π, ρ, σ), the image of this projection is the subcomplex This is easily seen to be isomorphic to the product ∆n 1 × ∆n 2 × · · · where n1, n2, . . . are the block sizes of K −1 (π), so it is topologically trivial since each factor ∆n i is. So the fiber in (23) is topologically trivial as well, and we can apply the poset fiber theorem.
The geometric realizations of the two simplicial complexes∆ Πn and Ω(∆ Πn) are homeomorphic (they are related by barycentric subdivision, as explained at the end of Section 5). Together with the previous proposition, this shows that ∆ Πn as a simplicial complex has the same topology as Ω( 2 Πn). It might be possible to use this property in order to get an alternative way to obtain the homotopy type of 2 Πn. Proving shellability of ∆ Πn only requires to find an appropriate total order on its maximal elements, which is potentially easier than ordering maximal chains of 2 Πn.
6 k-divisible noncrossing 2-partitions 6.1 k-divisible noncrossing partitions Let k ≥ 1 be an integer. The poset of k-divisible noncrossing partitions (of size n) was introduced by Edelman [13], as the (full) subposet of N C kn containing elements π such that the cardinality of each block is a multiple of k. We use an equivalent formulation, due to Armstrong [1, Chapter 3] in a more general context. It relies on the embedding of N Cn in Sn and identifies k-divisible noncrossing partitions with k-element chains in N Cn. The equivalence between Edelman's definition and Armstrong's definition is stated in [1,Section 4.3].
Note that it might seem unnatural to have ≥ rather than ≤ in the last condition above. We could change the definition of the relative Kreweras complement (by composing with the Kreweras complement) to have the inequality in the other way around. If π ≤ τ , then K(π, τ ) = 1n if and only if π = τ , and roughly speaking π and τ are close to each other if K(π, τ ) is close to 1n.
Let us mention some properties taken from [1]. Because we took different conventions, our poset N C (k) n is isomorphic to the one denoted "N C (k) (An−1)" in [1].

k-divisible noncrossing partitions in the sense of Edelman
Let us record Edelman's definition: Definition 87 ( [13]). We define N C [k] n as the subposet of N C kn containing elements all of whose blocks have cardinality divisible by k.
There is a bijection between N C [k] n and k-trees with kn+1 vertices, obtained as a restriction of the bijection β from Section 1.1. It is natural to introduce the following: Definition 88. We define 2 Π [k] n as the subposet of 2 Π kn containing elements all of whose blocks have cardinality divisible by k.
The action of S kn on 2 Π kn stabilizes 2 Π [k] n , and the orbits are naturally indexed by N C n . For the next proposition, recall that χ denote the Frobenius characteristic map, which sends a Sn-character (equivalently, a Sn-set) to a symmetric function of degree n.
Proposition 89. The symmetric function χ( 2 Π [k] n ) is the image of χ( 2 Π (k) n ) by the algebra morphism that sends the ith homogeneous symmetric function hi to h ki .
Proof. First note that we have: where we sum over k-trees with kn + 1 vertices, the product is over internal vertices v of T , and deg(v) is the number of descendants of v. This follows from Proposition 12, using trees rather than noncrossing partitions (via the bijection β). So, it remains to show: This is clear from the interpretation in terms of k-parking trees.

Perspectives
Let us just mention some further questions arising from this work. First, as said in Remark 33, it would be interesting to investigate if our criterion for shellability is equivalent to CLshellability or if there exists a poset satisfying our criterion but not CL-shellable. It would be moreover very interesting to find posets where our criterion is particularly suited (besides 2 Πn). Also, there should be a generalisation of Edelman's poset to other finite reflection groups. Indeed, in this context there is an associated noncrossing partition lattice, and a noncrossing parking space. New methods might be needed to prove shellability in this general setting.
It would be very interesting to have a bijective proof of Proposition 76 in terms of planar trees (avoiding the relative Kreweras complement), which could be adapted to parking trees to get Proposition 81. The first step towards this proof is to find an appropriate order on the tree representation of N C (k) n .