A symmetric unimodal decomposition of the derangement polynomial of type
The derangement polynomial for the symmetric group enumerates derangements by the number of excedances. The derangement polynomial for the hyperoctahedral group is a natural type analogue. A new combinatorial formula for this polynomial is given in this paper. This formula implies that decomposes as a sum of two nonnegative, symmetric and unimodal polynomials whose centers of symmetry differ by a half and thus provides a new transparent proof of its unimodality. A geometric interpretation, analogous to Stanley’s interpretation of as the local -polynomial of the barycentric subdivision of the simplex, is given to one of the summands of this decomposition. This interpretation leads to a unimodal decomposition of the Eulerian polynomial of type whose summands can be expressed in terms of the Eulerian polynomial of type . The various decomposing polynomials introduced here are also studied in terms of recurrences, generating functions, combinatorial interpretations, expansions and real-rootedness.
Key words and phrases:Signed permutation, Eulerian polynomial, derangement polynomial, excedance, barycentric subdivision, local -vector, -vector
1. Introduction and results
The derangement polynomial of order is an interesting -analogue of the number of derangements (elements without fixed points) in the symmetric group . It is defined by the formula
where is the number of excedances (see Section 2 for missing definitions) of and is the set of derangements in . The polynomial , first studied by Brenti  in the context of symmetric functions, has a number of pleasant properties. For instance, it has symmetric and unimodal coefficients  (see also [4, Section 4] [23, Section 5] ) and only real roots . It can also be expressed as
where is the -th Eulerian polynomial.
We will be concerned with a natural analogue of for the hyperoctahedral group of signed permutations, introduced and studied independently by Chen, Tang and Zhao  and by Chow . It is defined by the formula
where is the number of type excedances of , introduced by Brenti , and is the set of derangements in .
of (1.2), where is the -th Eulerian polynomial of type . Our first main result is the following combinatorial formula for .
for , where , and the sum ranges over all and over all sequences of nonnegative integers which sum to .
Chow [16, Section 4] gave an additional proof of the unimodality of by expressing it as a sum of certain nonnegative unimodal polynomials, defined by a symmetric function identity, of a common mode. Theorem 1.1 implies that can be written as a sum of two polynomials with nonnegative, symmetric and unimodal coefficients, whose centers of symmetry differ by a half, and thus provides a new proof of its unimodality, as we now explain. Since has degree and zero constant term, it can be written uniquely in the form
where and are polynomials of degrees at most and , respectively, satisfying
(see, for instance, [6, Lemma 2.4] for this elementary fact). For the first few values of we have
The following information for the polynomials and and for can be derived from (1.5).
for , where the sums range over all and over all sequences (respectively, ) of nonnegative integers which sum to . Moreover, and are -nonnegative, meaning there exist nonnegative integers and such that
In particular, and are symmetric and unimodal, with center of symmetry and , respectively, and is unimodal with a peak at .
Much of the motivation behind this paper comes from the theory of subdivisions and local -vectors, developed by Stanley , and its extension . We recall that the local -vector is a fundamental enumerative invariant of a simplicial subdivision (triangulation) of the simplex. An example by Stanley (see [25, Proposition 2.4]) shows that is equal to the local -polynomial of the (first) simplicial barycentric subdivision of the -dimensional simplex. This fact gives a geometric interpretation to and another proof of its symmetry and unimodality.
Our second main result provides a type analogue to this interpretation. To state it, we introduce the following notation. We denote by the simplicial barycentric subdivision of the cubical barycentric subdivision of the -dimensional simplex (Figure 1 shows this subdivision for ). We also introduce the ‘half Eulerian polynomials’
for the group , where and are the sets of signed permutations of length with positive and negative, respectively, last entry, and set and (the set has appeared in the context of major indices for classical Weyl groups; see [7, page 613]).
The polynomial is equal to the local -polynomial of the simplicial subdivision (in particular, has nonnegative, symmetric and unimodal coefficients). Moreover, we have
We should point out that it is Theorem 1.3 and the methods of [3, 25] which led the authors to suspect that formula (1.5) holds. Indeed, it follows from the relevant definitions and some more work (see Section 6) that the local -polynomial of is equal to the right-hand side of (1.15). By exploiting the symmetry of this polynomial and certain recurrence relations for that and for (see Section 7), one can show that the local -polynomial of is equal to , as defined by the decomposition (1.6). A formula for the change in the local -vector of a simplicial subdivision of the simplex after further subdivision [3, Proposition 3.6] (see also Proposition 5.3) can then be used to produce equation (1.9). This suggested that (1.10), and hence (1.5), hold as well.
The structure and other results of this paper are as follows. Section 2 provides the necessary background on (signed) permutations, simplicial complexes and subdivisions. Section 3 proves Theorem 1.1 and Corollary 1.2. A bijective proof of Theorem 1.1, as well as one using generating functions, is given and the exponential generating functions of and are computed. Section 4 gives a combinatorial interpretation to the coefficients of these polynomials. Section 5 proves the main properties of the relative local -vector, a generalization of the concept of local -vector which was introduced in [3, Section 3] (and, in a variant form, in ) and derives a monotonicity property for local -vectors. These results were stated without proof in . As an example (used in one of the proofs of Theorem 1.3), the relative local -vector of the barycentric subdivision of the simplex is computed. Section 6 gives two proofs of Theorem 1.3. A first step towards these proofs is to interpret as the -polynomial of the simplicial complex (Proposition 6.1). Given that, one proof uses the theory of (relative) local -vectors, as discussed earlier, while the other uses recurrences and generating functions.
Section 7 studies the polynomials and . A simple relation between the two is shown to hold (Lemma 7.1). Using its interpretation as the -polynomial of and the theory of local -vectors, a simple formula for (hence one for and one for the Eulerian polynomial ) in terms of the Eulerian polynomial is proven (Proposition 7.2). Using this formula, it is shown that and are real-rooted, hence unimodal and log-concave, and a new proof of the unimodality of is deduced. Recurrences and generating functions for and , as well as for and , are also given and a third proof of Theorem 1.3 is deduced.
2. Permutations and subdivisions
This section fixes notation and includes background material on (signed) permutations, simplicial complexes and their subdivisions. For more information on these topics, the reader is referred to [8, 9, 25, 26, 27].
Throughout this paper, denotes the set of nonnegative integers. For each positive integer we set and . We denote by the cardinality, and by the set of all subsets, of a finite set .
A permutation of a finite set is a bijective map . We denote by the set of all permutations of and set . Suppose that has elements, which are totally ordered by . A permutation can be represented as the sequence , or as the word , or as a disjoint union of cycles [27, Section 1.3]. The standard cycle form is defined by requiring that (a) each cycle is written with its largest element (with respect to the total order ) first and (b) the cycles are written in increasing order of their largest element [27, page 23].
Given , an element is called an excedance of (with respect to ) if and an inverse excedance if . The element is called a descent (respectively, ascent) of if and (respectively, ). The number of excedances (respectively, inverse excedances, descents or ascents) of will be denoted by (respectively, by , or ). The th Eulerian polynomial [27, Section 1.4] is defined by the formulas
Clearly, these sums depend only on and not on or the choice of total order .
The previous definitions apply in particular to (with the standard choice of obtained by setting for ). We will denote by the set of all derangements (permutations without fixed points) in .
2.2. Signed permutations
For the purposes of this paper, it will be convenient to define a signed permutation of as a choice of a subset of such that for and permutation . We will represent such a permutation as the sequence , or as the word , or as a disjoint union of cycles. We will find it convenient to define the standard cycle form of using the total order on which is the reverse of the one inherited from the natural total order on . Thus, cycles of will be written with their smallest element first and in decreasing order of their smallest element. We will say that is a derangement if there is no such that . We will denote the set of all signed permutations of by and the set of all derangements in by .
Given as before, we say that is a -descent (respectively, -ascent) of if (respectively, ), where by convention. The th Eulerian polynomial of type [11, Section 3] can be defined by
where stands for the number of -descents and for the number of -ascents of . Following Brenti [11, p. 431], we say that is a -excedance of if , or if and . We say that is an inverse -excedance of if , or if and . The number of -excedances of will be denoted by and that of inverse -excedances by . We then have and (see Theorem 3.15 and Corollary 3.16 in )
The th derangement polynomial of type is defined by (1.3). Since and the map which sends a permutation to its inverse induces an involution on which preserves fixed points, we have
Let be a polynomial with real coefficients. We recall that is unimodal (and has unimodal coefficients) if there exists an index such that for and for . Such an index is called a peak. The polynomial is said to be log-concave if for and to have internal zeros if there exist indices such that and . We will say that is symmetric (and that it has symmetric coefficients) if there exists an integer such that for . The center of symmetry of is then defined to be (this is well-defined provided is nonzero).
We will say that is real-rooted if all its complex roots are real. It is well-known (see, for instance, ) that if is a real-rooted polynomial with nonnegative coefficients, then is log-concave and unimodal, with no internal zeros. The following theorem, first proved by Edrei , gives a necessary and sufficient condition for a polynomial with nonnegative real coefficients to be real-rooted.
() Let be a polynomial with for every and set for all negative integers . Then is real-rooted if and only if every minor of the lower triangular matrix is nonnegative.
A (nonzero) symmetric polynomial can be written (uniquely) in the form
for some polynomial . We say that is -nonnegative if for every . Clearly, every -nonnegative polynomial is unimodal. For classes of -nonnegative polynomials which appear in combinatorics we refer the reader, for instance, to  and references therein.
2.4. Simplicial complexes
An (abstract) simplicial complex on the ground set is a collection of subsets of such that implies (all simplicial complexes considered in this paper will be assumed to be finite). The elements of are called faces. The dimension of a face is equal to one less than its cardinality. The dimension of is the maximum dimension of its faces. Faces of dimension 0 and 1 are called vertices and edges, respectively. A facet of is a face which is maximal with respect to inclusion. The complex is said to be pure if all its facets have the same dimension. The face poset of a simplicial complex is the set of nonempty faces of , partially ordered by inclusion.
The open star of a face is the collection of all faces of containing . The link of a face in is the subcomplex of defined as . Suppose that and are simplicial complexes on disjoint ground sets. The simplicial join of and is the simplicial complex whose faces are the sets of the form , where and . The order complex [8, Section 9.3] [27, Section 3.8] of a (finite) partially ordered set is defined as the simplicial complex of chains (totally ordered subsets) of .
All topological properties or invariants of mentioned in the sequel will refer to those of its geometric realization [8, Section 9.1]. For example, is a simplicial ball if is homeomorphic to a ball. For a simplicial -dimensional ball , we denote by the subcomplex consisting of all subsets of the -dimensional faces which are contained in a unique facet of . We call the boundary and the interior of .
Let be a simplicial complex. A (topological) simplicial subdivision of [25, Section 2] is a simplicial complex together with a map such that the following hold for every : (a) the set is a subcomplex of which is a simplicial ball of dimension ; and (b) the interior of is equal to . The subcomplex is called the restriction of to . The face is called the carrier of . The subdivision is called quasi-geometric [25, Definition 4.1 (a)] if no face of has the carriers of its vertices contained in a face of of smaller dimension. Moreover, is called geometric [25, Definition 4.1 (b)] if there exists a geometric realization of which geometrically subdivides a geometric realization of , in the way prescribed by . Clearly, all geometric subdivisions (such as the barycentric subdivisions considered in this paper) are quasi-geometric.
We now describe two common ways to subdivide a simplicial complex . The order complex of the face poset , denoted by , consists of the chains of nonempty faces of . This complex is naturally a (geometric) simplicial subdivision of , called the barycentric subdivision, where the carrier of a chain of nonempty faces of is defined as the maximum element of .
Given a face we set , where is a new vertex added and . Then is a simplicial complex which is a simplicial subdivision of , called the stellar subdivision of on .
2.6. Face enumeration
Let be a -dimensional simplicial complex. We denote by the number of -dimensional faces of . A fundamental enumerative invariant of is the -polynomial, defined by
The -polynomial of is defined by
For the importance of -polynomials, the reader is referred to [26, Chapter II]. For the simplicial join of two simplicial complexes we have .
Let be a simplicial subdivision of a -dimensional simplex . The polynomial defined by
is the local -polynomial of (with respect to ) [25, Definition 2.1]. The sequence is the local -vector of (with respect to ).
The following theorem summarizes some of the main properties of local -vectors (see Theorems 3.2 and 3.3 and Corollary 4.7 in ). For the definition of regular subdivision we refer the reader to [25, Definition 5.1].
For every simplicial subdivision of a pure simplicial complex we have
The local -polynomial is symmetric for every simplicial subdivision of the simplex , i.e. we have for .
The local -polynomial has nonnegative coefficients for every quasi-geometric simplicial subdivision of the simplex .
The local -polynomial has unimodal coefficients for every regular simplicial subdivision of the simplex .
3. Proof of the main formula
This section gives two proofs of Theorem 1.1, one bijective and one using generating functions, and deduces Corollary 1.2. As a byproduct of the second proof, the exponential generating functions of and are computed.
First proof of Theorem 1.1.
Let us denote by the collection of sequences of permutations, where and for , such that is a weak ordered partition of with nonempty for and is a derangement of . We will describe a one-to-one correspondence such that
for every , where and stands for or , if is even or odd, respectively. Given this, using (2.4) and recalling that there are weak ordered partitions of satisfying for , we get
and the proof follows.
To define , consider a derangement and let be the standard cycle form of . Then there is an index such that all elements of are positive and the first (smallest) element of is negative. We define as the product of and as the set of all elements which appear in these cycles, so that is a derangement. The remaining cycles form a word whose first element is negative. This word decomposes uniquely as a product of subwords so that for , all elements of are negative if is odd and positive if is even. We define as the set of absolute values of the elements of and as the permutation which corresponds to the word . For instance, if and in standard cycle form, then in cycle form, and , , , as sequences. We set and leave it to the reader to verify that the map is a well defined bijection.
For the second proof of Theorem 1.1 we set
and (see [10, Proposition 5])
Second proof of Theorem 1.1.
and similarly that
and conclude that
Proof of Corollary 1.2.
Theorem 1.1 shows that for every . From the symmetry properties and of the Eulerian and derangement polynomials for it follows that and satisfy (1.7) and (1.8), respectively. The uniqueness of the defining properties of and imply that and for every . This proves equations (1.9) and (1.10).
Since the polynomials and have nonnegative and symmetric coefficients and only real roots, we can write
for some polynomials and with nonnegative coefficients. Explicit combinatorial interpretations to these coefficients are known (see, for instance, [19, Theorem 5.6] and [4, Section 4]). Equations (1.9), (1.10), (3.5) and (3.6) imply explicit combinatorial formulas for the polynomials