Gamma-positivity of derangement polynomials and binomial Eulerian polynomials for colored permutations☆
Introduction
For any positive integer n, let be the set . Denote by the set of permutations of . Given a permutation , an index i () is a descent (resp. excedance) of π if (resp. ). Let (resp. ) denote the number of descents (resp. excedances) of π. Recall that π is a derangement if for all , and denote by the set of derangements in . The polynomials are known as the Eulerian polynomials and derangement polynomials, respectively.
Recall that a polynomial with nonnegative integer coefficients is said to be γ-positive, if it admits an expansion of the form where are nonnegative integers. Gamma-positivity directly implies palindromicity and unimodality and appears widely in combinatorial and geometric contexts, see [2], [3], [4], [5], [7], [11], [12], [13], [14], [15], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. A common interesting property of Eulerian polynomials and derangement polynomials is that they are γ-positive.
It is known (cf. [27]) that the Eulerian polynomials are the h-polynomials of dual permutohedra. Postnikov, Reiner, and Williams [28, Section 10.4] proved that the h-polynomials of dual stellohedra equal the binomial transformations of the Eulerian polynomials, and provided the combinatorial interpretation where
For nonnegative integers m and n, let . For positive integers n and r, an r-colored permutation, introduced by Steingrímsson [33], [34], is a pair , where and , usually denoted as . Denote by the set of r-colored permutations. Given an r-colored permutation , an index is said to be descent (resp. ascent) of w, i.e., (resp. ) if either (resp. ) or and ( and ), with the assumption that and . An index is said to be an excedance of w if either or and . Denote by , and the number of descents, ascent and excedances in w, respectively. Denote by and the set of colored permutations with the last coordinate of zero color and nonzero color, respectively, in other words, Athanasiadis [3] introduced the binomial Eulerian polynomials for colored permutations where Note that the case of Eq. (1.5) reduces to Eq. (1.1) by using the symmetry of and .
Athanasiadis [3, Theorem 1.2] showed that can be expressed as where and are γ-positive polynomials with centers of symmetry and , respectively. Such a decomposition (1.6) is called the symmetric decomposition of polynomials by Brändén and Solus [6]. Recently, Haglund and Zhang [18] proved that both and have only real roots.
Even though combinatorial interpretations of and have already been found in [18], no combinatorial interpretation of and is known similar to classical Eulerian polynomials. Our first aim is to give such an interpretation, which could be seen as a generalization of interpretation (1.2) for .
Definition 1.1 For all positive integers n, r with , we define
We obtain the following interpretations for and .
Theorem 1.2 For all positive integers with , we have Remark 1.1 When , the combinatorial interpretation (1.9) reduces to the interpretation (1.2) by using a simple bijection and symmetry of .
From Table 1 and Table 2, we have and .
For a permutation , we call () a double descent (resp. double ascent, peak, valley) of w if (resp. , , ), where we use the convention , and . Denote by (resp. , , ) the set of double descents (resp. double ascents, peaks, valleys) of w. Denote by (resp. , , ) the number of double descents (resp. double ascents, peaks, valleys) of w. The γ-positivity formula of Postnikov, Reiner, and Williams [28, Theorem 11.6] in the case of the stellohedron asserts that where The interpretations of above γ-coefficients can also be found in [31] and [23]. Recently Athanasiadis [3, Corollary 3.3] gave similar γ-positivity expansions for and using the similar property of the derangement polynomials for r-colored permutations (see (1.18)). We derive the following colored permutation analogues of (1.12) by using Theorem 1.2 and the well-known Modified Foata-Strehl action (see Section 3). Theorem 1.3 For all positive integers with , let We have
Remark 1.2 The above theorem generalizes Athanasiadis's result for permutations in , see [3, proposition 8.6]. Moreover, the above combinatorial interpretations of γ-coefficients give an answer to [3, Problem 8.7].
A colored permutation is called a derangement if it has no fixed points of color 0, and denote by the subset consisting of derangements in . Follow Chow and Mansour [10, Section 3], the following colored permutation analogues of the derangement polynomials are defined by either one of the formulas
By the principle of inclusion-exclusion, we have for all , where
The real-rootedness of derangement polynomials of type B was proved by Chen, Tang, and Zhao [8], and by Chow [9], independently. Athanasiadis [1] showed that can be expressed as where and are γ-positive polynomials with centers of symmetry and , respectively. Recently, Gustafsson and Solus [17] proved that both and have only real roots, see also [6].
In [17], Gustafsson and Solus defined the bad elements (see Definition 2.1), we give the following variant version.
Definition 1.4 Given a colored permutation , an element is said to be star bad with respect to w if for it holds that for every , for every , and and have the zero color,
with the convention and . Let be the set of star bad elements with respect to and denote .
For example, let , we have and .
Combining Gustafsson and Solus's combinatorial interpretations of and (see Lemma 2.2) with Definition 1.4, we give the following variant interpretations.
Proposition 1.5 For positive integers n and r, we have
For example, when and , we have From Proposition 1.5, we have
The following γ-positivity expansions for and were established by Athanasiadis [1, Theorem 1.3].
Theorem 1.6 Athanasiadis For all positive integers , let We have Remark 1.3 Athanasiadis [1, Theorem 1.3] gave the γ-expansions of and using Rees product construction and geometric simplicial complexes. Using Proposition 1.5, we give a combinatorial proof asked by Athanasiadis [2, Problem 2.29].
From Table 3 and Table 4, by Theorem 1.6 we have and .
The rest of this paper is organized as follows. In Section 2, we prove the Proposition 1.5. In Section 3, we introduce the Modified Foata–Strehl group action on colored permutations (called colored valley-hopping) and prove Theorem 1.6. In Section 4 and Section 5, we present the proof of Theorem 1.2 and Theorem 1.3, respectively.
Section snippets
Proof of Proposition 1.5
In this section, we prove Proposition 1.5 by constructing a bijection from Gustafsson and Solus's combinatorial interpretation (see Lemma 2.2) to (1.19) and (1.20). Definition 2.1 Gustafsson and Solus Given a colored permutation , an element is said to be bad with respect to w if for it holds that for every , for every , and and have the same color,
with the convention and . Let be the set of bad elements with respect to and denote .
Proof of Theorem 1.6
In this section, recall the Modified Foata–Strehl action (MFS-action for short) on permutations, we prove Theorem 1.6 by defining the MFS action on colored permutations.
Definition 3.1 MFS-action Let with boundary condition , for any , the x-factorization of π reads , where (resp. ) is the maximal contiguous subword immediately to the left (resp. right) of x whose letters are all larger than x. Following Foata and Strehl [14] we define the action by Note that if x
Proof of Theorem 1.2
In this section, using the combinatorial interpretation of and in Proposition 1.5 we give the combinatorial interpretations of , , and by the definitions (4.1), (4.2), (4.6) and (4.7).
Following [3, Section 3], we set where and , and in view of Equations (1.17) and (1.18), we have
Combinatorial interpretations of and are
Proof of Theorem 1.3
In this section, we prove Theorem 1.3 by combining the colored valley hopping on colored permutations with the combinatorial interpretations in Theorem 1.2.
Define For example, let , since and , we have .
Lemma 5.1 For all positive integers with , we have Proof For , we define . Using the definitions of
Acknowledgement
The author thanks Prof. Jiang Zeng for his helpful suggestions. The author also thanks the anonymous referees for his/her careful reading and helpful comments.
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