Signed Mahonian polynomials for major and sorting indices

Abstract

We derive some new signed Mahonian polynomials over the complex reflection group $G(r,1,n)=C_r\wr \mathfrak {S}_n$, where the “sign” is taken to be any of the 2r 1-dim characters and the “Mahonian” statistics are the ${\textsf {lmaj}}$ defined by Bagno and the ${\textsf {sor}}$ defined by Eu et al. Various new signed Mahonian polynomials over Coxeter groups of types $B_n$ and $D_n$ are obtained as well. We also investigate the signed counting polynomials on G(r, 1, n) for those statistics with the distribution $[r]_q[2r]_q\cdots [nr]_q$.

1 Introduction

1.1 Signed Mahonian

Let $\mathfrak {S}_n$ be the symmetric group of $\{1,2,\ldots , n\}$. The inversion and $major$ statistics of a permutation $\pi = \pi _1\pi _2\cdots \pi _n \in \mathfrak {S}_n$ are defined, respectively, by

$$\begin{aligned} {\textsf {inv}}(\pi ):&=\left| \left\{ (i,j): i<j \text{ and } \pi _i>\pi _j \right\} \right| ,\\ {\textsf {maj}}(\pi ):&= \sum _{\pi _i>\pi _{i+1}} i . \end{aligned}$$

A fundamental result of MacMahon [18] states that ${\textsf {inv}}$ and ${\textsf {maj}}$ have the same distribution over $\mathfrak {S}_n$ together with the generating function

$$\begin{aligned} \sum _{\pi \in \mathfrak {S}_n}q^{{\textsf {inv}}(\pi )}= \sum _{\pi \in \mathfrak {S}_n}q^{{\textsf {maj}}(\pi )}=[1]_q[2]_q\cdots [n]_q, \end{aligned}$$

(1.1)

where $[k]_q:=1+q+q^2+\cdots +q^{k-1}$ is the q-analogue of k. To celebrate this result, any statistic equidistributed with ${\textsf {inv}}$ on $\mathfrak {S}_n$ is called Mahonian.

The motivation of this paper is the following “signed Mahonian” identity [21], obtained by Gessel and Simion.

$$\begin{aligned} \sum _{\pi \in \mathfrak {S}_n}(-1)^{{\textsf {inv}}(\pi )}q^{{\textsf {maj}}(\pi )} = [1]_q[2]_{-q}[3]_q[4]_{-q}\cdots [n]_{(-1)^{n-1} q}. \end{aligned}$$

(1.2)

1.2 Extensions to Coxeter groups

One can regard $\mathfrak {S}_n$ as the Coxeter group of type $A_{n-1}$. For a Coxeter group, the length function $\ell (\pi )$ is the minimum number of generators required to express a group element $\pi$. Note that in the type $A_{n-1}$ case the length function $\ell _A(\pi )$ is exactly ${\textsf {inv}}(\pi )$, where the subscript A is to emphasize the type. Therefore, in a Coxeter group or a complex reflection group G, we call a statistic Mahonian if it is equidistributed with the length function.

On the other hand, (1.1) and (1.2) can be seen as

$$\begin{aligned} \sum _{\pi \in \mathfrak {S}_n} \chi (\pi ) q^{{\textsf {maj}}(\pi )}, \end{aligned}$$

where $\chi$ is one of the two 1-dim characters of $\mathfrak {S}_n$, namely $\chi =1$ or $\chi (\pi )=(-1)^{\ell _A(\pi )}$. Therefore, from this point of view, one may consider the polynomial

$$\begin{aligned} \sum _{\pi \in G} \chi ({\pi }) q^{{\textsf {stat}}({\pi })}, \end{aligned}$$

(1.3)

where $\chi$ is any 1-dim character of G and ${\textsf {stat}}$ is some Mahonian statistic.

There have been many works extending (1.2) to other Coxeter groups or reflection groups. In the case of type $B_n$, Adin and Roichman [3] defined the Mahonian index so called flag major, ${\textsf {fmaj}}$, and Adin, Gessel and Roichman [2] obtained the signed Mahonian polynomials with respect to ${\textsf {fmaj}}$ together with any of the four 1-dim characters ($1, (-1)^{\ell _B(\pi )}, (-1)^{{\textsf {neg}}(\pi )}, (-1)^{{\textsf {inv}}(|\pi |)}$) of $B_n$. Here, $\ell _B(\pi )$ refers to the length function of $\pi =\pi _1\pi _2\cdots \pi _n\in B_n$ which has the following combinatorial interpretation

$$\begin{aligned} \ell _B(\pi )={\textsf {inv}}(\pi )-\sum _{i\in {\textsf {Neg}}(\pi )}\pi _i, \end{aligned}$$

(1.4)

where ${\textsf {Neg}}(\pi ):=\{i: \pi _i<0\}$, ${\textsf {neg}}(\pi )$ is the cardinality of ${\textsf {Neg}}(\pi )$, and $|\pi |:=|\pi _1||\pi _2|\cdots |\pi _n|\in \mathfrak {S}_n$. Meanwhile, Fire [13] obtained another set of identities by considering the Mahonian statistics ${\textsf {Fmaj}}$ and ${\textsf {nmaj}}$ together with $\chi =(-1)^{\ell _B(\pi )}$.

For the case of type $D_n$, there are several Mahonian statistics in literature. For example, ${\textsf {dmaj}}, {\textsf {Dmaj}}$ and flag major index ${\textsf {fmaj}}$. There are only two 1-dim characters (1 and $(-1)^{\ell _D(\pi )}$) on $D_n$ and the signed Mahonian problems have been studied by Biagioli [6] in the case of ${\textsf {Dmaj}}$ and by Biagioli and Caselli [8] in the case of ${\textsf {fmaj}}$.

The definitions of relative statistics for types $B_n$ and $D_n$ are given in Sects. 3 and 4, respectively.

1.3 Extensions to complex reflection group

One highlight of this paper is to derive new signed Mahonian polynomials on the complex reflection group G(r, 1, n) (denoted by G(r, n) for short throughout the paper), which is isomorphic to the wreath product $C_r\wr \mathfrak {S}_n$, where $C_r$ is the cyclic group of order r. G(r, n) is also called the group of r-colored permutations. Note that $G(1,n)=\mathfrak {S}_n$ and $G(2,n)=B_n$. We will derive the signed Mahonian polynomials by taking $\chi =\chi _{a,b}$ to be any of the 2r 1-dim characters and ${\textsf {stat}}$ the length function $\ell$ or the Mahonian statistic ${\textsf {lmaj}}$ defined by Bagno [4].

Main Theorem A

(Theorem 2.3) For $a=0,1$ and $b=0,1,\ldots , r-1$, we have the following signed Mahonian polynomials:

$$\begin{aligned} \sum _{\pi \in G(r,n)} \chi _{a,b}(\pi )q^{\ell (\pi )}=\prod _{k=1}^n[k]_{(-1)^aq}\left( 1+(-1)^{a(k-1)}\omega ^bq^k[r-1]_{\omega ^bq}\right) , \end{aligned}$$

and

$$\begin{aligned} \sum _{\pi \in G(r,n)} \chi _{a,b}(\pi )q^{{\textsf {lmaj}}(\pi )}=\prod _{k=1}^n[k]_{(-1)^{a(k-1)}q}\left( 1+(-1)^{a(k-1)}\omega ^bq^k[r-1]_{\omega ^bq}\right) , \end{aligned}$$

where $\omega$ is a primitive rth root of unity.

It is worth mentioning that Biagioli and Caselli [8] obtained a closed form of the polynomial $\sum _{\pi \in G(r,n)}\chi (\pi )q^{{\textsf {fmaj}}(\pi )}$ for any 1-dim character $\chi$ of G(r, n) and the flag major index ${\textsf {fmaj}}$ defined by Adin and Roichman [3]. However this ${\textsf {fmaj}}$ is not Mahonian if $r\ge 3$ (i.e., is not equidistributed with $\ell$), and hence it does not equal ${\textsf {lmaj}}$ and therefore the result does not overlap ours. Another interesting statistic ${\textsf {rmaj}}$, called root major index, on G(r, n) defined by Haglund, Loehr, and Remmel [16] is equidistributed with ${\textsf {fmaj}}$, and then is not Mahonian. Hence, ${\textsf {rmaj}}$ is not taken into consideration in this paper. See Fig. 1 for the relationship between these “major” statistics on G(r, n). ${\textsf {stat}}_1\longrightarrow {\textsf {stat}}_2$ means ${\textsf {stat}}_1$ is extended to ${\textsf {stat}}_2$. For example, ${\textsf {nmaj}}$ is exactly ${\textsf {lmaj}}$ when $r=2$.

Fig. 1

The relationship between various major indices

Another interesting Mohonian statistic is the sorting index ${\textsf {sor}}$, which is implicitly given for type $A_{n-1}$ in a bijective proof on $\mathfrak {S}_n$ by Foata and Han [14], defined for Coxeter groups by Petersen [19], and then generalized by Eu et al. [11] for complex reflection groups. Our second class of signed Mahonian polynomials takes ${\textsf {sor}}$ into consideration. The definition of ${\textsf {sor}}$ will be given later in Sect. 2.3.

Main Theorem B

(Theorem 2.4) For $a=0,1$ and $b=0,1,\ldots , r-1$, we have the following signed Mahonian polynomials:

$$\begin{aligned}&\sum _{\pi \in G(r,n)} \chi _{a,b}(\pi )q^{{\textsf {sor}}(\pi )} \\&\qquad = \prod _{k=1}^n \left( 1 + (-1)^a[k-1]_q\left( q + \omega ^{br}q^k[r-1]_q \right) + \omega ^b q^{2k-1}[r-1]_{\omega ^b q} \right) , \end{aligned}$$

where $\omega$ is a primitive rth root of unity.

1.4 Statistics with distribution $[r]_q[2r]_q\cdots [nr]_q$

In addition to Mahonian statistics, there are several known statistics on G(r, n) having the distribution

$$\begin{aligned}{}[r]_q[2r]_q[3r]_q\cdots [nr]_q. \end{aligned}$$

For example, ${\textsf {fmaj}}, {\textsf {fmaf}}, {\textsf {rmaj}}$ and ${\textsf {rinv}}$ are some of them (definitions will be given later). In the same spirit, one can formally consider the signed counting polynomial

$$\begin{aligned} \sum _{\pi \in G(r,n)} (-1)^{{\textsf {stat}}_1(\pi )} q^{{\textsf {stat}}_2(\pi )} \end{aligned}$$

by taking any two of these four statistics. It turns out that except for two cases we do have nice closed forms. See Table 1 for a brief summary.

Table 1 All derived signed counting polynomials, where $*$ indicates that the closed forms are valid for even r

The rest of the paper is organized as follows. The proofs of the two main results (Main Theorem A – B) are given in Sect. 2. Some new signed Mahonian results of type $B_n$ and $D_n$ are collected in Sects. 3 and 4, respectively. Section 5 is devoted to those statistics on G(r, n) with $[r]_q[2r]_q\cdots [nr]_q$ as the distributions. Finally, we propose a brief conclusion in Sect. 6.

2 Signed Mahonian on G(r, n)

2.1 Colored permutation group

Let r, n be positive integers. The group of colored permutations of n digits with r colors, denoted by G(r, n), is the wreath product $C_r\wr \mathfrak {S}_n$ of the cyclic group $C_r$ with $\mathfrak {S}_n$. It is useful to describe the elements in two ways. First, G(r, n) is generated by the set of generators $\mathcal {S}=\{s_0,s_1,\ldots ,s_{n-1}\}$, which satisfies the following relations:

$$\begin{aligned} s_0^r&= 1, \nonumber \\ s_i^2&= 1, i=1,2,\ldots ,n-1, \nonumber \\ (s_is_j)^2&= 1, |i-j|>1, \nonumber \\ (s_is_{i+1})^3&= 1, i=1,2,\ldots ,n-2,\nonumber \\ (s_0s_1)^{2r}&= 1. \end{aligned}$$

(2.1)

In fact, one can realize the generator $s_i$ ($i\ne 0$) as the transposition $(i,i+1)$ and $s_0$ as the mapping

$$\begin{aligned} s_0(i)=\left\{ \begin{array}{ll} \omega \cdot 1, &{} \text {if }i=1,\\ i, &{} \text {otherwise,}\\ \end{array}\right. \end{aligned}$$

where $\omega$ is a primitive rth root of unity. We can also write an element $\pi \in G(r,n)$ as an ordered pair $(z,\sigma )$ or a sequence $\sigma _1^{[z_1]}\,\sigma _2^{[z_2]}\cdots \sigma _n^{[z_n]}$ with the convention $\sigma _i^{[0]}=\sigma _i$, where $z=(z_1,z_2,\ldots ,z_n)$ is an n-tuple of integers in $\mathbb {Z}_r\cong C_r$ and $\sigma \in \mathfrak {S}_n$, such that $\sigma _i=|\pi _i|$ and ${\pi _i}/{|\pi _i|}=\omega ^{z_i}$. For example, $\omega 2\,\,1\,\,5\,\,\omega ^3 4\,\,\omega ^2 3\in G(4,5)$ can be written as ((1, 0, 0, 3, 2), 21543) or $2^{[1]}\,1\,5\,4^{[3]}\,3^{[2]}$. Note that $G(1,n)=\mathfrak {S}_n$ and $G(2,n)=B_n$. Readers are referred to [3, 4, 16] for more information.

Let $\ell (\pi )$ be the length of $\pi \in G(r,n)$ with respect to $\mathcal {S}$. Bagno [4] gave the following combinatorial interpretation of $\ell (\pi )$:

$$\begin{aligned} \ell (\pi )={\textsf {inv}}(\pi )+\sum _{z_i>0}(|\pi _i|+z_i-1), \end{aligned}$$

(2.2)

where ${\textsf {inv}}(\pi ):=|\{(i,j):\,i<j\text { and }\pi _i>\pi _j\}|$ with respect to the linear order

$$\begin{aligned} (n^{[r-1]}<\cdots<n^{[1]})<\cdots< (1^{[r-1]}<\cdots<1^{[1]})< (1<\cdots <n). \end{aligned}$$

(2.3)

By defining ${\textsf {maj}}(\pi ):=\sum _{\pi _i>\pi _{i+1}}i$ with respect to the above linear order, a Mahonian statistic ${\textsf {lmaj}}$ on G(r, n) was found by Bagno [4], defined by

$$\begin{aligned} {\textsf {lmaj}}(\pi ):={\textsf {maj}}(\pi )+\sum _{z_i>0}(|\pi _i|+z_i-1). \end{aligned}$$

(2.4)

It has the generating function

$$\begin{aligned} \sum _{\pi \in G(r,n)}q^{\ell (\pi )}=\sum _{\pi \in G(r,n)}q^{{\textsf {lmaj}}(\pi )}= \prod _{k=1}^n[k]_q(1+q^k[r-1]_q). \end{aligned}$$

(2.5)

2.2 Character and signed Mahonian of $\ell$ and ${\textsf {lmaj}}$

First we characterize all 1-dim characters of G(r, n) in terms of the length function. For $\pi =(z,\sigma )\in G(r,n)$ let ${\textsf {col}}(\pi ):=\sum _{i=1}^n z_i$.

Lemma 2.1

G(r, n) has 2r 1-dim characters, which can be expressed as

$$\begin{aligned} \chi _{a,b}(\pi )=(-1)^{a(\ell (\pi )-{\textsf {col}}(\pi ))}\omega ^{b\,{\textsf {col}}(\pi )}, \end{aligned}$$

(2.6)

for $a=0,1$ and $b=0,1,\cdots , r-1$.

Proof

Let $\chi$ be any 1-dim character. We first look at the values $\chi (s_i)$, $i=1,2,\ldots ,n-1$. Since, by (2.1), $s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$, $\chi (s_i)$ and $\chi (s_{i+1})$ should be the same; therefore, $\chi (s_1)=\chi (s_2)=\cdots =\chi (s_{n-1})$. Moreover, $s_i$ is an involution, then $\chi (s_i)=(-1)^a$ for some $a=0,1$. Since $s_0$ is of order r, $\chi (s_0)=\omega ^b$ for some $b=0,1,\ldots , r-1$. Then there are 2r non-isomorphic choices for $\chi$, which are denoted by $\chi _{a,b}$ for $a=0,1$ and $b=0,1,\ldots , r-1$. By the definitions of $\mathcal {S}$ and $\ell$, the number of occurrences of $s_0$ needed to express $\pi$ is exactly $\sum _{i=1}^n z_i={\textsf {col}}(\pi )$. Hence the expression of $\chi _{a,b}$ in (2.6) fits the requirements. It is easy to verify that each $\chi _{a,b}$ is indeed a group homomorphism. This completes the proof. $\square$

It should be emphasized that the characterization of the 1-dim characters of G(r, n) is not original, but has been studied in [8] as the form:

$$\begin{aligned} \chi _{a,b}(\pi )=(-1)^{a\,{\textsf {inv}}(|\pi |)}\omega ^{b\,{\textsf {col}}(\pi )}, \end{aligned}$$

(2.7)

where $|\pi |=\sigma$. The two expressions in (2.6) and (2.7) are similar; however, the former one is more helpful to derive our two main results. In fact, by the definition of $\ell$, (2.6) can be obtained directly from (2.7) by showing

$$\begin{aligned} (-1)^{{\textsf {inv}}(|\pi |)} = (-1)^{{\textsf {inv}}(\pi )+\sum _{z_i>0}(|\pi _i|-1)}. \end{aligned}$$

(2.8)

The proof of the identity above is similar to the one of Lemma 3.3 and is omitted here.

To prove our signed Mahonian results, we need some more preparations. The key idea is to uniquely decompose a colored permutation $\pi =\sigma _1^{[z_1]}\,\sigma _2^{[z_2]}\cdots \sigma _n^{[z_n]}\in G(r,n)$ as $\pi =\tau \rho$, a product of a linear ordered $\tau$ of $\sigma _1^{[z_1]},\sigma _2^{[z_2]},\ldots ,\sigma _n^{[z_n]}$ and a permutation $\rho \in \mathfrak {S}_n$, where $\rho _i=j$ if and only if $\pi _i$ is the jth smallest element among $\{\sigma _1^{[z_1]},\sigma _2^{[z_2]},\ldots ,\sigma _n^{[z_n]}\}$. For example,

$$\begin{aligned} 2\,4^{[2]}\,1^{[1]}\,3^{[2]}=(4^{[2]}\,3^{[2]}\,1^{[1]}\,2)~~(4\,1\,3\,2). \end{aligned}$$

In general, let $U_{r,n}:=\{\tau \in G(r,n):\,\tau _1<\cdots <\tau _n\}$ and then $G(r,n)=U_{r,n}\cdot \mathfrak {S}_n$. Here we adopt the linear order (2.3). It is obvious that ${\textsf {col}}(\pi )={\textsf {col}}(\tau )$ if $\pi =\tau \rho \in U_{r,n}\cdot \mathfrak {S}_n$. Note that we also have ${\textsf {inv}}(\pi )={\textsf {inv}}(\rho )$ and ${\textsf {maj}}(\pi )={\textsf {maj}}(\rho )$. The bivariate generating function

$$\begin{aligned} F(t,q):=\sum _{\tau \in U_{r,n}} t^{\sum _{z_i>0}(|\tau _i|-1)}q^{{\textsf {col}}(\tau )} \end{aligned}$$

is essential, and we have the following formula.

Lemma 2.2

We have

$$\begin{aligned} F(t,q)=\prod _{k=1}^{n}(1+t^{k-1}q+t^{k-1}q^2 +\cdots +t^{k-1}q^{r-1})=\prod _{k=1}^{n}(1+t^{k-1}q[r-1]_q). \end{aligned}$$

Proof

For $1\le k\le n$, we define elements $\psi _k$ of the group algebra of G(r, n) by

$$\begin{aligned} \psi _k=1+s_{k-1}\cdots s_1s_0+s_{k-1}\cdots s_1(s_0)^2+\cdots +s_{k-1}\cdots s_1(s_0)^{r-1}. \end{aligned}$$

We first claim that

$$\begin{aligned} \psi _1\psi _2\cdots \psi _n=\sum _{\tau \in U_{r,n}}\tau . \end{aligned}$$

Clearly, $\psi _1=\sum _{\tau \in U_{r,1}}\tau$. For the inductive step, suppose the claim is true for $n-1$ and identify the elements of $U_{r,n-1}$ with the set $\{\tau \in U_{r,n}:\,\tau _n=n\}$. Given such a permutation $\tau =\tau _1\cdots \tau _{n-1}\,n$, we have

$$\begin{aligned} \tau \cdot \psi _n=\tau _1\cdots \tau _{n-1}\,n+n^{[1]}\tau _1 \cdots \tau _{n-1}+\cdots +n^{[r-1]}\tau _1\cdots \tau _{n-1}. \end{aligned}$$

(2.9)

Moreover, it is obvious that $\tau \cdot \psi _n=\tau '\cdot \psi _n$ if and only if $\tau =\tau '$ and the induction is completed.

Define the linear mapping $\Psi :\mathbb {Z}[U_{r,n}]\rightarrow \mathbb {Z}[q,t]$ given by

$$\begin{aligned} \Psi (\tau ) := t^{\sum _{z_i>0}(|\tau _i|-1)}q^{{\textsf {col}}(\tau )}. \end{aligned}$$

By construction, we have $\Psi (\psi _k)=1+t^{k-1}q+t^{k-1}q^2+\cdots +t^{k-1}q^{r-1}=1+t^{k-1}q[r-1]_q$ for $k\ge 1$.

We now prove that $\Psi (\psi _1\cdots \psi _n)=\Psi (\psi _1)\cdots \Psi (\psi _n)$. Again, suppose for induction that the assertion is true for $n-1$. It suffices to show that the following holds:

$$\begin{aligned} \Psi (\psi _1\cdots \psi _{n-1}\psi _{n})=\Psi (\psi _1\cdots \psi _{n-1})\Psi (\psi _n). \end{aligned}$$

By (2.9), we have

$$\begin{aligned} \Psi (\tau \cdot \psi _n)&=\Psi (\tau _1\cdots \tau _{n-1}n)+\Psi (n^{[1]}\tau _1\cdots \tau _{n-1})+\cdots +\Psi (n^{[r-1]}\tau _1\cdots \tau _{n-1})\\&=\Psi (\tau )+t^{n-1}q\Psi (\tau )+\cdots +t^{n-1}q^{r-1}\Psi (\tau )\\&=\Psi (\tau )\left( 1+t^{n-1}q[r-1]_q\right) =\Psi (\tau )\Psi (\psi _n). \end{aligned}$$

Thus,

$$\begin{aligned} \Psi (\psi _1\cdots \psi _{n-1}\psi _{n})&=\Psi \left( \sum _{\tau \in U_{r,n},\tau _n=n}\tau \cdot \psi _n\right) =\sum _{\tau \in U_{r,n},\tau _n=n}\Psi (\tau \cdot \psi _n)\\&=\Psi (\psi _n)\sum _{\tau \in U_{r,n},\tau _n=n}\Psi (\tau ) =\Psi (\psi _n)\Psi (\psi _1\cdots \psi _{n-1}), \end{aligned}$$

as desired. Hence, the result follows. $\square$

Now we are ready to present the first main result of this paper.

Theorem 2.3

For $a=0,1$ and $b=0,1,\ldots , r-1$, we have the following signed Mahonian identities

$$\begin{aligned} \sum _{\pi \in G(r,n)} \chi _{a,b}(\pi )q^{\ell (\pi )} = \prod _{k=1}^n[k]_{(-1)^aq}\left( 1+(-1)^{a(k-1)}\omega ^bq^k[r-1]_{\omega ^bq}\right) , \end{aligned}$$

(2.10)

and

$$\begin{aligned} \sum _{\pi \in G(r,n)} \chi _{a,b}(\pi )q^{{\textsf {lmaj}}(\pi )} = \prod _{k=1}^n[k]_{(-1)^{a(k-1)}q} \left( 1+(-1)^{a(k-1)}\omega ^bq^k[r-1]_{\omega ^bq}\right) . \end{aligned}$$

(2.11)

Proof

In what follows, let $({\textsf {Stat}},{\textsf {stat}})=(\ell ,{\textsf {inv}})$ or $({\textsf {lmaj}},{\textsf {maj}})$. Then,

$$\begin{aligned}&\sum _{\pi \in G(r,n)} \chi _{a,b}(\pi )q^{{\textsf {Stat}}(\pi )}\\&\qquad \qquad \qquad =\sum _{\pi \in G(r,n)}(-1)^{a({\textsf {inv}}(\pi )+\sum _{z_i>0}(|\pi _i|-1))}\omega ^{b\,{\textsf {col}}(\pi )}q^{{\textsf {stat}}(\pi )+\sum _{z_i> 0}(|\pi _i|+z_i-1)}\\&\qquad \qquad \qquad =\sum _{\pi \in G(r,n)}\left( (-1)^aq\right) ^{\sum _{z_i>0}(|\pi _i|-1)} (\omega ^bq)^{{\textsf {col}}(\pi )}{(-1)^a}^{{\textsf {inv}}(\pi )}q^{{\textsf {stat}}(\pi )}\\&\qquad \qquad \qquad =\sum _{\tau \in U_{r,n}}\left( (-1)^aq\right) ^{\sum _{z_i>0}(|\tau _i|-1)}(\omega ^bq)^{{\textsf {col}}(\tau )}\sum _{\rho \in \mathfrak {S}_n}{(-1)^a}^{{\textsf {inv}}(\rho )}q^{{\textsf {stat}}(\rho )}\\&\qquad \qquad \qquad = F((-1)^aq,\omega ^bq)\sum _{\rho \in \mathfrak {S}_n}{(-1)^a}^{{\textsf {inv}}(\rho )}q^{{\textsf {stat}}(\rho )}. \end{aligned}$$

Notice that if ${\textsf {stat}}={\textsf {inv}}$, the last summation above turns out to be $\displaystyle \sum _{\rho \in \mathfrak {S}_n}\left( (-1)^{a}q\right) ^{{\textsf {inv}}(\rho )}$. By (1.1), the corresponding signed Mahonian identities for $\ell$ are

$$\begin{aligned} \prod _{k=1}^n[k]_{(-1)^aq}\left( 1+(-1)^{a(k-1)}\omega ^bq^k[r-1]_{\omega ^bq}\right) . \end{aligned}$$

If ${\textsf {stat}}={\textsf {maj}}$, that summation turns out to be the distribution of major index over $\mathfrak {S}_n$ when $a=0$ or the signed Mahonian for ${\textsf {maj}}$ over $\mathfrak {S}_n$ when $a=1$. By (1.1) and (1.2), the corresponding signed Mahonian polynomials for ${\textsf {lmaj}}$ are

$$\begin{aligned} \prod _{k=1}^n[k]_{(-1)^{a(k-1)}q} \left( 1+(-1)^{a(k-1)}\omega ^bq^k[r-1]_{\omega ^bq}\right) . \end{aligned}$$

$\square$

2.3 Signed Mahonian of the sorting index

Another way to represent elements of G(r, n) is by means of “bijections”. Let

$$\begin{aligned} \Sigma :=\{1,\ldots ,n,\bar{1},\ldots ,\bar{n},\bar{\bar{1}}, \ldots ,\bar{\bar{n}},\ldots ,1^{[r-1]},\ldots ,n^{[r-1]}\}, \end{aligned}$$

then $\pi =(z,\sigma )\in G(r,n)$ can be viewed as the bijection on $\Sigma$ such that $\pi (i)=\sigma _i^{[z_i]}$ for $i\in [n]$ and $\pi (\bar{i})=\overline{\pi (i)}$ for $i\in \Sigma$. For example, $\bar{2}\bar{\bar{4}}1\bar{3}\bar{5}=(24135,(1,2,0,1,1))\in G(3,5)$ can be represented as the bijection

$$\begin{aligned} \left( \begin{array}{ccccccccccccccc} \bar{\bar{1}} &{}\quad \bar{\bar{2}} &{}\quad \bar{\bar{3}} &{}\quad \bar{\bar{4}} &{}\quad \bar{\bar{5}} &{}\quad \bar{1} &{}\quad \bar{2} &{}\quad \bar{3} &{}\quad \bar{4} &{}\quad \bar{5} &{}\quad 1 &{}\quad 2 &{}\quad 3 &{}\quad 4 &{}\quad 5 \\ 2 &{}\quad \bar{4} &{}\quad \bar{\bar{1}} &{}\quad 3 &{}\quad 5 &{}\quad \bar{\bar{2}} &{}\quad 4 &{}\quad \bar{1} &{}\quad \bar{\bar{3}} &{}\quad \bar{\bar{5}} &{}\quad \bar{2} &{}\quad \bar{\bar{4}} &{}\quad 1 &{}\quad \bar{3} &{}\quad \bar{5} \end{array} \right) . \end{aligned}$$

For $1\le i<j\le n$ and $0\le t<r$, let $(i^{[t]},j)$ be the transposition of swapping $\pi (i^{[t]})$ with $\pi (j)$, $\pi (i^{[t+1]})$ with $\pi (\bar{j}), \ldots , \pi (i^{[t+r-1]})$ with $\pi (j^{[r-1]})$. Also, for $1\le i\le n$ and $0<t<r$, let $(i^{[t]},i)$ be the action of adding t bars on $\pi (i),\pi (\bar{i}),\ldots ,\pi (i^{[r-1]})$. For example, if $\pi =\bar{2}\bar{\bar{4}}1\bar{3}\bar{5}\in G(3,5)$, then $\pi \cdot (\bar{2},4)=\bar{2}314\bar{5}$ and $\pi \cdot (\bar{\bar{5}})=\bar{2}\bar{\bar{4}}1\bar{3}5$. Formally, multiplying $\pi =\sigma _1^{[z_1]}\,\sigma _2^{[z_2]}\cdots \sigma _n^{[z_n]}\in G(r,n)$ on the right by $(i^{[t]},j)$, $i<j$, has the effect of replacing $\pi _j$ by $\sigma _i^{[z_i+t]}$ and $\pi _i$ by $\sigma _j^{[z_j-t]}$, while multiplying $\pi$ on the right by $(i^{[t]},i)$ has the effect of replacing $\pi _i$ by $\sigma _i^{[z_i+t]}$.

Given a permutation $\pi \in G(r,n)$, there is a unique expression

$$\begin{aligned} \pi =\left( i_1^{[t_1]},j_1\right) \left( i_2^{[t_2]},j_2\right) \cdots \left( i_k^{[t_k]},j_k\right) \end{aligned}$$

as a product of transpositions with $0<j_1<j_2<\cdots <j_k$ for some k. The sorting index, ${\textsf {sor}}(\pi )$, is given in [11] by

$$\begin{aligned} {\textsf {sor}}(\pi ) := \sum _{s=1}^k \left( j_s-i_s + \delta (t_s>0)\cdot \left( 2(i_s-1)+t_s \right) \right) , \end{aligned}$$

(2.12)

where $\delta (\mathsf {A})=1$ if the statement $\mathsf {A}$ is true, or $\delta (\mathsf {A})=0$ otherwise. For example, if $\pi =\bar{2}\bar{\bar{4}}1\bar{3}\bar{5}\in G(3,5)$, we have

$$\begin{aligned} \bar{2}\bar{\bar{4}}1\bar{3}\bar{5} {\mathop {\longrightarrow }\limits ^{~(\bar{\bar{5}},5)~}} \bar{2}\bar{\bar{4}}1\bar{3}5 {\mathop {\longrightarrow }\limits ^{~(\bar{2},4)~}} \bar{2}3145 {\mathop {\longrightarrow }\limits ^{~(2,3)~}} \bar{2}1345 {\mathop {\longrightarrow }\limits ^{~(\bar{\bar{1}},2)~}} \bar{1}2345 {\mathop {\longrightarrow }\limits ^{~(\bar{\bar{1}},1)~}} 12345, \end{aligned}$$

and $\pi =(\bar{\bar{1}},1)(\bar{\bar{1}},2)(2,3)(\bar{2},4)(\bar{\bar{5}},5)$, so ${\textsf {sor}}(\pi ) = \ \big (1-1+(0+2)\big ) + \big (2-1+(0+2)\big ) + \big (3-2+0\big ) + \big (4-2+(2\cdot 1+1)\big ) + \big (5-5+(2\cdot 4+2)\big ) = 21$. It has been proved in [11] that ${\textsf {sor}}$ has the same distribution with $\ell$ over G(r, n) and thus is Mahonian, that is,

$$\begin{aligned} \sum _{\pi \in G(r,n)}q^{{\textsf {sor}}(\pi )}=\sum _{\pi \in G(r,n)}q^{\ell (\pi )}= \prod _{k=1}^n[k]_q(1+q^k[r-1]_q). \end{aligned}$$

(2.13)

To generate elements of G(r, n), a simple way is to append the letter n to the end of elements of $G(r,n-1)$, then pick an index i and a color t arbitrarily, and apply the transposition $(i^{[t]},n)$. Formally, we define the elements $\phi _i$ of the group algebra of G(r, n) by $\phi _1:=1+(\bar{1},1)+\cdots +(1^{[r-1]},1)$ and for $2\le k\le n$

$$\begin{aligned} \phi _k:=1+\sum _{i=1}^{k-1}(i,k)+\sum _{t=1}^{r-1}\sum _{i=1}^{k}(i^{[t]},k). \end{aligned}$$

Thus, for $\pi =\pi _1\cdots \pi _{n-1}n\in G(r,n)$, we have

$$\begin{aligned} \pi \cdot \phi _n&= \ \pi _1\cdots \pi _{n-1}n + \pi _1\cdots n\pi _{n-1} + \cdots + n\pi _2\cdots \pi _{n-1}\pi _1 \\&\quad + \pi _1\cdots \pi _{n-1}n^{[r-1]} + \pi _1\cdots n^{[r-1]}\pi _{n-1}^{[1]} + \cdots + n^{[r-1]}\pi _2\cdots \pi _{n-1}\pi _1^{[1]} \\&\quad + \cdots \\&\quad + \pi _1\cdots \pi _{n-1}n^{[1]} + \pi _1\cdots n^{[1]}\pi _{n-1}^{[r-1]} + \cdots + n^{[1]}\pi _2\cdots \pi _{n-1}\pi _1^{[r-1]}. \end{aligned}$$

It has been proved in [11] that

$$\begin{aligned} \phi _1\phi _2\cdots \phi _n=\sum _{\pi \in G(r,n)}\pi . \end{aligned}$$

(2.14)

We are ready to derive the signed Mahonian polynomials with respect to the sorting index.

Theorem 2.4

For $a=0,1$ and $b=0,1,\ldots , r-1$, we have the following signed Mahonian identities

$$\begin{aligned}&\sum _{\pi \in G(r,n)} \chi _{a,b}(\pi )q^{{\textsf {sor}}(\pi )} \nonumber \\&\qquad \qquad \qquad = \prod _{k=1}^n \left( 1 + (-1)^a[k-1]_q\left( q + \omega ^{br}q^k[r-1]_q \right) + \omega ^b q^{2k-1}[r-1]_{\omega ^b q} \right) . \end{aligned}$$

(2.15)

Proof

For $a=0,1$ and $b=0,1,\ldots , r-1$, define the linear mapping $\Phi _{a,b}:\mathbb {Z}[G(r,n)]\rightarrow \mathbb {Z}[q]$ by

$$\begin{aligned} \Phi _{a,b}(\pi ):=\chi _{a,b}(\pi )q^{{\textsf {sor}}(\pi )}. \end{aligned}$$

Consider $\Phi _{a,b}(\phi _k)$ for $k\ge 1$. By the definition of ${\textsf {sor}}$, it is obvious that ${\textsf {sor}}\big ((i,k)\big )=k-i$ and ${\textsf {sor}}\big ((i^{[t]},k)\big )=k+i+t-2$ for $1\le i\le k$ and $1\le t< r$. Now, view $(i,k)=s_{k-1}s_{k-2}\cdots s_{i+1}s_is_{i+1}\cdots s_{k-2}s_{k-1}$. Since $\chi _{a,b}$ is a 1-dim character,

$$\begin{aligned} \chi _{a,b}\big ((i,k)\big ) = (-1)^{a((2k-2i-1)-0)}\cdot \omega ^{b\cdot 0} = (-1)^a, \end{aligned}$$

and thus

$$\begin{aligned} \Phi _{a,b}\left( 1+\sum _{i=1}^{k-1}(i,k) \right) = 1 + \sum _{i=1}^{k-1}(-1)^aq^{k-i} = 1+(-1)^aq[k-1]_q. \end{aligned}$$

(2.16)

Similarly, since

$$\begin{aligned}&(i^{[t]},k)\nonumber \\&\quad ={\left\{ \begin{array}{ll} s_{k-1}s_{k-2}\cdots s_1s_0^{r-t}s_1\cdots s_{i-2}s_{i-1}s_{i-2}\cdots s_1s_0^ts_1\cdots s_{k-2}s_{k-1}, &{} \text {if }1\le i<k; \\ s_{k-1}s_{k-2}\cdots s_1s_0^{t}s_1\cdots s_{k-2}s_{k-1}, &{} \text {if }i=k, \end{array}\right. } \end{aligned}$$

we have

$$\begin{aligned} \chi _{a,b}\big ((i^{[t]},k)\big ) = (-1)^{a((2k+2i+r-5)-r)}\cdot \omega ^{b\cdot r} = (-1)^a\omega ^{br}, \quad \forall \ 1\le i<k \end{aligned}$$

and

$$\begin{aligned} \chi _{a,b}\big ((k^{[t]},k)\big ) = (-1)^{a((2k+t-2)-t)}\cdot \omega ^{b\cdot t} = \omega ^{bt}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Phi _{a,b}\left( \sum _{t=1}^{r-1}\sum _{i=1}^{k}(i^{[t]},k) \right)&= \sum _{t=1}^{r-1}\left( \left( \sum _{i=1}^{k-1}(-1)^{a}\omega ^{br}q^{k+i+t-2} \right) + \omega ^{bt}q^{2k+t-2} \right) \nonumber \\&= \sum _{t=1}^{r-1}\left( (-1)^{a}\omega ^{br}q^{k+t-1}[k-1]_q + \omega ^{bt}q^{2k+t-2} \right) \nonumber \\&= (-1)^{a}\omega ^{br}q^{k}[k-1]_q\sum _{t=1}^{r-1}q^{t-1} + q^{2k-1}\sum _{t=1}^{r-1}\omega ^{bt}q^{t-1} \nonumber \\&= (-1)^{a}\omega ^{br}q^{k}[k-1]_q[r-1]_q + \omega ^b q^{2k-1}[r-1]_{\omega ^b q}. \end{aligned}$$

(2.17)

Combining (2.16) and (2.17) yields

$$\begin{aligned} \Phi _{a,b}(\phi _k) = 1 + (-1)^a[k-1]_q\left( q + \omega ^{br}q^k[r-1]_q \right) + \omega ^b q^{2k-1}[r-1]_{\omega ^b q}. \end{aligned}$$

By (2.14), it suffices to show $\Phi _{a,b}(\phi _1\cdots \phi _n)=\Phi _{a,b}(\phi _1)\cdots \Phi _{a,b}(\phi _n)$. Suppose for induction that the assertion is true for $n-1$. We aim to show the following holds:

$$\begin{aligned} \Phi _{a,b}(\phi _1\cdots \phi _{n-1}\phi _n)=\Phi _{a,b}( \phi _1\cdots \phi _{n-1})\Phi _{a,b}(\phi _n). \end{aligned}$$

Let $\pi =\pi _1\cdots \pi _{n-1}n\in G(r,n)$. Since $\pi (n)=n$, by the definition of ${\textsf {sor}}$, we have ${\textsf {sor}}(\pi \cdot (i,n))={\textsf {sor}}(\pi )+(n-i)$ for $1\le i<n$, and ${\textsf {sor}}(\pi \cdot (i^{[t]},n))={\textsf {sor}}(\pi )+(n+i+t-2)$ for $1\le i\le n$ and $1\le t<r$. Furthermore, by the fact that $\chi _{a,b}$ is an 1-dim character, we have $\chi _{a,b}(\pi \cdot (i,n))=\chi _{a,b}(\pi )\cdot \chi _{a,b}((i,n))$ for $1\le i<n$, and $\chi _{a,b}(\pi \cdot (i^{[t]},n))=\chi _{a,b}(\pi )\cdot \chi _{a,b}((i^{[t]},n))$ for $1\le i\le n$ and $1\le t<r$. Following a similar argument as in (2.16) – (2.17) derives

$$\begin{aligned} \Phi _{a,b}(\pi \cdot \phi _n) = \Phi _{a,b}(\pi )\Phi _{a,b}(\phi _n). \end{aligned}$$

Thus,

$$\begin{aligned} \Phi _{a,b}(\phi _1\cdots \phi _{n-1}\phi _n)&= \Phi _{a,b}\left( \sum _{\pi \in G(r,n),\pi (n)=n}\pi \cdot \phi _n \right) \\&= \sum _{\pi \in G(r,n),\pi (n)=n} \Phi _{a,b}(\pi \cdot \phi _n) \\&= \Phi _{a,b}(\phi _n) \sum _{\pi \in G(r,n),\pi (n)=n} \Phi _{a,b}(\pi )\\&= \Phi _{a,b}(\phi _n)\Phi _{a,b}(\phi _1\cdots \phi _{n-1}), \end{aligned}$$

as desired. $\square$

3 Signed Mahonian on $B_n$

For Coxeter groups of type $B_n=G(2,n)$, there are some Mahonian statistics other than $\ell _B$ and ${\textsf {lmaj}}$. We take a close look in this section.

3.1 Signed permutation group

Let $B_n$ be the signed permutation group of $\{1,2,\ldots ,n\}$, which consists of all bijections $\pi$ of $\{\pm 1,\ldots ,\pm n\}$ onto itself such that $\pi (-i)=-\pi (i)$. Elements in $B_n$ are centrally symmetric and hence can be simply written in the form $\pi =\pi _1\pi _2\cdots \pi _n$, where $\pi _i:=\pi (i)$. $B_n$ is also the Coxeter group of type $B_n$ with generators $s_0=(1,-1)$ and $s_i=(i,i+1)$ for $i=1,\ldots ,n-1$. Let $\ell _B$ be the length function of $B_n$ with respect to this set of generators. Recall that

$$\begin{aligned} \ell _B(\pi )={\textsf {inv}}(\pi )-\sum _{i\in {\textsf {Neg}}(\pi )}\pi _i. \end{aligned}$$

By denoting $-i$ by $i^{[1]}$, it is obvious that $B_n=G(2,n)$, and $\ell _B$ turns out to be a special case of $\ell$ with $r=2$ in (2.2).

The search of a Mahonian major statistic on $B_n$ was a long-standing problem and was first solved by Adin and Roichman [3] by introducing the flag major index, ${\textsf {fmaj}}$. So far there are at least three type $B_n$ Mahonian statistics in the literature, defined as follows.

• ${\textsf {fmaj}}$. The flag major index [3] is defined by

  $$\begin{aligned} {\textsf {fmaj}}(\pi ):=2\cdot {\textsf {maj}}_F(\pi )+{\textsf {neg}}(\pi ). \end{aligned}$$

• ${\textsf {Fmaj}}$. The F-major index [3] is defined by

  $$\begin{aligned} {\textsf {Fmaj}}(\pi ):=2\cdot {\textsf {maj}}(\pi )+{\textsf {neg}}(\pi ). \end{aligned}$$

• ${\textsf {nmaj}}$. The negative major index [1] is defined by

  $$\begin{aligned} {\textsf {nmaj}}(\pi ):={\textsf {maj}}(\pi )-\sum _{i\in {\textsf {Neg}}(\pi )}\pi _i. \end{aligned}$$

Here ${\textsf {maj}}$ is defined as before, while ${\textsf {maj}}_F$ is computed similarly but with respect to the flag-order:

$$\begin{aligned} \bar{1}<\cdots<\bar{n}<1<\cdots <n. \end{aligned}$$

For example, if $\pi =\bar{3}1\bar{6}2\bar{5}\bar{4}$, then ${\textsf {fmaj}}(\pi )=2\cdot 11+4=26$, ${\textsf {Fmaj}}(\pi )=2\cdot 6+4=16$, and ${\textsf {nmaj}}(\pi )=6-(-18)=24$. Note that ${\textsf {nmaj}}$ is exactly ${\textsf {lmaj}}$ with $r=2$.

In addition to major-type statistics on $B_n$, the type $B_n$ sorting index ${\textsf {sor}}_B$ is defined by Petersen [19]. Its definition can be reduced from 2.12 by simply plugging $r=2$.

To summarize, we from (2.5) have

$$\begin{aligned} \sum _{\pi \in B_n}q^{\ell _B(\pi )}= & {} \sum _{\pi \in B_n}q^{{\textsf {fmaj}}(\pi )}\nonumber \\= & {} \sum _{\pi \in B_n}q^{{\textsf {Fmaj}}(\pi )}=\sum _{\pi \in B_n}q^{{\textsf {nmaj}}(\pi )}\nonumber \\= & {} \sum _{\pi \in B_n}q^{{\textsf {sor}}_B(\pi )}=\prod _{k=1}^{n}[2k]_q. \end{aligned}$$

(3.1)

3.2 Signed Mahonian on $B_n$

The four 1-dim characters of $B_n$, say the trivial character 1, $(-1)^{{\textsf {neg}}(\pi )}$, $(-1)^{\ell _B(\pi )}$ and $(-1)^{{\textsf {inv}}(|\pi |)}$, can be obtained from Lemma 2.1 by letting $r=2$. More precisely,

$$\begin{aligned} \chi _{0,0}(\pi )&= 1, \\ \chi _{0,1}(\pi )&= (-1)^{{\textsf {neg}}(\pi )}, \\ \chi _{1,1}(\pi )&= (-1)^{\ell _B(\pi )}, \\ \chi _{1,0}(\pi )&= (-1)^{\ell _B(\pi )-{\textsf {neg}}(\pi )} = (-1)^{{\textsf {inv}}(\pi )-\big (\sum _{i\in {\textsf {Neg}}(\pi )}\pi _i\big ) - {\textsf {neg}}(\pi )} \\&= (-1)^{{\textsf {inv}}(\pi )+\sum _{i\in {\textsf {Neg}}(\pi )}(|\pi _i|-1)} {\mathop {=}\limits ^{(*)}} (-1)^{{\textsf {inv}}(|\pi |)}, \end{aligned}$$

where the proof of (*) is relegated to Lemma 3.3. The signed Mahonian polynomials of ${\textsf {fmaj}}$ together with the four 1-dim characters were obtained by Adin et al. [2], and the signed Mahonian polynomials of $\ell _B$ or ${\textsf {nmaj}}$ can be derived directly from Theorem 2.3 as follows.

Corollary 3.1

For $\ell _B$, we have the following signed Mahonian polynomials:

• 1. $\displaystyle \sum _{\pi \in B_n}(-1)^{\ell _B(\pi )}q^{\ell _B(\pi )}=\prod _{k=1}^n[2k]_{-q}$;

• 2. $\displaystyle \sum _{\pi \in B_n}(-1)^{{\textsf {neg}}(\pi )}q^{\ell _B(\pi )}=\prod _{k=1}^n[2]_{-q^k}[k]_q$;

• 3. $\displaystyle \sum _{\pi \in B_n}(-1)^{{\textsf {inv}}(|\pi |)}q^{\ell _B(\pi )}=\prod _{k=1}^n[2]_{(-1)^{k-1}q^k}[k]_{-q}$.

For ${\textsf {nmaj}}$, we have the following signed Mahonian polynomials:

• 1. $\displaystyle \sum _{\pi \in B_n}(-1)^{\ell _B(\pi )}q^{{\textsf {nmaj}}(\pi )} =\prod _{k=1}^n[2]_{(-q)^k}[k]_{(-1)^{k-1}q}$;

• 2. $\displaystyle \sum _{\pi \in B_n}(-1)^{{\textsf {neg}}(\pi )}q^{{\textsf {nmaj}}(\pi )}=\prod _{k=1}^n[2]_{-q^k}[k]_q$;

• 3. $\displaystyle \sum _{\pi \in B_n}(-1)^{{\textsf {inv}}(|\pi |)}q^{{\textsf {nmaj}}(\pi )}=\prod _{k=1}^n[2]_{(-1)^{k-1}q^k}[k]_{(-1)^{k-1}q}$.

The signed Mahonian polynomial of ${\textsf {sor}}_B$ can be obtained from Theorem 2.4 as follows.

Corollary 3.2

For ${\textsf {sor}}_B$, we have the following signed Mahonian polynomials:

• 1. $\displaystyle \sum _{\pi \in B_n}(-1)^{\ell _B(\pi )}q^{{\textsf {sor}}_B(\pi )}=\prod _{k=1}^n \left( 1-q[2k-1]_q \right)$;

• 2. $\displaystyle \sum _{\pi \in B_n}(-1)^{{\textsf {neg}}(\pi )}q^{{\textsf {sor}}_B(\pi )}=\prod _{k=1}^n \left( [2k-1]_q-q^{2k-1} \right)$;

• 3. $\displaystyle \sum _{\pi \in B_n}(-1)^{{\textsf {inv}}(|\pi |)}q^{{\textsf {sor}}_B(\pi )}=\prod _{k=1}^n \left( 1-q[2k-2]_q+q^{2k-1} \right)$.

In the rest of this subsection, we focus on the signed Mahonian polynomials of ${\textsf {Fmaj}}$. Following the same fashion as in G(r, n), we consider the decomposition $B_n=U_n\cdot \mathfrak {S}_n$, where $U_n=U_{2,n}=\{\tau \in B_n| \tau _1<\tau _2<\cdots <\tau _n\}$. It is obvious that ${\textsf {inv}}(\tau \rho )={\textsf {inv}}(\rho )$, ${\textsf {maj}}(\tau \rho )={\textsf {maj}}(\rho )$, and ${\textsf {Neg}}(\tau \rho )={\textsf {Neg}}(\tau )$ for $\tau \in U_n$ and $\rho \in \mathfrak {S}_n$. By letting $r=2$ in Lemma 2.2, we obtain that

$$\begin{aligned} F(t,q)\Big |_{r=2} = \sum _{\tau \in U_n}t^{\sum _{i\in {\textsf {Neg}}(\tau )}(|\tau _i|-1)}q^{{\textsf {neg}}(\tau )} = [2]_{t^{k-1}q}. \end{aligned}$$

(3.2)

Furthermore, by a similar argument as in Lemma 2.2, we have

$$\begin{aligned} B(t,q):=\sum _{\tau \in U_n}t^{{\textsf {neg}}(\tau )}q^{-\sum _{i\in {\textsf {Neg}}(\tau )}\tau _i}=\prod _{k=1}^n[2]_{tq^k}. \end{aligned}$$

(3.3)

The following lemma is essential to our main results in this section.

Lemma 3.3

For any signed permutation $\pi \in B_n$, we have

$$\begin{aligned} (-1)^{{\textsf {inv}}(|\pi |)}=(-1)^{{\textsf {inv}}(\pi )+\sum _{i\in {\textsf {Neg}}(\pi )}(|\pi _i|-1)}. \end{aligned}$$

Proof

If ${\textsf {Neg}}(\pi )=\emptyset$, then $|\pi |=\pi$ and the identity trivially holds. So we only need to consider the case of ${\textsf {neg}}(\pi )>0$. Assume $\pi _k=\max \{|\pi _i| : i\in {\textsf {Neg}}(\pi )\}$, and let $\pi '=\pi _1\cdots \pi _{k-1}|\pi _k|\pi _{k+1}\cdots \pi _n$. For example, if $\pi =\bar{2}73\bar{1}6\bar{5}4$, then $\pi _k=\bar{5}$ and $\pi '=\bar{2}73\bar{1}654$.

Let $s=|\{\pi _i: i<k \text { and }|\pi _i|<|\pi _k|\}|$ and $t=|\{\pi _i: i>k \text { and } |\pi _i|<|\pi _k|\}|$. One has $s+t=|\pi _k|-1$. By a simple calculation, we have

$$\begin{aligned} {\textsf {inv}}(\pi ')={\textsf {inv}}(\pi )-s+t={\textsf {inv}}(\pi )+|\pi _k|-1-2s, \end{aligned}$$

implying $(-1)^{{\textsf {inv}}(\pi ')}=(-1)^{{\textsf {inv}}(\pi )+|\pi _k|-1}.$ Hence, the result follows by considering iteratively all elements in ${\textsf {Neg}}(\pi ')$. $\square$

We are ready for the main result in this section. Note that Theorem 3.4(1) has appeared in [13, Theorem 5.7]. We still put it here for the sake of completeness.

Theorem 3.4

For ${\textsf {Fmaj}}$, we have the following signed Mahonian polynomials.

• 1. $\displaystyle \sum _{\pi \in B_n}(-1)^{\ell _B(\pi )}q^{{\textsf {Fmaj}}(\pi )}=\prod _{k=1}^n[2]_{(-1)^kq}[k]_{(-1)^{k-1}q^2}$.

• 2. $\displaystyle \sum _{\pi \in B_n}(-1)^{{\textsf {neg}}(\pi )}q^{{\textsf {Fmaj}}(\pi )} =\prod _{k=1}^n[2]_{-q}[k]_{q^2}$.

• 3. $\displaystyle \sum _{\pi \in B_n}(-1)^{{\textsf {inv}}(|\pi |)}q^{{\textsf {Fmaj}}(\pi )} =\prod _{k=1}^n[2]_{(-1)^{k-1}q}[k]_{(-1)^{k-1}q^2}$.

Proof

Recall that ${\textsf {Fmaj}}(\pi )=2\cdot {\textsf {maj}}(\pi )+{\textsf {neg}}(\pi )$ and $B_n=U_n\cdot \mathfrak {S}_n$.

• 1. By (1.2) and (3.3), the left-hand side is equal to

  $$\begin{aligned}&\sum _{\tau \in U_n}(-1)^{-\sum _{i\in {\textsf {Neg}}(\tau )}\tau _i}q^{{\textsf {neg}}(\tau )} \sum _{\rho \in \mathfrak {S}_n}(-1)^{{\textsf {inv}}(\rho )}q^{2\cdot {\textsf {maj}}(\rho )}\\&\quad = B(q,-1)\prod _{k=1}^n[k]_{(-1)^{k-1}q^2} = \prod _{k=1}^n[2]_{(-1)^kq}[k]_{(-1)^{k-1}q^2}. \end{aligned}$$

• 2. By (1.1) and (3.3), the left-hand side is equal to

  $$\begin{aligned} \sum _{\tau \in U_n}(-q)^{{\textsf {neg}}(\tau )} \sum _{\rho \in \mathfrak {S}_n}q^{2\cdot {\textsf {maj}}(\rho )} = B(-q,1)\prod _{k=1}^n[k]_{q^2} = \prod _{k=1}^n[2]_{-q}[k]_{q^2}. \end{aligned}$$

• 3. By (1.2), (3.2) and Lemmas 3.3, the left-hand side is equal to

  $$\begin{aligned}&\sum _{\pi \in B_n}(-1)^{{\textsf {inv}}(\pi )+\sum _{i\in {\textsf {Neg}}(\pi )}(|\pi _i|-1)}q^{2\cdot {\textsf {maj}}(\pi )+{\textsf {neg}}(\pi )}\\&\quad = \sum _{\tau \in U_n}(-1)^{\sum _{i\in {\textsf {Neg}}(\tau )}(|\tau _i|-1)}q^{{\textsf {neg}}(\tau )} \sum _{\rho \in \mathfrak {S}_n}(-1)^{{\textsf {inv}}(\rho )}q^{2\cdot {\textsf {maj}}(\rho )}\\&\quad = F(-1,q)\Big |_{r=2}\prod _{k=1}^n[k]_{(-1)^{k-1}q^2} = \prod _{k=1}^n[2]_{(-1)^{k-1}q}[k]_{(-1)^{k-1}q^2}. \end{aligned}$$

  $\square$

4 Signed Mahonian on $D_n$

4.1 Even-signed permutation group

The even-signed permutation group $D_n$ is the subgroup of $B_n$ defined by

$$\begin{aligned} D_n:=\{\pi \in B_n:\, {\textsf {neg}}(\pi )\text { is even}\}, \end{aligned}$$

which consists of those permutations with an even number of negatives among $\pi _1,\ldots ,\pi _n$. $D_n$ is known as the “Coxeter group of type $D_n$” which has the generators $s_0',s_1,\ldots ,s_{n-1}$, where $s_0'=(\bar{1},2)$ and $s_i=(i,i+1)$ for $i\ge 1$. Let $\ell _D$ be the corresponding length function of $D_n$. It is known [9, 17] that the generating function for the distribution of $\ell _D$ is

$$\begin{aligned} \sum _{\pi \in D_n} q^{\ell _D(\pi )}=[n]_q\prod _{k=1}^{n-1}[2k]_q=[2]_q[4]_q\cdots [2n-2]_q[n]_q \end{aligned}$$

(4.1)

and a combinatorial description of $\ell _D$ is

$$\begin{aligned} \ell _D(\pi )={\textsf {inv}}(\pi )-{\textsf {neg}}(\pi )-\sum _{i\in {\textsf {Neg}}(\pi )}\pi _i. \end{aligned}$$

Biagioli [5] defined the index

$$\begin{aligned} {\textsf {dmaj}}(\pi ):={\textsf {maj}}(\pi )-{\textsf {neg}}(\pi )-\sum _{i\in {\textsf {Neg}}(\pi )}\pi _i \end{aligned}$$

on $D_n$ and proved that it is Mahonian.

$D_n$ can also be generated by

$$\begin{aligned} \mathcal {T}_n:=\{t_{i,j}:\,1\le |i|<j\le n\} \cup \{t_{\bar{i},i}:\,1<i\le n\}, \end{aligned}$$

where $t_{i,j}=(i,j)$ for $1\le |i|<j\le n$ and $t_{\bar{i},i}=(\bar{1},1)(\bar{i},i)$ for $1<i\le n$. Then, any $\pi \in D_n$ has a unique factorization in the form

$$\begin{aligned} \pi =t_{i_1,j_1}t_{i_2,j_2}\cdots t_{i_k,j_k} \end{aligned}$$

with $0<j_1<j_2<\cdots <j_k\le n$ for some k. Petersen [19] defined the sorting index of type $D_n$ by

$$\begin{aligned} {\textsf {sor}}_D(\pi ):=\sum _{s=1}^{k} \left( j_s-i_s - 2\delta (i_s<0) \right) \end{aligned}$$

and proved that it is Mahonian. For example, $\pi =\bar{3}24\bar{5}1=t_{\bar{1},3}t_{3,4}t_{\bar{4},5}$ has the sorting index ${\textsf {sor}}_D(\pi )= (3-(-1)-2)+ (4-3)+ (5-(-4)-2)= 10$.

4.2 Signed Mahonian on $D_n$

The group $D_n$ has two 1-dim characters [20]: the trivial character 1 and $(-1)^{\ell _D(\pi )}$. In this subsection we derive the signed Mahonian polynomials $\sum _{\pi \in D_n } (-1)^{\ell _D(\pi )} q^{{\textsf {stat}}(\pi )}$, where ${\textsf {stat}}=\ell _D,{\textsf {dmaj}}$ or ${\textsf {sor}}_D$.

We first consider ${\textsf {stat}}=\ell _D$ or ${\textsf {dmaj}}$. Similar to the case of $B_n$, we consider the decomposition $D_n=U^D_n\cdot \mathfrak {S}_n$, where $U^D_n:=\{\pi \in U_n:\,{\textsf {neg}}(\pi )\text { is even}\}$.

Lemma 4.1

We have

$$\begin{aligned} B^D(q):=\sum _{\tau \in U^D_n}q^{\ell _D(\tau )}=\prod _{k=1}^{n-1}[2]_{q^k}. \end{aligned}$$

Proof

Let $[n]:=\{1,2,\ldots ,n\}$. Then $\sum _{\tau \in U^D_n}q^{\ell _D(\tau )}$ is equal to

$$\begin{aligned} \sum _{\begin{array}{c} T\subseteq [n],\\ |T|\text { is even} \end{array}} q^{\left( \sum _{i\in T}i\right) -|T|}= & {} \sum _{\begin{array}{c} T\subseteq \{0,1,\ldots ,n-1\},\\ |T|\text { is even} \end{array}} q^{\sum _{i\in T}i}\\= & {} \sum _{\begin{array}{c} T\subseteq [n-1],\\ |T|\text { is odd} \end{array}}q^{\sum _{i\in T\cup \{0\}}i} +\sum _{\begin{array}{c} T\subseteq [n-1],\\ |T|\text { is even} \end{array}}q^{\sum _{i\in T}i} \\= & {} \sum _{T\subseteq [n-1]}q^{\sum _{i\in T}i} =\prod _{k=1}^{n-1}(1+q^k), \end{aligned}$$

as desired. $\square$

Theorem 4.2

For $D_n$, we have the following signed Mahonian polynomials.

• 1. $\displaystyle \sum _{\pi \in D_n}(-1)^{\ell _D(\pi )}q^{\ell _D(\pi )}=[n]_{-q}\prod _{k=1}^{n-1}[2k]_{-q}.$

• 2. $\displaystyle \sum _{\pi \in D_n}(-1)^{\ell _D(\pi )}q^{{\textsf {dmaj}}(\pi )}=[n]_{(-1)^{n-1}q}\prod _{k=1}^{n-1}[2]_{(-q)^k}[k]_{(-1)^{k-1}q}.$

Proof

• 1. It is obtained by replacing q by $-q$ in (4.1).

• 2. Recall that $D_n=U^D_n\cdot \mathfrak {S}_n$. By Lemma 4.1 and (1.2), the left-hand side is equal to

  $$\begin{aligned} \sum _{\tau \in U^D_n}(-q)^{-\left( \sum _{i\in {\textsf {Neg}}(\tau )}\tau _i\right) -{\textsf {neg}}(\tau )} \sum _{\rho \in \mathfrak {S}_n}(-1)^{{\textsf {inv}}(\rho )}q^{{\textsf {maj}}(\rho )}=B^D(-q)\prod _{k=1}^n [k]_{(-1)^{k-1}q}, \end{aligned}$$

  as desired. $\square$

Remark 4.3

Note that Biagioli and Caselli [7] defined two other Mahonian indices over $D_n$, namely the flag major index ${\textsf {fmaj}}$ and D-major index ${\textsf {Dmaj}}$. The signed Mahonian polynomials of ${\textsf {fmaj}}$ and ${\textsf {Dmaj}}$ were derived in [8] and [6], respectively.

In what follows, we consider ${\textsf {sor}}_D$. Define elements $\theta _i$ of the group algebra of $D_n$ by $\theta _1:=1$ and for $2\le k\le n$

$$\begin{aligned} \theta _k:=1+\sum _{i=1}^{k-1}t_{i,k}+\sum _{i=1}^{k}t_{\bar{i},k}. \end{aligned}$$

It has been shown in [19] that

$$\begin{aligned} \theta _1\theta _2\cdots \theta _n = \sum _{\pi \in D_n}\pi . \end{aligned}$$

(4.2)

Theorem 4.4

For ${\textsf {sor}}_D$, we have the following signed Mahonian polynomials:

$$\begin{aligned} \sum _{\pi \in D_n}(-1)^{\ell _D(\pi )}q^{{\textsf {sor}}_D(\pi )} = \prod _{k=2}^n \left( 1-(q+q^{k-1})[k-1]_q+q^{2k-2} \right) . \end{aligned}$$

Proof

Define the linear mapping $\Theta :\mathbb {Z}[D_n]\rightarrow \mathbb {Z}[q]$ by

$$\begin{aligned} \Theta (\pi ):=(-1)^{\ell _D(\pi )}q^{{\textsf {sor}}_D(\pi )}. \end{aligned}$$

We first claim that, for $k\ge 2$,

$$\begin{aligned} \Theta (\theta _k) = 1-(q+q^{k-1})[k-1]_q+q^{2k-2}. \end{aligned}$$

(4.3)

Note that $\Theta (\theta _1)=1$.

By the definition of ${\textsf {sor}}_D$, one has ${\textsf {sor}}_D(t_{i,k})=k-i$ for $1\le i<k$, and ${\textsf {sor}}_D(t_{\bar{i},k})=k+i-2$. Following the same argument as in (2.16), we have

$$\begin{aligned} \Theta \left( 1+\sum _{i=1}^{k-1}t_{i,k} \right) = 1-q[k-1]_q. \end{aligned}$$

Next, observe that

$$\begin{aligned} t_{\bar{i},k} = {\left\{ \begin{array}{ll} s_{i-1}s_{i-2}\cdots s_1s_{k-1}s_{k-2}\cdots s_2 s'_0 s_2 \cdots s_{k-2} s_{k-1} s_1\cdots s_{i-2} s_{i-1}, &{} \text {if } 1\le i<k; \\ s_{k-1}s_{k-2}\cdots s_2 s'_0 s_1 s_2 \cdots s_{k-2} s_{k-1}, &{} \text {if } i=k. \end{array}\right. } \end{aligned}$$

Since generators $s_0',s_1,\ldots ,s_{n-1}$ are involutions, the expression of $t_{\bar{i},k}$ above yields

$$\begin{aligned} (-1)^{\ell _D(t_{\bar{i},k})} = {\left\{ \begin{array}{ll} (-1)^{2(i-1+k-2)+1}=-1, &{} \text {if } 1\le i<k; \\ (-1)^{2(k-1)}=1, &{} \text {if } i=k. \end{array}\right. } \end{aligned}$$

This implies that $\Theta (t_{\bar{i},k})=-q^{k+i-2}$ for $1\le i<k$, and $\Theta (t_{\bar{k},k})=q^{2k-2}$. Therefore,

$$\begin{aligned} \Theta (\theta _k)&= 1 - q[k-1]_q - \sum _{i=1}^{k-1}q^{k+i-2} + q^{2k-2}, \end{aligned}$$

and thus (4.3) follows.

By (4.2) it suffices to show $\Theta (\theta _1\cdots \theta _n)=\Theta (\theta _1)\cdots \Theta (\theta _n)$. We omit the rest of the proof since it can be dealt with in the same fashion as we did in the proof of Theorem 2.4. $\square$

5 Signed counting on another class of statistics on G(r, n)

In this section, we investigate four equidistributed statistics on G(r, n): ${\textsf {fmaj}}$, ${\textsf {rmaj}}$, ${\textsf {fmaf}}$ and ${\textsf {rinv}}$, the last of which is new. Hence we can consider the polynomial

$$\begin{aligned} \sum _{\pi \in G(r,n)}(-1)^{{\textsf {stat}}_1(\pi )}q^{{\textsf {stat}}_2(\pi )} \end{aligned}$$

by taking two of these four statistics. It turns out that among the possible 12 cases in many of which we can have nice closed forms.

5.1 Equidistributed statistics

Throughout this section both ${\textsf {inv}}$ and ${\textsf {maj}}$ are calculated with respect to the linear order

$$\begin{aligned} \left( 1^{[r-1]}<\cdots<n^{[r-1]}\right)<\cdots<\left( 1^{[1]}<\cdots<n^{[1]}\right)<(1<\cdots <n). \end{aligned}$$

Let $\pi =(z, \sigma )\in G(r,n)$ and define the following statistics:

• ${\textsf {fmaj}}$. The flag major index [3] is defined by

  $$\begin{aligned} {\textsf {fmaj}}(\pi ):=r\cdot {\textsf {maj}}(\pi )+\sum _{i=1}^n z_i. \end{aligned}$$

• ${\textsf {rmaj}}$. The root major index[16] is defined by

  $$\begin{aligned} {\textsf {rmaj}}(\pi ):={\textsf {maj}}(\pi )+\sum _{i=1}^n z_i\cdot \sigma _i. \end{aligned}$$

• ${\textsf {fmaf}}$. The flag maf index[12] is defined by

  $$\begin{aligned} {\textsf {fmaf}}(\pi ):=r\cdot \sum _{j=1}^k(i_j-j)+{\textsf {fmaj}}(\tilde{\pi }), \end{aligned}$$

  where $\{i_1,i_2,\ldots ,i_k\}$ is the fix set ${\textsf {Fix}}(\pi ):=\{i:\pi _i=i\}$ and $\tilde{\pi }\in G(r,n-k)$ is obtained from $\pi$ by deleting fixed terms and then renumbering the remaining terms with order preserved.

For example, for $\pi =2^{[1]}\,1^{[3]}\,5\,4\,3^{[2]} \in G(4,5)$, we have ${\textsf {fmaj}}(\pi )=38$, ${\textsf {rmaj}}(\pi )=19$ and ${\textsf {fmaf}}(\pi )=34$, with $\tilde{\pi } = 2^{[1]} 1^{[3]} 43^{[2]}$.

These three statistics have the same distribution over G(r, n) and the following generating function (see [12, 16])

$$\begin{aligned} \sum _{\pi \in G(r,n)} q^{{\textsf {fmaj}}(\pi )}=\sum _{\pi \in G(r,n)} q^{{\textsf {fmaf}}(\pi )} =\sum _{\pi \in G(r,n)} q^{{\textsf {rmaj}}(\pi )}=\prod _{k=1}^n[rk]_q. \end{aligned}$$

(5.1)

Note that the statistic ${\textsf {fmaj}}$ on G(r, n) is a generalization of ${\textsf {fmaj}}$ on $B_n$. However, it is no longer equidistributed with the length function for $r\ge 3$.

Now we define a new statistic root inversion index ${\textsf {rinv}}$ on G(r, n).

Definition 5.1

Given $\pi =(z,\sigma )\in G(r,n)$, the root inversion number of $\pi$ is

$$\begin{aligned} {\textsf {rinv}}(\pi ):={\textsf {inv}}(\pi )+\sum _{i=1}^n z_i\cdot \sigma _i. \end{aligned}$$

For example, for $\pi =2^{[1]}\,1^{[3]}\,5\,4\,3^{[2]} \in G(4,5)$ we have ${\textsf {rinv}}(\pi )=16$. Similar to the fact that ${\textsf {inv}}$ and ${\textsf {maj}}$ have a symmetric joint distribution

$$\begin{aligned} \sum _{\pi \in \mathfrak {S}_n}t^{{\textsf {inv}}(\pi )}q^{{\textsf {maj}}(\pi )}=\sum _{\pi \in \mathfrak {S}_n}q^{{\textsf {inv}}(\pi )}t^{{\textsf {maj}}(\pi )} \end{aligned}$$

(5.2)

over $\mathfrak {S}_n$ [15], we will prove that the statistics ${\textsf {rinv}}$ and ${\textsf {rmaj}}$ have a symmetric joint distribution over G(r, n).

Theorem 5.2

The joint distribution of ${\textsf {rmaj}}$ and ${\textsf {rinv}}$ is symmetric over G(r, n). That is,

$$\begin{aligned} \sum _{\pi \in G(r,n)}t^{{\textsf {rinv}}(\pi )}q^{{\textsf {rmaj}}(\pi )}= \sum _{\pi \in G(r,n)}q^{{\textsf {rinv}}(\pi )}t^{{\textsf {rmaj}}(\pi )}. \end{aligned}$$

Proof

Recall that $Z(\pi )=\sum z_i$ for $\pi =(z,\sigma )\in G(r,n)$. Let $\hat{Z}(\pi ):=\sum z_i\cdot \sigma _i$ and therefore $\hat{Z}(\pi )=\hat{Z}(\tau )$ if $\pi =\tau \rho \in U_{r,n}\cdot \mathfrak {S}_n$.

$$\begin{aligned}&\sum _{\pi \in G(r,n)}t^{{\textsf {rinv}}(\pi )}q^{{\textsf {rmaj}}(\pi )} =\sum _{\pi \in G(r,n)}t^{{\textsf {inv}}(\pi )+\hat{Z}(\pi )}q^{{\textsf {maj}}(\pi )+\hat{Z}(\pi )}\\&\quad =\sum _{\tau \in U_{r,n},\,\rho \in \mathfrak {S}_n}t^{{\textsf {inv}}(\rho )+\hat{Z}(\tau )}q^{{\textsf {maj}}(\rho )+\hat{Z}(\tau )} =\sum _{\tau \in U_{r,n}}(tq)^{\hat{Z}(\tau )}\sum _{\rho \in \mathfrak {S}_n}t^{{\textsf {inv}}(\rho )}q^{{\textsf {maj}}(\rho )}\\&\quad {\mathop {=}\limits ^{(*)}}\sum _{\tau \in U_{r,n}}(tq)^{\hat{Z}(\tau )}\sum _{\rho \in \mathfrak {S}_n}q^{{\textsf {inv}}(\rho )}t^{{\textsf {maj}}(\rho )} =\sum _{\tau \in U_{r,n},\,\rho \in \mathfrak {S}_n}q^{{\textsf {inv}}(\rho )+\hat{Z}(\tau )}t^{{\textsf {maj}}(\rho )+\hat{Z}(\tau )}\\&\quad =\sum _{\pi \in G(r,n)}q^{{\textsf {inv}}(\pi )+\hat{Z}(\pi )}t^{{\textsf {maj}}(\pi )+\hat{Z}(\pi )} =\sum _{\pi \in G(r,n)}q^{{\textsf {rinv}}(\pi )}t^{{\textsf {rmaj}}(\pi )}, \end{aligned}$$

where $(*)$ is because of the symmetric joint distribution of ${\textsf {maj}}$ and ${\textsf {inv}}$ over $\mathfrak {S}_n$. $\square$

5.2 Signed counting polynomials

The following bivariate generating function, which is an extension of (3.3), will play a key role in our discussion. The proof is again similar to that of Lemma 2.2 and is omitted.

Lemma 5.3

We have

$$\begin{aligned} R(t,q):=\sum _{\tau \in U_{r,n}}t^{Z(\tau )}q^{\hat{Z}(\tau )}=\prod _{k=1}^n[r]_{tq^k}. \end{aligned}$$

We are ready for the main results of this section. First we have ${\textsf {rmaj}}$ v.s. ${\textsf {fmaj}}$.

Theorem 5.4

$$\begin{aligned} \sum _{\pi \in G(r,n)}t^{{\textsf {rmaj}}(\pi )}q^{{\textsf {fmaj}}(\pi )} =\prod _{k=1}^{n}[r]_{t^kq}[k]_{tq^r}. \end{aligned}$$

Proof

The left-hand side is equal to

$$\begin{aligned} \sum _{\tau \in U_{r,n}}t^{\hat{Z}(\tau )}q^{Z(\tau )}\sum _{\rho \in \mathfrak {S}_n}t^{{\textsf {maj}}(\rho )}q^{r\cdot {\textsf {maj}}(\rho )}, \end{aligned}$$

and the result follows by Lemma 5.3 and (1.1). $\square$

Then we have ${\textsf {rinv}}$ v.s. ${\textsf {rmaj}}$:

Theorem 5.5

$$\begin{aligned} \sum _{\pi \in G(r,n)}(-1)^{{\textsf {rinv}}(\pi )}q^{{\textsf {rmaj}}(\pi )} = \sum _{\pi \in G(r,n)}(-1)^{{\textsf {rmaj}}(\pi )}q^{{\textsf {rinv}}(\pi )} = \prod _{k=1}^n[r]_{(-q)^{k}}[k]_{(-1)^{k-1}q}. \end{aligned}$$

Proof

The left-hand side is equal to

$$\begin{aligned} \sum _{\tau \in U_{r,n}}(-q)^{\hat{Z}(\tau )}\sum _{\rho \in \mathfrak {S}_n}(-1)^{{\textsf {inv}}(\rho )}q^{{\textsf {maj}}(\rho )}, \end{aligned}$$

and the result follows by Theorem 5.2, Lemma 5.3 and (1.2). $\square$

And we have ${\textsf {rinv}}$ v.s. ${\textsf {fmaj}}$:

Theorem 5.6

$$\begin{aligned} \sum _{\pi \in G(r,n)}(-1)^{{\textsf {rinv}}(\pi )}q^{{\textsf {fmaj}}(\pi )} = \prod _{k=1}^n[r]_{(-1)^kq}[k]_{(-1)^{k-1}{q^r}} \end{aligned}$$

and

$$\begin{aligned} \displaystyle \sum _{\pi \in G(r,n)}(-1)^{{\textsf {fmaj}}(\pi )}q^{{\textsf {rinv}}(\pi )} = \prod _{k=1}^n[r]_{-q^k}[k]_{(-1)^{(k-1)r}q}. \end{aligned}$$

Proof

By (1.2) and Lemma 5.3,

$$\begin{aligned}&\sum _{\pi \in G(r,n)}(-1)^{{\textsf {rinv}}(\pi )}q^{{\textsf {fmaj}}(\pi )} =\sum _{\tau \in U_{r,n}}(-1)^{\hat{Z}(\tau )}q^{Z(\tau )}\sum _{\rho \in \mathfrak {S}_n}(-1)^{{\textsf {inv}}(\rho )}q^{r\cdot {\textsf {maj}}(\rho )}\\&\quad = R(q, -1)\prod _{k=1}^{n}[k]_{(-1)^{k-1}{q^r}} =\prod _{k=1}^n[r]_{(-1)^kq}[k]_{(-1)^{k-1}{q^r}}. \end{aligned}$$

Similarly,

$$\begin{aligned}&\sum _{\pi \in G(r,n)}(-1)^{{\textsf {fmaj}}(\pi )}q^{{\textsf {rinv}}(\pi )}= \sum _{\tau \in U_{r,n}}(-1)^{Z(\tau )}q^{\hat{Z}(\tau )}\sum _{\rho \in \mathfrak {S}_n}(-1)^{r\cdot {\textsf {maj}}(\rho )}{q^{{\textsf {inv}}(\rho )}}\\&\quad {\mathop {=}\limits ^{(*)}} R(-1, q)\prod _{k=1}^{n}[k]_{(-1)^{(k-1)r}q} =\prod _{k=1}^n[r]_{-q^k}[k]_{(-1)^{(k-1)r}q}. \end{aligned}$$

$(*)$: the second factor comes from (1.1) (or (1.2) and (5.2)) if r is even (or odd). $\square$

The ${\textsf {fmaf}}$ statistic is more elusive. In the following we will prove signed counting results for even r when $({\textsf {stat}}_1,{\textsf {stat}}_2)=({\textsf {fmaf}},{\textsf {rinv}}), ({\textsf {fmaf}}, {\textsf {rmaj}}),({\textsf {fmaf}},{\textsf {fmaj}})$ or $({\textsf {fmaj}},{\textsf {fmaf}})$.

We need some preparations. For $\pi =(z,\sigma )\in G(r,n)$, denote the cardinality of ${\textsf {Fix}}(\pi )$ by ${\textsf {fix}}(\pi )$. Then ${\textsf {fmaf}}(\pi )$ can be expressed as

$$\begin{aligned} {\textsf {fmaf}}(\pi )&=r\cdot \left( \sum _{i\in {\textsf {Fix}}(\pi )}i-{{\textsf {fix}}(\pi )+1\atopwithdelims ()2}\right) +{\textsf {fmaj}}(\tilde{\pi })\nonumber \\&=r\cdot \left( \sum _{i\in {\textsf {Fix}}(\pi )}i-{{\textsf {fix}}(\pi )+1\atopwithdelims ()2}+{\textsf {maj}}(\tilde{\pi })\right) +Z(\tilde{\pi }). \end{aligned}$$

(5.3)

Theorem 5.7

When r is even, we have

$$\begin{aligned} \sum _{\pi \in G(r,n)}(-1)^{{\textsf {fmaf}}(\pi )}q^{{\textsf {rinv}}(\pi )}= \sum _{\pi \in G(r,n)}(-1)^{{\textsf {fmaf}}(\pi )}q^{{\textsf {rmaj}}(\pi )}=\prod _{k=1}^{n}[r]_{-q^k}[k]_q. \end{aligned}$$

Proof

Let $({\textsf {Stat}},{\textsf {stat}})$ denote $({\textsf {rinv}},{\textsf {inv}})$ or $({\textsf {rmaj}},{\textsf {maj}})$. Since r is even, by (5.3), we have

$$\begin{aligned} \sum _{\pi \in G(r,n)}(-1)^{{\textsf {fmaf}}(\pi )}q^{{\textsf {Stat}}(\pi )}&=\sum _{\pi \in G(r,n)}(-1)^{Z(\tilde{\pi })}q^{{\textsf {Stat}}(\pi )} \\&=\sum _{\tau \in U_{r,n}}(-1)^{Z(\tau )}q^{\hat{Z}(\tau )}\sum _{\rho \in \mathfrak {S}_n}q^{{\textsf {stat}}(\rho )}, \end{aligned}$$

and the result follows by Lemma 5.3 and (1.1). $\square$

Theorem 5.8

When r is even, we have

$$\begin{aligned} \sum _{\pi \in G(r,n)}(-1)^{{\textsf {fmaj}}(\pi )}q^{{\textsf {fmaf}}(\pi )} =\sum _{\pi \in G(r,n)}(-1)^{{\textsf {fmaf}}(\pi )}q^{{\textsf {fmaj}}(\pi )}=\prod _{k=1}^n[rk]_{-q}. \end{aligned}$$

Proof

Let ${\textsf {stat}}_1$ be one of ${\textsf {fmaj}}$ and ${\textsf {fmaf}}$ and ${\textsf {stat}}_2$ be the other. Since r is even, by (5.3) and (5.1), we have

$$\begin{aligned}&\sum _{\pi \in G(r,n)}(-1)^{{\textsf {stat}}_1(\pi )}q^{{\textsf {stat}}_2(\pi )} =\sum _{\pi \in G(r,n)}(-1)^{Z(\pi )}(-1)^{Z(\pi )}(-q)^{{\textsf {stat}}_2(\pi )}\\&\quad =\sum _{\pi \in G(r,n)}(-q)^{{\textsf {stat}}_2(\pi )}=\prod _{k=1}^n[rk]_{-q}. \end{aligned}$$

Note that $Z(\tilde{\pi })=Z(\pi )$ by definition. $\square$

Note that the results for $({\textsf {rmaj}}, {\textsf {fmaf}})$ and $({\textsf {rinv}}, {\textsf {fmaf}})$, as well as the odd r cases in Theorems 5.7 and 5.8 are not known. The authors would like to know if these missing results can be completed.

6 Concluding notes

In this paper, we proposed many new signed Mahonian polynomials for $B_n$, $D_n$ and $G(r,n)=G(r,1,n)$. In summary, for type $B_n$ we derived all signed Mahonian polynomials with respect to the statistics $\ell _B$, ${\textsf {nmaj}}$, ${\textsf {Fmaj}}$ and ${\textsf {sor}}_B$, while for type $D_n$ we derived polynomials with respect to $\ell _D$, ${\textsf {dmaj}}$ and ${\textsf {sor}}_D$. For G(r, n) we first derived a closed formula for the 1-dim characters of G(r, n) in terms of the length function, and as an application of it we obtained signed Mahonian polynomials with respect to $\ell$, ${\textsf {lmaj}}$ and ${\textsf {sor}}$. This is novel in the sense that the better-known ${\textsf {fmaj}}$ statistic is not equidistributed with the length function if $r\ge 3$ while ${\textsf {lmaj}}$ is Mahonian.

Caselli [10] introduced the concept of projective reflection groups G(r, p, s, n), which is a generalization of complex reflection groups G(r, p, n), and Biagioli and Caselli [8] derived signed polynomials with respect to the statistic ${\textsf {fmaj}}$. Hence, a natural question is to see whether one can have a Mahonian statistic ${\textsf {lmaj}}$ or ${\textsf {sor}}$ on G(r, p, n) or G(r, p, s, n) and have corresponding signed Mahonian polynomials.

In the last section we considered the signed polynomials on G(r, n) with respect to a pair of statistics with the same distribution $[r]_q[2r]_q\cdots [nr]_q$, where the ‘sign’ is decided by the parity of the first statistic. It would be interesting to complete the missing pieces in the last section, especially the $({\textsf {rmaj}}, {\textsf {fmaf}})$ and $({\textsf {rinv}}, {\textsf {fmaf}})$ cases. Also it is natural to investigate the bivariate generating functions, or even better, the multivariate generating functions with respect to these statistics. We leave these questions to the interested readers.