The poset of bipartitions
Bipartitional relations were introduced by Foata and Zeilberger in their characterization of relations which give rise to equidistribution of the associated inversion statistic and major index. We consider the natural partial order on bipartitional relations given by inclusion. We show that, with respect to this partial order, the bipartitional relations on a set of size form a graded lattice of rank . Moreover, we prove that the order complex of this lattice is homotopy equivalent to a sphere of dimension . Each proper interval in this lattice has either a contractible order complex, or it is isomorphic to the direct product of Boolean lattices and smaller lattices of bipartitional relations. As a consequence, we obtain that the Möbius function of every interval is , , or . The main tool in the proofs is discrete Morse theory as developed by Forman, and an application of this theory to order complexes of graded posets, designed by Babson and Hersh, in the extended form of Hersh and Welker.
Key words and phrases:bipartitions, set partitions, order complex, Möbius function, homotopy equivalence, discrete Morse theory
2000 Mathematics Subject Classification:Primary 06A07; Secondary 05A18 06A06 55P15
The poset of partitions of the set , where the order is defined by refinement, is a classical object in combinatorics. Various aspects of this poset have been studied in the literature (cf. [21, Ch. 3]). In particular, its Möbius function has been computed by Schützenberger and by Frucht and Rota independently (cf. [19, p. 359]), and the homotopy type of its order complex is a wedge of spheres. (The latter follows from the well-known fact that is a geometric lattice, and from Björner’s result  that geometric lattices are shellable.)
Closely related, and more relevant to the present work, is the poset of ordered partitions of . It has a much simpler structure; for example, all intervals in this poset are isomorphic to products of Boolean lattices.
Bipartitional relations (bipartitions, for short) were introduced by Foata and Zeilberger , who showed that these are the relations for which the (appropriately generalized) major index and inversion number are equidistributed on all rearrangement classes. Han [11, Th. 5] showed that these bipartitional relations can be axiomatically characterized as the relations for which and its complement are transitive. (Cf. [6, 17] for further work on questions of this kind.)
Bipartitional relations on carry a natural poset structure, the partial order being defined by inclusion of relations. We denote the corresponding poset of bipartitions by . Figure 1 shows the Hasse diagram of . The poset contains the poset of ordered partitions of and its dual as subposets, and therefore can be considered as a common extension of the two. It turns out that the richness of the structure of the poset of bipartitions is comparable to that of the lattice of partitions. To begin with, is a graded lattice of total rank (see Theorem 4.1 and Corollary 5.3), although it is neither modular (cf. Example 7.7) nor geometric, in fact not even Cohen–Macaulay (cf. Corollary 9.4). Furthermore, the Möbius function of each interval is , , or (see Definition 10.1, Corollaries 9.5 and 10.3, and Theorem 10.4 for the precise statement of which intervals take which Möbius function values). We show this by proving the stronger result that the order complex of is homotopy equivalent to a sphere (see Theorem 9.3), and each proper interval is either the direct product of Boolean lattices and smaller lattices of bipartitions, or has a contractible order complex (see Proposition 10.2 and Theorem 10.4). The proofs of these facts form the most difficult part of our paper. They are essentially based on an adaptation of the Gray code of permutations due to Johnson  and Trotter  and on work of Babson and Hersh  (in the extended form by Hersh and Welker ) constructing a discrete Morse function in the sense of Forman [8, 9, 10] for the order complex of a graded poset. The former is needed to decompose into a union of distributive lattices in a shelling-like manner. This decomposition is then refined using the well-known -shelling of distributive lattices in order to obtain an enumeration of the maximal chains of to which the results of Babson, Hersh, and Welker apply. (As Example 9.1 shows, our enumeration of the maximal chains of is not a poset lexicographic order in the sense of , so that we do indeed need the extended form observed in .) We remark that our “two-phase” decomposition is similar in spirit as constructions by Hanlon, Hersh and Shareshian  and by Hersh and Welker . It would be interesting to see whether there is a uniform framework for this type of shelling-like decompositions. However, we have not been able to find such a generalization.
This paper is organized as follows. The next two sections are of preliminary nature. Namely, Section 2 reviews basic facts on bipartitional relations, while Section 3 outlines the basic ideas of the construction of Babson and Hersh. Here we observe that the proofs of their main results are actually applicable to a larger class of enumerations of maximal chains, which we call “enumerations growing by creating skipped intervals.” In Section 4, we provide the proof that is a lattice, and we show that it is graded and compute its rank function in Section 5. The purpose of Section 6 is to show that may be written as union of distributive lattices, each indexed by a permutation, where the proof of distributivity is deferred to Section 7. We begin Section 8 by reviewing the Johnson–Trotter algorithm and an easy generalization to enumerating all elements in a direct product of symmetric groups. We continue by using these enumerations to decompose the order complex of , and the order complex of certain intervals in it, in a shelling-like manner. Section 9 forms the core of our article. Here we construct an enumeration of the maximal chains of that refines the “J–T decomposition” introduced in Section 8, and to which the results of Babson and Hersh are adaptable, as reviewed in Section 3. Finally, in Section 10, we outline how the argument of the preceding section may be modified to handle the case of proper intervals of as well.
2. Definition and elementary properties of bipartitional relations
In Definition 2.1 below, we formally introduce bipartitional relations. This definition is (essentially) taken from Han . Subsequently, we shall provide a different way to see bipartitional relations, namely in terms of ordered bipartitions. Historically, bipartitional relations were originally defined by Foata and Zeilberger in [7, Def. 1] in the latter way, and Han showed in [11, Th. 5] the equivalence with a condition which, in its turn, is equivalent to the transitivity condition that we use for defining bipartitional relations as given below.
A relation on a finite set is a bipartitional relation, if both and are transitive. We denote the set of bipartitional relations on X by .
Note that, by definition, the complement of a bipartitional relation is also a bipartitional relation. Following , we say that are incomparable, if either both and belong to , or none of them does. We will use the notation for such pairs.
Lemma 2.2 (Han).
The incomparability relation is an equivalence relation.
As it was first observed by Han in , every bipartitional relation induces a linear order on the -incomparability classes as follows. For we set if and only if but . The -incomparability classes form a set partition of and we may order them by to obtain an ordered partition of . An ordered partition of is an ordered list of pairwise disjoint nonempty subsets , such that is the union of the sets . Every bipartitional relation may be represented by a unique pair of an ordered partition of and a vector (cf. [11, Th. 5]), as follows. We set
In fact, the ’s must be the -equivalence classes, numbered in such a way that if and only if for every and . We must set if and only if for all .
For example, the bipartitional relation has two -equivalence classes: and . Since and , we must have and . Moreover, implies , whereas and imply .
Following , we call the ordered partition together with the vector an ordered bipartition, and we write it as . We call the blocks satisfying underlined (and, consequently, we call the blocks satisfying nonunderlined). Furthermore, we call the ordered bipartition defining via (2.1) the ordered bipartition representation of . On the other hand, every relation defined by an ordered bipartition representation in the way above is bipartitional: the transitivity of is clear, and the transitivity of is evident from the following trivial observation.
If is represented by the ordered bipartition of , then is represented by the ordered bipartition .
We will use the notation111The letter has no specific significance here, but we selected it in tribute to the ubiquitous letter in Foata and Zeilberger’s article . to denote the bipartitional relation defined by its ordered bipartition representation . For example, the bipartitional relation from above may also be given as .
Frequently, we shall write this ordered bipartition in a suggestive manner, where we physically underline the elements of underlined blocks. For example, the above bipartitional relation will also be written in the form .
3. Discrete Morse matching via chain enumeration
Discrete Morse Theory, developed by Forman [8, 9, 10], is a combinatorial theory that helps to determine the homotopy type of a simplicial complex, considered as a -complex. Roughly speaking, in this theory a Morse function on the faces of a simplicial complex induces a Morse matching, which in its turn enables one to perform a sequence of elementary collapses and find a smaller, homotopy equivalent -complex. Only the unmatched faces of the simplicial complex “survive” the collapsing; the subcomplexes induced by these faces are the critical cells, from which the homotopy type of the complex can (hopefully) be read off. In our paper, we shall not need to know exact definitions of all these ingredients. For our purpose it will suffice to keep in mind that one of the primary goals is to identify the critical cells. For a detailed description of the theory we refer the reader to the above cited sources.
In this paper we will use a method developed by Babson and Hersh , in the extended form of Hersh and Welker  (which incorporates a correction to  pointed out in [14, 20]). This method is designed to find the homotopy type of the order complex of a graded partially ordered set with minimum element and maximum element . Recall that the order complex of a partially ordered set is the simplicial complex whose vertices are the elements of and whose faces are the chains of . Babson and Hersh  find a Morse matching on the Hasse diagram of the poset of faces of , the order relation being defined by inclusion, by fixing an enumeration of the maximal chains of , which they call poset lexicographic order. It was observed by Hersh and Welker [15, Theorem 3.1] that the key property of a poset lexicographic order that is used in all proofs of Babson and Hersh in  is that the enumeration of maximal chains considered grows by creating skipped intervals (which is implicit in [1, Remark 2.1]). They call this property the crossing condition, originally introduced by Hersh . The following definition is easily seen to be equivalent to this crossing condition.
Let be a graded poset of rank with a unique minimum element and a unique maximum element . An enumeration of all maximal chains of grows by creating skipped intervals if for every maximal chain there is a family of intervals with elements , none of the intervals contained in another, with the following property: a chain contained in a maximal chain is also contained in a maximal chain for some if and only if the set of ranks of is disjoint from at least one interval in .
In the main result of Babson and Hersh , a second interval system, which is derived from the -intervals, plays a crucial role. This interval system is called -intervals . The process of finding the system of -intervals is given in [1, p. 516] and may be extended without any change to enumerations of maximal chains that grow by creating skipped intervals as follows.
Consider an enumeration of all maximal chains of a graded poset of rank with and that grows by creating skipped intervals. Let be a maximal chain whose associated interval system satisfies
We define the associated -intervals as the output of the following process:
Initialize by setting and .
Let be the interval in whose left end point is the least. Add to , and remove it from .
Replace each interval in by the intersection . Define the “new” to be the resulting new family of intervals.
Delete from those intervals which are not minimal with respect to inclusion.
Repeat steps (1)–(3) until . The output of the algorithm is .
Our wording differs slightly from the one used by Babson and Hersh, since they consider the families and as families of subsets of , whereas we consider them as families of subsets of .
Theorem 3.3 (Babson–Hersh).
Let be a graded poset of rank with and , and let be an enumeration of its maximal chains that grows by creating skipped intervals. Then, in the Morse matching constructed by Babson and Hersh in [1, paragraphs above Th. 2.1], each maximal chain contributes at most one critical cell. The chain contributes a critical cell exactly when the union of all intervals listed in equals . If a maximal chain contributes a critical cell, then the dimension of this critical cell is one less than the number of intervals listed in .
Suppose is a simplicial complex with a discrete Morse function. Then is homotopy equivalent to a CW complex with exactly one cell of dimension for each critical cell of dimension . In particular, if there is no critical cell then is contractible.
We point out that Babson and Hersh modify Forman’s conventions by including the empty face in the range of the Morse function, see the second paragraph after Definition 1.1 in . As a consequence, a vertex might be matched to the empty face, something which is impossible in the setup of Forman. The term “critical cell” is thus slightly more restrictive in  than in [8, 9, 10] in that such a vertex would be a critical cell according to Forman but not according to Babson and Hersh.
4. The lattice of bipartitional relations
In this section, we formally define the order relation on the set of bipartitional relations, and we prove that the so defined poset is a lattice (see Theorem 4.1). At the end of this section, we record an auxiliary result concerning the lattice structure of in Lemma 4.4, which will be needed later in Section 6 in the proof of Lemma 6.2.
Let and be two bipartitional relations in . We define if and only if as subsets of . In this manner, becomes a partially ordered set.
For any finite set , the poset is a lattice.
We remind the reader that a pair belongs to the transitive closure of a relation if there exists a chain with such that , and for .
For every there exists a smallest bipartitional relation with respect to inclusion containing both and , that is, a join . This join is given by the transitive closure of .
Let denote the transitive closure of . Every bipartitional relation containing both and contains also by transitivity. We only need to show that is bipartitional. It is clearly transitive, only the transitivity of remains to be seen.
Assume by way of contradiction that and belong to the complement of but for some . By the definition of , there exists a sequence such that , , , and for every we have or . Without loss of generality we may assume that we have . We cannot have since this implies , in contradiction with , , and the transitivity of (where, as before, denotes the complement ). By induction on , we see that belongs to , for . In particular, we have . The pair cannot belong to , otherwise we have and, by the transitivity of , also . On the other hand, by the transitivity of the relation , we obtain from and that , in contradiction with our assumption. ∎
We may represent any relation as a directed graph on the vertex set by drawing an edge exactly when . If we represent as a directed graph, we obtain that if and only if there is a directed path such that each edge belongs to the graph representing . By the transitivity of and , a shortest such path is necessarily -alternating in the sense that every second edge belongs to , the other edges belonging to . There is no bound on the minimum length of such a shortest path, as is shown in the following example.
Let and consider the bipartitional relation
where each block has two elements, except possibly for the rightmost block, which is a singleton if is odd. Consider also
where each block has two elements, except for the leftmost block, which is always a singleton, and possibly for the rightmost block which is a singleton if is even. It is easy to verify that
The shortest -alternating path from to is , since if .
On the other hand, if only belongs to but does not, then the shortest -alternating path from to has length .
Let and be bipartitional relations on . If for some we have and then already belongs to .
Assume, by way of contradiction, that the shortest -alternating path from to satisfies . Then, because of , belongs to and . Since , the pair also belongs to and . Thus and . We claim that we may replace with and with in the -alternating path and obtain a -alternating path . Indeed, and imply that and belong to the same nonunderlined block of . Hence, if , then belongs to a block of to the “right” of the block containing , whence . Similarly, if , then and yield . The proof that may be replaced with is analogous. We obtain that there is a -alternating path from to , implying , in contradiction to our assumption. Therefore we must have . ∎
5. Cover relations and rank function
In this section we describe the cover relations in the bipartition lattice . This description will allow us to show that is a graded poset, and to give an explicit formula for the rank function.
Let be bipartitional relations. Then covers if and only if its ordered bipartition representation may be obtained from the ordered bipartition representation of in one of the three following ways:
join two adjacent underlined blocks of ,
separate a nonunderlined block of into two adjacent nonunderlined blocks, or
change a nonunderlined singleton block of into an underlined singleton block.
Moreover, is a graded poset, with rank function
Proof of Theorem 5.1.
First we show that the ordered bipartition representation of must come from the ordered bipartition representation of in one of the three ways mentioned in the statement. For that purpose, assume that covers . Let us compare the restrictions of and to every block . Note that the restriction of a bipartitional relation on to a subset of is also bipartitional.
Case 1. properly contains for some . In this case we must have . The relation given by
is a bipartitional relation, properly containing , and contained in . In fact, its ordered bipartition representation may be obtained from by replacing with the ordered bipartition representation of . Since covers , we must have .
If contains no underlined block then merging two adjacent blocks of yields a bipartitional relation on satisfying . Since covers and, hence, covers , we must have . Therefore is obtained from by an operation of type (ii).
If contains an underlined block, then by changing this block to nonunderlined we may obtain a bipartional relation properly contained in and still containing . Hence must be . The only case when there is no bipartition on strictly between and is when , and is obtained from by an operation of type (iii).
Case 2. for all . In this case every -equivalence class is contained in some -equivalence class, and this containment is proper for at least one of the ’s, since otherwise we must have . Hence the situation of Case 1 applies to at least one of the blocks of and . (Clearly, must cover ). Thus, by the already proven case, the ordered bipartition representation of must be obtained from the ordered bipartition representation of by an operation of type (ii) or (iii). Here we may exclude an operation of type (iii), since we are not allowed to have the -equivalence classes (which are the same as the -equivalence classes) to coincide with the -equivalence classes. Therefore is obtained from by an operation of type (ii), which by Lemma 2.3 is equivalent to saying that is obtained from by an operation of type (i).
It is easy to see that the function given in (5.1) assigns zero to the empty bipartitional relation , and increases by exactly one every time we perform one of the operations (i), (ii), or (iii). By the already established part of the statement, increases by one on every cover relation, and so is a graded poset with rank function . On the other hand, every operation of type (i), (ii), or (iii) on a bipartitional relation must yield a bipartitional relation covering , since the rank function has increased by exactly one. ∎
If then has rank .
6. -compatible bipartitions
The purpose of this section is to introduce the notion of compatibility of bipartitional relations with a given ordered partition (the latter having been defined in the paragraph after Lemma 2.2). This notion will be of crucial importance for the subsequent structural analysis of in the subsequent sections. As a first application, we use it in Proposition 6.4 to give a criterion to decide when and are bipartitional relations given by their ordered bipartition representations.
We call an ordered partition compatible with the bipartitional relation , if for every we have
Equivalently, if , then every is the union of consecutively indexed ’s. A particular case arises if consists of singleton blocks only. In this case, given that , there is a permutation of the elements of such that . By abuse of terminology, we shall often say in this case that “the ordered partition is a permutation,” and the bipartitional relation is compatible with such an ordered partition if and only if the elements of may be listed in such an order that placing these lists one after the other in increasing order of blocks gives the left-to-right reading of the permutation . For any ordered partition , we denote the subposet of -compatible bipartitions in by . The Hasse diagram of is shown in Figure 2.
The next lemma shows that this subposet is also a sublattice.
Let be an ordered partition of . If and are -compatible bipartitional relations then so are and .
Let and assume but for some and . By Lemma 4.4, we have . Without loss of generality we may assume . Since is -compatible, we obtain . Hence is also -compatible. The other half of the statement follows by duality, since any bipartitional relation is -compatible if and only if its complement is -compatible. ∎
Using Theorem 5.1 we may deduce the following fact.
Let be a maximal chain in , where . Then there is a unique ordered partition which is compatible with all elements of the chain. This ordered partition is a permutation.
For the statement is trivially true. Assume and let and be two different elements of . Consider the smallest for which contains at least one of and . Such an exists since , and it is positive since . We claim that exactly one of and will belong to . In fact, does not contain any of them, so and belong to the same nonunderlined -equivalence class. is obtained from by one of the operations described in Theorem 5.1. Since at least one of and was added, this operation can only be the separation of the -equivalence class of and into two nonunderlined blocks. Such an operation adds exactly one of and . Let us set if and , respectively if and .
We want to construct an ordered partition which is compatible with all ’s. If , this implies that belongs to an earlier block of than . There is at most one such ordered partition: the permutation, induced by the relation , provided that is a linear order.
We are left to show that is indeed a linear order. Clearly, for distinct and exactly one of and holds. We only need to show the transitivity of the relation . Assume by way of contradiction that , and hold for some . Then we have
for some . By the cyclic symmetry of the list we may assume that either or .
If , then, since and , the transitivity of the relation implies , which is in contradiction with .
On the other hand, if , then since and , the transitivity of the relation implies , which is in contradiction with . ∎
Proposition 6.3 allows us to characterize when and are bipartitional relations given by their ordered bipartition representation.
Let be bipartitional relations represented as and . Then is contained in if and only if the following three conditions are satisfied:
there is an ordered partition that is also a permutation which is compatible with both and ,
every underlined is contained in some underlined ,
every nonunderlined is contained in some nonunderlined .
Assume first that is contained in . Then there is a maximal chain in containing both and . By Proposition 6.3 there is an ordered partition compatible with every element of , and this ordered partition is a permutation, so condition (i) is satisfied. Consider an underlined block . For every we have and so since . Hence is contained in some . The proof of condition (iii) is analogous.
We are left to show that whenever is not contained in , at least one of the given conditions is violated. Assume and consider an ordered pair . If holds as well then and are contained in the same underlined block in the representation of . Thus condition (ii) is violated since . Similarly implies a violation of condition (iii). We are left with the case where , , , and . Now condition (i) is violated. Indeed, let be an arbitrary ordered partition that is also a permutation, satisfying and . By definition, if is compatible with then we must have while compatibility with requires just the opposite, . ∎
7. The distributivity of the sublattice of -compatible bipartitions
In this section we introduce a representation of all -compatible bipartitions, where is an arbitrary fixed permutation. We will use this representation to show that is a distributive lattice, for all ordered partitions . Without loss of generality, we may assume and, for the moment, we may even assume that . The analogous results for an arbitrary finite set and an arbitrary permutation may be obtained by renaming the elements.
Let be a -compatible bipartitional relation, represented as , such that the elements in each block are listed in increasing order. We define the code of as the vector where each is an element of the set , given by the following rule:
For example, the code of the bipartitional relation is . Evidently, the ordered bipartition representing may be uniquely reconstructed from its code, we only need to determine which vectors are valid codes of bipartitional relations.
The definition of the code of is inspired by formula (5.1) giving the rank of . According to this formula, we may compute of a -compatible bipartional relation as follows. We take the ordered bipartition representation of , where we list the elements in increasing order. For the first element in each nonunderlined block we increase by , and we associate no contribution to the other elements in nonunderlined blocks. For the last element in each underlined block we increase by , and for each other element of an underlined block we increase by . Thus we could equivalently define a code where the ordered list of weights is replaced by the list . The rank of is the sum of the coordinates in this “simpler code.” The list of weights is obtained from by the linear transformation . Thus, even for the code we have chosen, is a linear function of the sum of the coordinates in its code. Our choice of code has two “advantages” over the “more obvious” code described above:
The description of a valid code in Corollary 7.3 below involves very simple linear inequalities with integer bounds.
For our code, the code of is obtained by simply taking the negative of the code of .
In the end, it is only a matter of taste whether one prefers the list of weights or the list , and the results below may be easily transformed to fit the reader’s preference.
A vector is the code of a -compatible bipartitional relation if and only if the following conditions are satisfied:
if for some then ;
if for some then .
The necessity of the conditions above is obvious.
Conversely, given a vector satisfying the conditions above, we may find a unique ordered bipartition representing a relation whose code is , as follows:
Start the first block with if and with if . Continue reading the ’s, left to right.
For , if , start a new nonunderlined block with . Note that rule (iv) prevents us from starting a nonunderlined block without ending a preceding underlined block.
For , if then add a nonunderlined to the nonunderlined block that is currently being written (by condition (iii)).
For , if then end an underlined block with . This block is a singleton if , and so belongs to a preceding nonunderlined block, or if , and so ends the preceding underlined block.
For , if , then add an underlined to the current underlined block if , and start a new underlined block with if .
Clearly the above process yields the only whose code is , and conditions (i) through (iv) guarantee that the process never halts with an error. ∎
Lemma 7.2 may be rephrased in terms of inequalities as follows.
A vector is the code of a -compatible bipartitional relation if and only if it satisfies , and for .
Let and be -compatible bipartitional relations with codes respectively . Then if and only if holds for .
Assume that and . Since and are both -compatible, by Proposition 6.4, is contained in if and only if every underlined is contained in some underlined and every nonunderlined is contained in some nonunderlined . It suffices to show that this is equivalent to for all .
Assume first, and consider the possible values of , for a fixed . Let be the block of containing . If , then is automatically true. If then cannot be , otherwise the nonunderlined block containing has a smaller element in , whereas the least element of the nonunderlined block is . Only could contain , but it does not. This contradiction shows that . If then is an element in an underlined block of . This block must be contained in some underlined . In other words, belongs to an underlined block in showing . Finally, if , then is the subset of some underlined . This is contained in some underlined , for which we must have . Thus, is not the last element in , forcing .
For the converse, assume, by way of contradiction, that for