Gamma-positivity of derangement polynomials and binomial Eulerian polynomials for colored permutations

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Abstract

The binomial Eulerian polynomials, first introduced in work of Postnikov, Reiner and Williams, are γ-positive polynomials and can be interpreted as h-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations and proved that they can be written as the sums of two γ-positive polynomials. In this paper, we find combinatorial interpretations of Athanasiadis' γ-positive polynomials, which leads to an alternative proof of their γ-positivity expansions using the method of group actions on colored permutations. Two results are presented. The first one is to give the γ-coefficients of the symmetric decompositions of binomial Eulerian polynomials for colored permutations, which answers a problem of Athanasiadis (2020) [3]. The second one is to give a direct combinatorial proof on the γ-expansions in the symmetric decompositions of colored derangement polynomials, which answers another problem asked by Athanasiadis (2018) [2].

Introduction

For any positive integer n, let [n] be the set {1,2,,n}. Denote by Sn the set of permutations of [n]. Given a permutation π=π1π2πnSn, an index i (1in1) is a descent (resp. excedance) of π if πi>πi+1 (resp. πi>i). Let desπ (resp. excπ) denote the number of descents (resp. excedances) of π. Recall that π is a derangement if πii for all i[n], and denote by Dn the set of derangements in Sn. The polynomialsAn(t):=πSntdes(π)anddn(t):=πDntexc(π) are known as the Eulerian polynomials and derangement polynomials, respectively.

Recall that a polynomial h(t) with nonnegative integer coefficients is said to be γ-positive, if it admits an expansion of the formh(t)=i=0n/2γiti(1+t)n2i, where γi 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 transformationsA˜n(t)=1+tm=1n(nm)Am(t) of the Eulerian polynomials, and provided the combinatorial interpretationA˜n(t)=πQn+1tdes(π), whereQn:={πSn:π1<π2<<πm=nfor somem}.

For nonnegative integers m and n, let [m,n]={m,m+1,,n}. For positive integers n and r, an r-colored permutation, introduced by Steingrímsson [33], [34], is a pair πc, where πSn and c[0,r1]n, usually denoted as π1c1π2c2πncn. Denote by ZrSn the set of r-colored permutations. Given an r-colored permutation w:=πcZrSn, an index i[n] is said to be descent (resp. ascent) of w, i.e., πici>πi+1ci+1 (resp. πici<πi+1ci+1) if either ci>ci+1 (resp. ci<ci+1) or ci=ci+1 and πi>πi+1 (ci=ci+1 and πi<πi+1), with the assumption that πn+1:=n+1 and cn+1:=0. An index i[n] is said to be an excedance of w if either πi>i or πi=i and ci>0. Denote by des(w), asc(w) and exc(w) the number of descents, ascent and excedances in w, respectively. Denote by (ZrSn)+ and (ZrSn) the set of colored permutations wZrSn with the last coordinate of zero color and nonzero color, respectively, in other words,(ZrSn)+={wZrSn:cn=0},(ZrSn)={wZrSn:cn>0}. Athanasiadis [3] introduced the binomial Eulerian polynomials for colored permutationsA˜n,r(t)=m=0n(nm)tnmAm,r(t), whereAn,r(t):=wZrSntdes(w). Note that the r=0 case of Eq. (1.5) reduces to Eq. (1.1) by using the symmetry of An(t) and A˜n(t).

Athanasiadis [3, Theorem 1.2] showed that A˜n,r(t) can be expressed asA˜n,r(t)=A˜n,r+(t)+A˜n,r(t), where A˜n,r+(t) and A˜n,r(t) are γ-positive polynomials with centers of symmetry n2 and n+12, 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 A˜n,r+(t) and A˜n,r(t) have only real roots.

Even though combinatorial interpretations of A˜n,r+(t) and A˜n,r(t) have already been found in [18], no combinatorial interpretation of A˜n,r+(t) and A˜n,r(t) 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 A˜n(t).

Definition 1.1

For all positive integers n, r with r1, we defineQn,r+:={πc(ZrSn)+:cm=0,πm+1=n>πm+2>>πnandcm+1==cn=0for somem[0,n1]},Qn,r:={πc(ZrSn)+:cm>0,πm+1>πm+2>>πn=1andcm+1==cn=0for somem[1,n1]}.

For example, let w1=54136142273(Z5S7)+, since c5=c6=c7=0 and π6=7>π7, we have w1Q7,5+. Let w2=54617223431(Z5S7)+, since c4>0, c5=c6=c7=0 and π5>π6>π7=1, we have w2Q7,5.

We obtain the following interpretations for A˜n,r+(t) and A˜n,r(t).

Theorem 1.2

For all positive integers n,r with r1, we haveA˜n,r+(t)=wQn+1,r+tdes(w),A˜n,r(t)=wQn+1,rtdes(w).

Remark 1.1

When r=1, the combinatorial interpretation (1.9) reduces to the interpretation (1.2) by using a simple bijection π1π2πnπnπ2π1 and symmetry of A˜n(t).

From Table 1 and Table 2, we have A˜2,2+(t)=t2+5t+1 and A˜2,2(t)=3t2+3t.

For a permutation w=π1c1π2c2πncnZrSn, we call πici (1in) a double descent (resp. double ascent, peak, valley) of w if πi1ci1>πici>πi+1ci+1 (resp. πi1ci1<πici<πi+1ci+1, πi1ci1<πici>πi+1ci+1, πi1ci1>πici<πi+1ci+1), where we use the convention π0=0, πn+1=n+1 and c0=cn+1=0. Denote by Dd(w) (resp. Da(w), Pk(w), Val(w)) the set of double descents (resp. double ascents, peaks, valleys) of w. Denote by dd(w) (resp. da(w), pk(w), val(w)) 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 thatA˜n(t)=k=0n2|γ˜n,i|ti(1+t)n2i, whereγ˜n,i:={πSn:dd(π)=0,des(π)=i}. 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 A˜n,r+(t) and A˜n,r(t) 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 n,r with r1, letΓ˜n,r,i+:={w(ZrSn)+:dd(w)=0,des(w)=i},Γ˜n,r,i:={w(ZrSn):dd(w)=0,des(w)=i}. We haveA˜n,r+(t)=i=0n/2|Γ˜n,r,i+|ti(1+t)n2i,A˜n,r(t)=i=1(n+1)/2|Γ˜n,r,i|ti(1+t)n+12i.

Remark 1.2

The above theorem generalizes Athanasiadis's result for permutations in Z2Sn, see [3, proposition 8.6]. Moreover, the above combinatorial interpretations of γ-coefficients give an answer to [3, Problem 8.7].

For example, when n=2 and r=2, we haveΓ˜2,2,0+={12}andΓ˜2,2,1+={112,211,21},Γ˜2,2,1={211,121,1121}. From Theorem 1.3, we haveA˜2,2+(t)=(1+t)2+3tandA˜2,2(t)=3t(1+t).

A colored permutation wZrSn is called a derangement if it has no fixed points of color 0, and denote by Dn,r the subset consisting of derangements in ZrSn. Follow Chow and Mansour [10, Section 3], the following colored permutation analogues of the derangement polynomials are defined by either one of the formulasdn,r(t):=wDn,rtexc(w)=k=0n(1)nk(nk)Ak,r(t).

By the principle of inclusion-exclusion, we haveAn,r(t)=k=0n(nk)dk,r(t) for all nN, where d0,r(t):=1

The real-rootedness of derangement polynomials of type B dn,2(t) was proved by Chen, Tang, and Zhao [8], and by Chow [9], independently. Athanasiadis [1] showed that dn,r(t) can be expressed asdn,r(t)=dn,r+(t)+dn,r(t), where dn,r+(t) and dn,r(t) are γ-positive polynomials with centers of symmetry n2 and n+12, respectively. Recently, Gustafsson and Solus [17] proved that both dn,r+(t) and dn,r(t) 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 w=πcZrSn, an element i[n] is said to be star bad with respect to w if for πj=i it holds that

  • (1)

    πj<πkc for every k>j,

  • (2)

    πj1<πkc for every k>j1, and

  • (3)

    πj and πj1 have the zero color,

with the convention π0=0 and c0=0. Let Sw be the set of star bad elements with respect to wZrSn and denote bad(w):=|Sw|.

For example, let w=13452216Z3S6, we have Sw={1,3,4} and bad(w)=3.

Combining Gustafsson and Solus's combinatorial interpretations of dn,r+(t) and dn,r(t) (see Lemma 2.2) with Definition 1.4, we give the following variant interpretations.

Proposition 1.5

For positive integers n and r, we havedn,r+(t)=w(ZrSn)+Sw=tdes(w),dn,r(t)=w(ZrSn)Sw=tdes(w).

For example, when n=2 and r=2, we have{w(Z2S2)+:Sw=}={112,21,211},{w(Z2S2):Sw=}={1121,2111}. From Proposition 1.5, we haved2,2+(t)=3tandd2,2(t)=t(1+t).

The following γ-positivity expansions for dn,r+(t) and dn,r(t) were established by Athanasiadis [1, Theorem 1.3].

Theorem 1.6 Athanasiadis

For all positive integers n,r, letϒn,r,i+:={w(ZrSn)+:da(w)=0,asc(w)=i},ϒn,r,i:={w(ZrSn):da(w)=0,asc(w)=i1}. We havedn,r+(t)=i=1n/2|ϒn,r,i+|ti(1+t)n2i,dn,r(t)=i=1(n+1)/2|ϒn,r,i|ti(1+t)n+12i.

Remark 1.3

Athanasiadis [1, Theorem 1.3] gave the γ-expansions of dn,r+(t) and dn,r(t) 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 d3,2+(t)=7t(1+t) and d3,2(t)=t(t+1)2+11t2.

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 w=πcZrSn, an element i[n] is said to be bad with respect to w if for πj=i it holds that

  • (1)

    πj<πk for every k>j,

  • (2)

    πj1<πk for every k>j1, and

  • (3)

    πj and πj1 have the same color,

with the convention π0=0 and c0=0. Let Sw be the set of bad elements with respect to wZrSn and denote bad(w):=|Sw|.

For example, let w

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 πSn with boundary condition π(0)=π(n+1)=0, for any x[n], the x-factorization of π reads π=w1w2xw3w4, where w2 (resp. w3) 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 φx byφx(π)=w1w3xw2w4. Note that if x

Proof of Theorem 1.2

In this section, using the combinatorial interpretation of dn,r+(t) and dn,r(t) in Proposition 1.5 we give the combinatorial interpretations of An,r+(t), An,r(t), A˜n,r+(t) and A˜n,r(t) by the definitions (4.1), (4.2), (4.6) and (4.7).

Following [3, Section 3], we setAn,r+(t):=k=0n(nk)dk,r+(t),An,r(t):=k=0n(nk)dk,r(t), where d0,r+(t):=1 and d0,r(t):=0, and in view of Equations (1.17) and (1.18), we haveAn,r(t)=An,r+(t)+An,r(t).

Combinatorial interpretations of An,r+(t) and An,r(t) 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.

DefineQn,r:={w(ZrSn)+,πm+1=n>πm+2>>πn=1andcm=cm+1==cn=0for some m}. For example, let w=5361422731(Z4S7)+, since π5=7>π6>π7=1 and c4=c5=c6=c7=0, we have wQ7,4.

Lemma 5.1

For all positive integers n,r with r2, we haveA˜n,r+(t)=wQn+2,rtdes(w)1.

Proof

For w=π1c1πncnπn+1cn+1Qn+1,r, we define β(w)=(π11)c1(πn1)cn. Using the definitions of Qn,r+

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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    Supported by the Israel Science Foundation, grant no. 1970/18.

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