Brook Mg
email: gary.brook@nwlh.nhs.uk

The effect of electronic patient records (EPR) on the time taken to treat patients with genital Chlamydia infection

Keywords: Chlamydia, Electronic Patient Records, Electronic Case Records Word count: 1937, excluding abstract, figures and tables

Objectives:

To measure the time taken to recall and treat patients with untreated Chlamydia attending a sexual health clinic before and after the introduction of EPR.

Methods

: 52 consecutive qualifying patients were identified for Jan-Mar 2007 (paper case records) and 2009 (EPR). For each the patient time intervals were measured between each of following dates: first attendance, first positive result received, first attempted patient contact and attendance for treatment. Results: Between 2007 and 2009, the median time taken to treat a patient after receipt of a positive Chlamydia result fell by 11.5 days (median 15 days in 2007, 3.5 days in 2009). The time between first attendance and treatment was reduced by 9.5 days (median 21 days in 2007, 11.5 days in 2009) despite results taking 2 days longer to arrive in 2009. The proportion of patients treated within 2 weeks of a positive result rose from 38% in 2007 to 94% in 2009. Compared with paper notes, EPR decreased the time to recall, by eliminating three time-delaying patient recall processes. By

ensuring more accurate patient recall information, EPR also lead to a higher proportion of patients successfully recalled by telephone (2 26 6/ /4 44 4 , ,5 59 9% % v ve er rs su us s 4 46 6/ /5 52 2, , 8 88 8% % i in n 2 20 00 07 7 a an nd d 2 20 00 09 9 r re es sp pe ec ct ti iv ve el ly y) ), , l le ea ad di in ng g t to o e ea ar rl li ie er r t tr re ea at tm me en nt t. .

Conclusions:

The 'time to treat' interval was dramatically reduced following the introduction of EPR. Clinics using paper notes should consider switching to EPR as a means of improving STI recall efficiency.

Introduction

The longer an STI goes untreated, the more risk there is of onward transmission and of clinical complications. [START_REF] White | Vicious and virtuous circles in the dynamics of infectious disease and the provision of health care: gonorrhea in Britain as an example[END_REF] Any system that makes patient recall more efficient and effective therefore serves to reduce the spread and complications of STIs. [START_REF] White | Vicious and virtuous circles in the dynamics of infectious disease and the provision of health care: gonorrhea in Britain as an example[END_REF] Theoretically, EPR are more efficient to use than paper-based records for recall due to instant access and the use of failsafe mechanisms which ensure better patient contact information. [START_REF] Brook | Implementation of Electronic Patient Records (EPR) in a Sexual Health Clinic[END_REF] This clinic changed from paper-based records to EPR in August 2007 [START_REF] Brook | Implementation of Electronic Patient Records (EPR) in a Sexual Health Clinic[END_REF] and thus was in a position to compare any subsequent efficiencies. We also switched from routine genital swabs to urine-based Chlamydia and gonorrhoea testing for asymptomatic patients between the two time periods, The number of men with Chlamydia who had been treated as asymptomatic NSU on the day of attendance was therefore reduced in 2009 compared with 2007. This study looks at time taken to recall and treat the two groups of asymptomatic patients with Chlamydia in each of these two years.

Key messages

Rapid recall and treatment of patients with a diagnosed STI such as Chlamydia reduces the risk of onward transmission and complications.

EPR improves patient recall through instant access to case notes and increased accuracy of recall information, leading to early treatment of patients with Chlamydia.

Sexual health clinics without EPR should move towards such a system as a means of improving their efficiency and effectivenss.

Methods

Two periods were chosen: January -March 2007 (paper note system) and January -March 2009 (EPR, Lilie Sexual Health Management System, Blithe Computer Systems Ltd.). For each period, 52 consecutive patients were identified who had a diagnosis of genital Chlamydia infection and had not been treated on the day the test was taken. For each, the following dates were noted: first attendance, first positive result, first attempted patient contact, day of patient treatment. The time intervals between each date were measured. We also recorded the methods of contact and gender of patients and assessed the affect of each on recall time.

The result handling and patient recall mechanisms for each time period were described and each step compared to ascertain their effects on recall efficiency.

Statistics

Sample size calculation: A minimum of 49 patients was required for each year to detect a difference of the mean times taken to treat patients after result receipt of 1 day (assuming a mean of 4 and 3 days and standard deviation of 2 days for 2007 and 2009 respectively) using a 5% alpha and 80% statistical power level. [START_REF]Sample size calculator. Two samples using average values[END_REF] Differences between the groups of time intervals were measured using the Mann-Whitney test. [START_REF] Lowry | Whitney test[END_REF] Differences in proportions of factors between the two years were measured using the Chi-Square test. [START_REF] Preacher | Calculation for the Chi-square test[END_REF]

Results

The processes for recall of patients with a positive Chlamydia test are shown in figures 1 and 2. There are seven steps leading to patient recall using paper notes in 2007, each with inherent possible delays. In 2009 the EPR system had eliminated three of these steps (steps 2-4). In addition, the EPR system allowed for electronic prompts to ensure that all staff ensure that the patients address and telephone details are correct.

The major recall times and other results comparing 2007 and 2009 are given in Table 1. The majority of patients (>80%) were in the age range 16-30 years. shows that the major inefficiencies of the paper records system were the long times taken find notes and the inaccurate contact information they contained, leading to delays in contacting the patients. EPR allowed for much more efficient working in these areas.

The need to treat patients quickly is self-evident. Epidemiological modelling has shown that the rapid treatment of patients with diagnosed infection reduces the incidence of acute STIs, [START_REF] White | Vicious and virtuous circles in the dynamics of infectious disease and the provision of health care: gonorrhea in Britain as an example[END_REF] as well as preventing clinical complications. The use of paper case records in clinics mitigates against this as they lead to inherent inefficiencies, such as delays in finding the notes and difficulties in keeping the patient contact information reliably updated. [START_REF] Brook | Implementation of Electronic Patient Records (EPR) in a Sexual Health Clinic[END_REF] EPR overcomes these problems by, amongst other things, ensuring instant access to the case record and prompting staff members to ensure accurate patient contact information. Automated letters and text messages can also be generated from EPR systems and it has been previously demonstrated that patient-recall efficiency is also seen with automated texting services. [START_REF] Menon-Johansson A S | Texting decreases the time to treatment for genital Chlamydia trachomatis infection[END_REF] In this study, most patients in the EPR arm were contacted on the day of the receipt of the result and treated within 3.5 days. By comparison, it took an average of 7 days to contact the patient and a further 8 days to treat them in the paper notes arm. How does this compare with any available outcome targets? In 2004, national outcome standards suggested that 70% of patients (50% in London) should be treated for Chlamydia within 4 weeks of diagnosis. [START_REF] Low | Developing national outcome standards for the management of gonorrhoea and genital chlamydia in genitourinary medicine clinics[END_REF] National Chlamydia guidelines in the UK and US have not set time-to-treat standards [START_REF] Horner | UK National Guideline for the Management of Genital Tract Infection with Chlamydia trachomatis[END_REF][START_REF]Sexually transmitted diseases treatment guidelines 2006:Chlamydial Infections in Adolescents and Adults[END_REF] although the English National Chlamydia Screening Programme suggests that 50% and 80% of patients should be treated within 14 and 30 days respectively.

[10] We far exceeded these standards in 2009, as we treated 94% and 98% of initially untreated patients within 14 and 30 days of a positive result.

The strength of this study is that it compares patients from the same cohort, at two multiple prompts to ensure that telephone numbers were correct. In the paper notes telephone numbers were more frequently incorrect and so more recall was by letter.

The different proportion of patients recalled by letter/telephone in the two periods are therefore a direct result of the change to EPR and could not be seen as a bias in the study.

We suggest that with the increasing use of non-invasive Chlamydia screening methods such as urine sampling, and more effective patient recall, new national standards should be set for the time taken to recall and treat infected patients.

This study shows that an EPR system is superior to paper-based case records in enabling the efficient and effective treatment of patients with STIs such as Chlamydia. EPR also has several other inherent advantages over paper record. [START_REF] Brook | Implementation of Electronic Patient Records (EPR) in a Sexual Health Clinic[END_REF].

  96 and 45 patients with Chlamydia were excluded from the analysis in 2007 and 2009 respectively, having been treated at the initial day of attendance. There are more men (35%) in the 2009 sample than in 2007 (8%), as less men in 2009 had been treated on the initial day of attendance (as asymptomatic NSU) following the introduction of urine-based Chlamydia/gonorrhoea testing in late 2007.

  le ep ph ho on ne e r re ec ca al ll l, , 2007 Vs 2009 2009, patients successfully recalled by telephone were seen/contacted less quickly for the intervals results to treatment, results to first attempted contact and attendance to treatment . This was due to the longer time delays between results arrival and the first attempted contact in 2007 as a result of the time taken for paper notes to be sought by administration staff. Discussion This study demonstrates that since the introduction of EPR, the time taken to treat patients after receipt of a positive Chlamydia result has dramatically reduced. In comparison with 2007, patients in 2009 were contacted a median of 7 days sooner and treated a median of 11.5 days earlier following the positive result. The overall time from first attendance to treatment was also reduced by a median of 9.5 days in 2009, even though results took a median 2 days longer to return from the laboratory. This decrease in time to treat has led to a large increase in the proportion of patients treated within 2 weeks of the result, rising from 37% in 2007 to 94% in 2009. The reasons for this are clear. Analysis of the clinic systems for patient recall used in 2007 and 2009

  time periods, in a normal clinical situation and is therefore a reflection of standard clinical practice. It is also the first paper to demonstrate how EPR can improve the efficiency of patient recall within the setting of a sexual health clinic. Potential weaknesses are that other changes unrelated to EPR might have occurred between the two time periods that led to these improvements. We have tried to eliminate confounding factors by looking for bias related to gender and mode of communication. There were more men eligible for the study in 2009, as in 2007 the majority of men with Chlamydia were treated as NSU at first attendance, when urethral swabs were used routinely. In 2009 most asymptomatic men had a urinebased test leading to more of them requiring later recall. Although more patients had a successful telephone recall in 2009, this was inherently due to the EPR as it has

Table 1 :

 1 Overall differences between 2007 and 2009

	Median (range) time	2007 (N=52)	2009 (N=52)	Difference	P value
	intervals measured in days:			in median
				days
	Result to treatment	15 (1-112)	3.5 (0-30)	-11.5	<0.0001
	Result to 1st attempted contact 7 (1-20)	0 (0-7)	-7	<0.0001
	1st attendance to treatment	21 (6-117)	11.5 (3-39)	-9.5	<0.0001
	1st attendance to 1st attempted	13 (6-45)	8 (3-12)	-5	<0.0001
	contact
	1 1 s st t A At tt te en nd da an nc ce e t to o R Re es su ul lt t	5 5 ( (1 1--3 32 2) )	7 7 ( (3 3--1 12 2) )	+ +2 2	<0.0001
	O

Ot th he er r R Re es su ul lt ts s

  Sub-group analyses were performed to look in more detail at the factors influencing recall times and times to treatment (table2). Analysis of the data comparing recall times for men and women showed no statistical differences in 2009. More people had to be contacted by letter in 2007 as the recorded telephone numbers in the notes were more likely to be inaccurate in 2007 compared with 2009. In 2007, patients recalled by letter took longer than those successfully recalled by telephone for the intervals: result to treatment and attendance to treatment

	P Pr ro op po or rt ti io on n t tr re ea at te ed d w wi it th hi in n 1 14 4	1 16 6/ /4 46 6* * ( (3 37 7% %) )	4 49 9/ /5 52 2 ( (9 94 4% %) )	0.0067
	d da ay ys s o of f t th he e r re es su ul lt t
	P Pr ro op po or rt ti io on n t tr re ea at te ed d w wi it th hi in n 3 30 0	3 39 9/ /4 46 6* * ( (8 85 5% %) )	5 50 0/ /5 52 2 ( (9 96 6% %) )	0.66
	d da ay ys s o of f t th he e t te es st t
	P Pr ro op po or rt ti io on n t tr re ea at te ed d w wi it th hi in n 3 30 0	4 40 0/ /4 46 6* * ( (8 87 7% %) )	5 51 1/ /5 52 2 ( (9 98 8% %) )	0.68
	d da ay ys s o of f t th he e r re es su ul lt t
	F Fe em ma al le es s i in n e ea ac ch h g gr ro ou up p	4 48 8/ /5 52 2 ( (9 92 2% %) )	3 34 4/ /5 52 2 ( (6 65 5% %) )	0.25
	S Su uc cc ce es ss sf fu ul l t te el le ep ph ho on ne e r re ec ca al ll l	2 26 6/ /4 44 4 ( (5 59 9% %) )* *	4 46 6/ /5 52 2 ( (8 88 8% %) )	0.20
	S Su uc cc ce es ss sf fu ul l l le et tt te er r r re ec ca al ll l	1 18 8/ /4 44 4 ( (4 41 1% %) )* *	6 6/ /5 52 2 ( (1 12 2% %) )	0.019
	*-Some data points missing

Table 2 .

 2 Sub-group analyses

	Time intervals	Median (range) in	Median (range)	Difference in	P value
		days	in days	median days
	Result to treatment, Women Vs	Women 3 (0-30)	Men 5.5 (0-11)	2	0.97
	men 2009
	Result to 1st attempted contact	Women 0 (0-7)	Men 0 (0-5)	0	0.79
	Women Vs men 2009
	1st attendance to treatment,	Women 10.5 (3-39) Men 13 (8-19)	2.5	0.25
	Women Vs men 2009
	1st attendance to 1st attempted	Women 7 (3-14)	Men 8.5 (6-12)	1.5	0.08
	contact, Women Vs men 2009
	Result to treatment, Letter	Letter 17 (6-112)	P Ph ho on ne e 11 (1-83) --6 6	0.022
	recall Vs phone 2007
	Result to 1st attempted contact	L Le et tt te er r 7 7 ( (2 2--1 13 3) )	P Ph ho on ne e 7 7 ( (1 1--2 20 0) )	0 0	0 0. .8 87 7
	Letter recall Vs phone 2007
	1st attendance to treatment,	L Le et tt te er r 21 (12-117)	P Ph ho on ne e 17 (6-91) --4 4	0.024
	Letter recall Vs phone 2007
	1st attendance to Results,	L Le et tt te er r 4 4 ( (1 1--9 9) )	P Ph ho on ne e 5 5 ( (1 1--1 11 1) )	1 1	0 0. .1 10 09 9
	Letter recall Vs phone 2007
	Result to treatment T Te el le ep ph ho on ne e	2007 11 (1-83)
	r re ec ca al ll l, , 2007 Vs 2009

  In theory all patients would be recalled on the day the result is received without the intervention of a person. Future research should look at how additional automated recall technology can be added to current EPR systems and what effect they would have on patient recall for treatment. Appropriate use of technology greatly improves our ability to treat patients rapidly and we should strive to use all available methods, for the good of our patients and the betterment of public health.Fig 2. STI Recall Using EPR Notes Results arrive at the same time each day so time is set aside for patient recall.Prompt fields on the EPR system ensure an accurate telephone number Patient address in EPR system used. Automatic one-click recall letters generated.

	Clinics still running paper-based case records should strongly consider switching to EPR. What research should be performed in the future? EPR and other technology, such as automated patient recall following the electronic receipt of results, might in future 1. Positive STI result arrives from Lab. 2. Request to reception to find the patients' notes 3. Notes sought by reception 4. Notes returned to Health Advisers 5. Health Adviser attempts to contact patient by phone 7. Patient returns for treatment Reception or Health Adviser may be Notes busy Some notes cannot be found readily Health Adviser has to make time to deal with the notes as they return Telephone number may not have Address may not have been updated 1. Positive STI result is received from the lab. directly by the Health Adviser 2. Health Adviser has immediate access to EPR and attempts to contact patient by phone/text 3. Patients who cannot be contacted by phone have a letter written further reduce the time to treat interval. Fig 1. STI Recall Using Paper Notes been updated in the notes 6. Patients who cannot be contacted Time taken to print the letters. by phone have a letter written 4. Patient returns for treatment

Acknowledgments: We would like to acknowledge the efforts of the clinic Health Advisers who were key to the success of the patient recall.

Competing Interests: GB has performed training and lecturing on the use of EPR for Blithe Computer Systems Ltd. in return for educational donations to the clinic. TB, LS, SS-None.

Funding: None "The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd to permit this article (if accepted) to be published in STI and any other BMJPGL products and sublicences such use and exploit all subsidiary rights, as set out in our licence http://sti.bmjjournals.com/ifora/licence.pdf)".


Riccardo Biagioli

Fabrizio Caselli

ENUMERATING PROJECTIVE REFLECTION GROUPS

Keywords: Reflection groups, characters, permutation statistics, generating functions

Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r, p, s, n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r, p, n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r, p, s, n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r, p, s, n), as distribution of one-dimensional characters and computation of Hilbert series of invariant algebras, are also treated.

Introduction

The study of permutation statistics, and in particular of those depending on up-down or descents patterns, is a very classical subject of study in algebraic combinatorics that goes back to the early 20th century to the work of MacMahon [START_REF] Macmahon | Combinatorial Analysis[END_REF], and has found a new interest in the last decade after the fundamental work of Adin and Roichman [START_REF] Adin | The Flag Major Index and Group Actions on Polynomial Rings[END_REF]. In their paper they defined a new statistic, the flag major index that extend to the Weyl group of type B the concept of major index, which is a classical and well studied statistic over the symmetric group. This paper opened the way to several others concerning the study of statistics over classical Weyl groups, wreath products, and some types of complex reflection groups. Now we have a wide picture of several combinatorial and algebraic properties holding for all these families of groups.

Recently, the second author introduced a new class of groups called projective reflection groups [START_REF] Caselli | Projective reflection groups[END_REF]. Among them, there is an infinite family denoted G(r, p, s, n) where r, p, s, n are integers such that p and s divide r, and ps divides rn. They include the infinite family of irreducible complex reflection groups, in fact G(r, p, 1, n) = G(r, p, n). In this context is fundamental a notion of duality. Let G = G(r, p, s, n), then his dual group is G * = G(r, s, p, n), where the parameters p and s are interchanged. In [START_REF] Caselli | Projective reflection groups[END_REF] it is shown that much of the representation theory of reflection groups can be extended to projective reflection groups, and that the combinatorics of the group G = G(r, p, s, n) is strictly related to the invariant theory of the dual group G * .

In this paper we continue that study. We introduce two descent numbers des and fdes, and a color sum col over G(r, p, s, n), that together with the flag major index fmaj [START_REF] Caselli | Projective reflection groups[END_REF] allow the extension to the groups G(r, p, s, n) of classical results in permutation enumeration.

We show that the polynomial χ(g)q fmaj(g) , where χ denotes any one-dimensional character of G(r, p, s, n) and the sum is over g ∈ G(r, p, s, n), admits a nice product formula. This generalizes a classical result of Gessel and Simion [START_REF] Wachs | An involution for signed Eulerian numbers[END_REF] for the symmetric group, and the main results of Adin-Gessel-Roichman [3] for the Coxeter group of type B, and of the first author [START_REF] Biagioli | Signed Mahonian polynomials for classical Weyl groups[END_REF] for the type D case.

The enumeration of permutations by number of descents and major index yields a remarkable q-analogue of a well-known identity for the Eulerian polynomials, usually called Carlitz's identity. This identity has been generalized in several ways and to several groups, see e.g. [START_REF] Adin | Descent Numbers and Major Indices for the Hyperoctahedral Group[END_REF][START_REF] Bagno | Colored-Descent Representations of Complex Reflection Groups G(r, p, n)[END_REF][START_REF] Biagioli | Invariant algebras and major indices for classical Weyl groups[END_REF][START_REF] Biagioli | On some analogues of descent numbers and major index for the hyperoctahedral group[END_REF][START_REF] Biagioli | Enumerating wreath products via Garsia-Gessel bijections[END_REF][START_REF] Chow | On the descent numbers and major indices for the hyperoctahedral group[END_REF][START_REF] Reiner | Descents and one-dimensional characters for classical Weyl groups[END_REF]. All these extensions can be divided into two families, depending on the used descent statistic, either the number of geometric descents or the number of flag descents. We give a general method to compute the trivariate distribution of des (or fdes), fmaj and col over G(r, p, s, n). This unifies and generalizes all previous cited results.

We further exploit our method to compute the generating function of the six statistics des, ides, fmaj, ifmaj, col, icol over the group G(r, p, s, n), (ides(g) denotes des(g -1 ) an similarly for the others). A specialization of this identity leads us to the computation of the generating function of the Hilbert series of the diagonal invariant algebras of the groups G(r, p, s, n).

The definition of the above statistics depends on the particular order chosen on the colored integer numbers. By comparing our results with others in the literature, we deduce that a different choice of the order may lead to the same generating function. We conclude the paper by providing bijective explanations of this phenomenon.

Notation and preliminaries

In this section we collect the notations that are used in this paper as well as the preliminary results that are needed.

We let Z be the set of integer numbers and N be the set of nonnegative integer numbers. For a, b ∈ Z, with a ≤ b we let [a, b] = {a, a + 1, . . . , b} and, for n . For x ∈ Q, we denote ⌊x⌋ the biggest integer smaller than or equal to x. As usual for n ∈ N, we set [n] q def = 1 + q + q 2 + . . . + q n-1 , and P n the set of nondecreasing sequences of nonnegative integers (λ 1 , λ 2 , . . . , λ n ), that is partitions of length less than or equal to n. Let F ∈ C[[q 1 , q 2 , . . .]] be a formal power series with complex coefficients, and M a monomial. We denote by {F } M the power series obtained from F discarding all its homogeneous components not divisible by M .

∈ N we let [n] def = [1, n]. For r ∈ N, r > 0 we let R r : Z → [0, r -1] the map "residue module r", where R r (i) is determined by R r (i) ≡ i mod r.
The main subject of this work are the complex reflection groups [START_REF] Shephard | Unitary groups generated by reflections[END_REF], or simply reflection groups. The most important example of a complex reflection group is the group of permutations of [n], known as the symmetric group, that we denote by S n . We know by the work of Shephard-Todd [START_REF] Shephard | Finite unitary reflection groups[END_REF] that all but a finite number of irreducible reflection groups are the groups G(r, p, n), where r, p, n are positive integers with p|r, that we are going to describe. If A is a matrix with complex entries we denote by |A| the real matrix whose entries are the absolute values of the entries of A. The wreath product G(r, n) = G(r, 1, n) is given by all n × n matrices satisfying the following conditions: the nonzero entries are r-th roots of unity; there is exactly one nonzero entry in every row and every column (i.e. |A| is a permutation matrix). If p divides r then the reflection group G(r, p, n) is the subgroup of G(r, n) given by all matrices A ∈ G(r, n) such that det A det |A| is a r p -th root of unity.

A projective reflection group is a quotient of a reflection group by a scalar subgroup (see [START_REF] Caselli | Projective reflection groups[END_REF]). Observe that a scalar subgroup of G(r, n) is necessarily a cyclic group C s of order s, generated by ζ s I, for some s|r, where I denotes the identity matrix. It is also easy to characterize all possible scalar subgroups of the groups G(r, p, n): in fact the scalar matrix ζ s I belongs to G(r, p, n) if and only if s|r and ps|rn. In this case we let

G(r, p, s, n) def = G(r, p, n)/C s . (1)
If G = G(r, p, s, n) then the projective reflection group G * def = G(r, s, p, n), where the roles of the parameters p and s are interchanged, is always well-defined. We say that G * is the dual of G and we refer the reader to [START_REF] Caselli | Projective reflection groups[END_REF] for the main properties of this duality.

We sometimes think of an element g ∈ G(r, n) as a colored permutation. If the nonzero entry in the i-th

row of g ∈ G(r, n) is ζ ci r we let c i (g) def = R r (c i
) and say that c 1 (g), . . . , c n (g) are the colors of g. Now it is easy to see that an element g ∈ G(r, n) is uniquely determined by the permutation |g| ∈ S n defined by |g|(i) = j if g i,j = 0, and by its colors c i (g) for all i ∈ [n]. More precisely, when we consider an element of G(r, n) as a colored permutation we represent its elements either as couples (c 1 , . . . , c n ; σ), where σ = σ(1) • • • σ(n) is a permutation in S n and (c 1 , . . . , c n ) is the sequence of its colors, or in window notation as

g = [g(1), . . . , g(n)] = [σ(1) c1 , . . . , σ(n) cn ].
We call σ(i) the absolute value of g(i), denoted |g(i)|. When it is not clear from the context, we will denote c i by c i (g). Moreover, if c i = 0, it will be omitted in the window notation of g. In this notation the multiplication in

G(r, n) = {(c 1 , . . . , c n ; σ) | c i ∈ [0, r -1], σ ∈ S n } is given by (d 1 , . . . , d n ; τ )(c 1 , . . . , c n ; σ) = (R r (c 1 + d σ(1) ), . . . , R r (c n + d σ(n) ); τ σ). If we let col(g) def = c 1 (g) + • • • + c n (g)
, then for p|r we can simply express the reflection group G(r, p, n) by

G(r, p, n) = {(c 1 , . . . , c n ; σ) ∈ G(r, n) | col(g) ≡ 0 mod p},
and C s as the cyclic group generated by (r/s, . . . , r/s; id).

Descent-type statistics

In this section we recall the notions of descents, flag descents, and flag major index for the wreath products. Then we introduce analogous definitions for descents and flag descents for projective reflection groups.

Definition. In all the paper we use the following order, called color order

1 r-1 < . . . < n r-1 < . . . < 1 1 < . . . < n 1 < 0 < 1 < . . . < n. (2)
For g ∈ G(r, n) we define the descent set as

Des G (g) def = {i ∈ [0, n -1] | g(i) > g(i + 1)}, (3)
where g(0) := 0, and denote its cardinality by des G (g). A geometric interpretation of this set in terms of Coxeter-like generators and length can be given as in [19, §7]. If we consider only positive descents we obtain

Des A (g) def = Des G (g) \ {0}, (4)
and we denote des A (g) its cardinality. The flag major index [START_REF] Adin | The Flag Major Index and Group Actions on Polynomial Rings[END_REF] and the flag descent number [START_REF] Adin | Descent Numbers and Major Indices for the Hyperoctahedral Group[END_REF], [START_REF] Bagno | Colored-Descent Representations of Complex Reflection Groups G(r, p, n)[END_REF] are defined by fmaj(g) def = r maj(g) + col(g) ( 5)

fdes(g) def = r des A (g) + c 1 (g), (6)
where as usual the major index maj(g) = i∈DesA(g) i is the sum of all positive descents of g.

Following [11, §5], for g = (c 1 , . . . , c n ; σ) ∈ G(r, p, s, n) we let HDes(g)

def = {i ∈ [n -1] | c i = c i+1 , and σ(i) > σ(i + 1)} h i (g) def = #{j ≥ i | j ∈ HDes(g)} (7)
k i (g) def = R r/s (c n ) if i = n k i+1 + R r (c i -c i+1 ) if i ∈ [n -1].
We call the elements in HDes(g) the homogeneous descents of g. Note that the sequence (k 1 (g), . . . , k n (g)) is a partition such that g = (R r (k 1 (g)), . . . , R r (k n (g)); σ). Moreover it is characterized by the following property of minimality: if

β 1 ≥ . . . ≥ β n and g = (R r (β 1 ), . . . , R r (β n ); σ), then β i ≥ k i (g), for all i ∈ [n].
For g ∈ G(r, p, s, n), we let

λ i (g) def = r • h i (g) + k i (g). ( 8
)
The sequence λ(g) def = (λ 1 (g), . . . , λ n (g)) is a partition. The flag-major index for g ∈ G(r, p, s, n) is defined by [11, §5]

fmaj(g) def = |λ(g)| = n i=0 λ i (g). (9)
We define the descent number and the flag descent number of g ∈ G(r, p, s, n) respectively by des(g) def = sλ 1 (g) + rs r and ( 10)

fdes(g) def = λ 1 (g). (11)
Finally, for g = (c 1 , . . . , c n ; σ) ∈ G(r, p, s, n) we define the color of g by col(g)

def = n i=1 R r/s (c i ). ( 12
)
Note that the previous definitions do not depend on the particular representative of g ∈ G(r, p, s, n) chosen among its s lifts in G(r, n).

Remark. From [11, Lemma 5.1], it follows that for p = s = 1 the flag major index on G(r, p, s, n) defined in (9) coincides with the flag major index of Adin and Roichman for wreath products G(r, n) defined in [START_REF] Bagno | Colored-Descent Representations of Complex Reflection Groups G(r, p, n)[END_REF]. Moreover the definition of fdes is ( 11) is consistent with that in [START_REF] Biagioli | Signed Mahonian polynomials for classical Weyl groups[END_REF], and so

des(g) = λ 1 (g) + r -1 r = des A (g) + c 1 (g) + r -1 r = des G (g),
where the last equality holds since

c 1 (g) + r -1 r = 0, if c 1 (g) = 0; 1, otherwise.
These equalities give motivations to the definitions of the previous statistics.

We finish this section with an example. [START_REF] Adin | Descent Numbers and Major Indices for the Hyperoctahedral Group[END_REF]3,[START_REF] Biagioli | On some analogues of descent numbers and major index for the hyperoctahedral group[END_REF]. We have HDes(g) = {2, 5}, (h 1 , . . . , h 8 ) = (2, 2, 1, 1, 1, 0, 0, 0) and (k 1 , . . . , k 8 ) = (18, 13, 13, 9, 5, 5, 1, 0), and so fdes(g) = 6 • 2 + 18 = 30, des(g) = 15, and col(g) = 6.

Example 3.1. Let g = [2 2 , 7 3 , 6 3 , 4 5 , 8 1 , 1 1 , 5 3 , 3 2 ] ∈ G(6,

One-dimensional characters and flag major index

The generating function of the major index with the unique nontrivial one-dimensional character admits a nice factorization formula over the symmetric group, as shown in Theorem 4.2 by Gessel-Simion (see [START_REF] Wachs | An involution for signed Eulerian numbers[END_REF]). The same happens for the other classical Weyl groups, as proved by Adin-Gessel-Roichman [3] for the type B case, and by the first author [START_REF] Biagioli | Signed Mahonian polynomials for classical Weyl groups[END_REF] for the type D case. In this section we generalize these results to all projective groups of type G(r, p, s, n).

The irreducible representations of G(r, p, s, n) are classified in [11, §6]. In particular, for n > 2, the one-dimensional characters of G(r, p, s, n) are all of the form

χ ǫ,k (g) = ǫ inv(|g|) ζ k•c(g) r ,
where ǫ = ±1, and k ∈ [0, r p -1] with the further condition that s divides kn, and c(g) is the sum of the colors of any element in G(r, p, n) representing the class of g (in particular c(g) = col(g) if s = 1). As usual for σ ∈ S n , we denote by inv(σ)

def = |{(i, j) ∈ [n] × [n] | i < j and σ(i) > σ(j)}|, the number of its inversions,
and by sign(σ) def = (-1) inv(σ) its sign. Our main results is the following one.

Theorem 4.1. Let χ ǫ,k be a one dimensional character of G(r, p, s, n). Then g∈G(r,p,s,n) χ ǫ,k (g)q fmaj(g) = r p (ζ k q) p 2r p (ǫζ k q) p • • • (n -1)r p (ǫ n-2 ζ k q) p nr ps (ǫ n-1 ζ k q) p [p] n-m ζ k q [p] m ǫζ k q q p
where m = ⌊n/2⌋.

We recall that {F } q p is the polynomial obtained from F discarding all the homogeneous components in the variable q of degree not divisible by p. Hence for r = 2 and p = s = 1 in Theorem 4.1, we obtain [3, Theorems 5.1, 6.1, 6.2]; for r = s = 2 and p = 1 [START_REF] Biagioli | Signed Mahonian polynomials for classical Weyl groups[END_REF]Theorem 4.8].

In order to prove Theorem 4.1 we proceed in several steps passing through G(r, n) and G(r, 1, s, n) until the general case of G(r, p, s, n). The basic stone is given by the following results for the symmetric group. Theorem 4.2 (Gessel-Simion). We have

σ∈Sn sign(σ)q maj(σ) = [1] q [2] -q [3] q • • • [n] (-1) n-1 q .
The particular case of the Weyl groups [3]. In this paper the authors focus on the sum g∈Bn (-1) ℓ(g) q fmaj(g) , where ℓ(g) is the Coxeter length of g with respect to a given set of simple reflections, making use of a combinatorial interpretation of ℓ(g) due to Brenti [START_REF] Björner | Combinatorics of Coxeter Groups[END_REF]. Theorem 4.1 is then achieved in this case since (-1) ℓ(g) = (-1) inv(|g|) (-1) c(g) for all g ∈ B n . Although one can define an analogue of the Coxeter length for the wreath products G(r, n) (see e.g., [START_REF] Bagno | Euler-Mahonian parameter on colored permutation groups[END_REF], [START_REF] Reiner | Signed permutation statistics[END_REF]), this does not lead to a one-dimensional character of the group, and the corresponding sum does not factorize nicely. This is why we focus on one-dimensional characters, obtaining in particular also a new proof in the case of Weyl groups of type B that does not make use of the combinatorial interpretation of the length function.

B n def = G(2, n) is treated in
The following is a technical lemma which is the multivariate extension of a result appearing in [3, §5]. For r > 0 and k r-1 , k r-2 , . . . , k 0 ∈ N such that k r-1

+ k r-2 + • • • + k 0 = n (i.e. (k r-1 , . . . , k 0 ) is a composition of n) we let S(k r-1 , . . . , k 0 ) def = {σ ∈ S n | Des A (σ) ⊆ {k r-1 , k r-1 + k r-2 , . . . , k r-1 + k r-2 + • • • + k 1 }}. Lemma 4.3. We have σ∈S(kr-1,...,k0) sign(σ) =      0, if at least 2 of the k i 's are odd; ⌊ n 2 ⌋ ⌊ kr-1 2 ⌋, . . . , ⌊ k0 2 ⌋
, otherwise.

Proof. The window notation of σ ∈ S(k r-1 , . . . , k 0 ) can be split into r blocks of length k r-1 , . . . , k 0 , the entries in each block being increasing. Consider the following involution on S(k r-1 , . . . , k 0 )

σ → (2i -1, 2i)σ if i is the smallest integer such that 2i -1 and 2i are in different blocks, σ → σ if such i does not exist.
We recall that the window notation of (2i -1, 2i)σ is obtained from that of σ by exchanging 2i -1 and 2i, and so this map is actually an involution on S(k r-1 , . . . , k 0 ). Since σ and (2i -1, 2i)σ have opposite signs we can restrict our attention to those σ ∈ S(k r-1 , . . . , k 0 ) such that 2i -1 and 2i belong to the same block, for all i ∈ {1, . . . , ⌊ n 2 ⌋}. It is clear that if two among the k i 's are odd such σ does not exist and the result follows in this case. We can thus assume that at most one of the k i 's is odd, and in this case all entries of the window notation of σ are uniquely determined by the even entries. This is because an odd entry j different from n belongs to the same block of j + 1 and, if n is odd, n necessarily belongs to the unique block of odd length. The number of even entries is ⌊ n 2 ⌋ and these should be combined into r parts of cardinality ⌊ kr-1 2 ⌋, . . . , ⌊ k0 2 ⌋. It remains to show that all these elements have even sign. We first note that there are no inversions of the form (σ -1 (2j -1), σ -1 (2j)). Then we observe that, for all i < 2j -1 < n, (σ -1 (2j -1), σ -1 (i)) is an inversion if and only if (σ -1 (2j), σ -1 (i)) is itself an inversion. This completes the proof if n is even. If n is odd we only have to make the final remark that there are an even number of inversions of the form (σ -1 (n), σ -1 (i)), since n appears in an odd position.

Theorem 4.4. We have

g∈G(r,n) sign(|g|)q fmaj(g) = [r] q [2r] -q [3r] q • • • [nr] (-1) n-1 q . Proof. We let U ⊂ G(r, n) be given by U = {g ∈ G(r, n) | g(1) < g(2) < • • • < g(n)} = {g ∈ G(r, n) | Des A (g) = ∅}. One can show that G(r, n) = U S n , with U ∩ S n = 1, i.e.
for each g ∈ G(r, n) there exist unique τ ∈ U and σ ∈ S n such that g = τ σ (see [START_REF] Biagioli | Enumerating wreath products via Garsia-Gessel bijections[END_REF]Proposition 4.1]). Moreover we have col(g) = col(τ ) and Des A (g) = Des A (σ). It follows that

g∈G(r,n) sign(|g|)q fmaj(g) = τ ∈U σ∈Sn sign(|τ σ|)q col(τ )+r•maj(σ) = τ ∈U sign(|τ |)q col(τ ) σ∈Sn sign(σ)q r•maj(σ) = τ ∈U sign(|τ |)q col(τ ) [1] q r [2] -q r [3] q r • • • [n] (-1) n-1 q r , (13)
by Theorem 4.2. Now we claim that

τ ∈U sign(|τ |)q col(τ ) = [r] m q 2 , if n = 2m is even; [r] q [r] m q 2 , if n = 2m + 1 is odd. (14)
In fact let

U (k r-1 , . . . , k 0 ) = {τ ∈ U | c j (τ ) = i if k r-1 + • • • + k i+1 < j ≤ k r-1 + • • • + k i }.
In other words the elements in U (k r-1 , . . . , k 0 ) are those in U having the first k r-1 entries colored with r -1, then k r-2 entries colored with r -2 and so on up to the last k 0 entries which are colored with 0.

It is clear that the map τ → |τ | is a bijection between U (k r-1 , . . . , k 0 ) and S(k r-1 , . . . , k 0 ). Then

τ ∈U sign(|τ |)q col(τ ) = kr-1,...,k0 τ ∈U(kr-1,...,k0) sign(|τ |)q col(τ ) = kr-1,...,k0
q iki σ∈S(kr-1,...,k0)

sign(σ) = kr-1,...,k0 ⌊ n 2 ⌋ ⌊ kr-1 2 ⌋, . . . , ⌊ k0 2 ⌋
q iki , by Lemma 4.3, where the last sum is taken over all r-tuples k r-1 , . . . , k 0 such that k r-1 + • • • + k 0 = n and at most one of the k i 's is odd. So, if n = 2m is even the k i 's are all even, we let k i = 2t i , and

τ ∈U sign(|τ |)q col(τ ) = tr-1,...,t0 tr-1+•••+t0=m m t r-1 , . . . , t 0 q 2iti = (1 + q 2 + • • • + q 2(r-1) ) m = [r] m q 2 .
If n = 2m + 1 is odd there is exactly one of the k i which is odd. We split the sum according to this odd entry: if k i is odd we let k i = 2t i + 1 and if it is even we let k i = 2t i . Then

τ ∈U sign(|τ |)q col(τ ) = r-1 j=0 tr-1,...,t0 tr-1+•••+t0=m m t r-1 , . . . , t 0 q j+ 2iti = r-1 j=0 q j tr-1,...,t0 tr-1+•••+t0=m m t r-1 , . . . , t 0 q 2iti = [r] q [r] m q 2 . Now one can easily check that [r] q 2 [2i -1] q r [2i] -q r = [(2i -1)r] q [2ir] -q . ( 15
)
If n is even the result follows immediately from Equations ( 13), [START_REF] Garsia | Permutation statistics and partitions[END_REF], and [START_REF] Macmahon | Combinatorial Analysis[END_REF]. If n is odd the result follows similarly using the further observation that [r] q [2m + 1] q r = [(2m + 1)r] q .

Now we face the problem of the quotient groups G(r, 1, s, n). We need a preliminary result.

Lemma 4.5. Let g ∈ G(r, 1, s, n) and g 0 , . . . , g s-1 be the s lifts of g in G(r, n). We have s-1 j=0 t λ1(gj ) q fmaj(gj ) = t λ1(g) q fmaj(g) [s] t r/s q nr/s .

Proof. We let g 0 be the unique lift of g satisfying c n (g 0 ) < r/s. Then the other lifts are given by

|g j | = |g 0 | and c i (g j ) = R r (c i (g 0 ) + jr/s) for all j ∈ [s -1] and for all i ∈ [n].
To compute fmaj we use definition [START_REF] Biagioli | Invariant algebras and major indices for classical Weyl groups[END_REF] with s = 1, since all lifts g 0 , . . . , g

s-1 ∈ G(r, n) = G(r, 1, 1, n).
It is clear that HDes(g j ) = HDes(g 0 ) and k i (g j ) = k i (g 0 ) + jr/s. Therefore fmaj(g j ) = fmaj(g 0 ) + njr/s and λ 1 (g j ) = λ 1 (g) + jr/s, so s-1 j=0 t λ1(gj ) q fmaj(gj ) = s-1 j=0 t λ1(g0)+jr/s q fmaj(g0)+jnr/s = t λ1(g0) q fmaj(g0) (1 + t r/s q nr/s + • • • + t (s-1)r/s q (s-1)nr/s ) = q fmaj(g0) [s] t r/s q nr/s , and the result follows since fmaj(g 0 ) = fmaj(g) and λ 1 (g 0 ) = λ 1 (g).

Corollary 4.6. We have

g∈G(r,1,s,n) sign(|g|)q fmaj(g) = [r] q [2r] -q [3r] q • • • [(n -1)r] (-1) n-2 q • [nr/s] (-1) n-1 q .
Proof. By Theorem 4.4 and Lemma 4.5 specialized to the case t = 1 we have

g∈G(r,1,s,n) sign(|g|)q fmaj(g) = g∈G(r,n) sign(|g|)q fmaj(g) /[s] q nr/s = [r] q [2r] -q [3r] q • • • [nr] (-1) n-1 q /[s] q nr/s ,
and the result easily follows since

[nr] (-1) n-1 q [s] q nr/s = 1 -((-1) n-1 q) nr 1 -(-1) n-1 q • 1 -q nr/s 1 -q nr = 1 -q nr 1 -(-1) n-1 q • 1 -q nr/s 1 -q nr = 1 -((-1) n-1 q) nr/s 1 -(-1) n-1 q = [nr/s] (-1) n-1 q ,
where we have used the fact that r/s is an integer and hence (n -1)nr/s is even.

Theorem 4.7. We have g∈G(r,p,s,n)

ǫ inv(|g|) q fmaj(g) = r p q p 2r p (ǫq) p • • • (n -1)r p (ǫ n-2 q) p nr ps (ǫ n-1 q) p [p] n-m q [p] m ǫq q p ,
where m = ⌊n/2⌋.

Proof. It is a consequence of [START_REF] Bagno | Colored-Descent Representations of Complex Reflection Groups G(r, p, n)[END_REF][START_REF] Caselli | Projective reflection groups[END_REF] that

g∈G(r,1,s,n) q fmaj(g) = [r] q [2r] q [3r] q • • • [(n -1)r] q • [nr/s] q ,
and therefore we may unify this equation and Corollary 4.6 by obtaining

g∈G(r,1,s,n) ǫ inv(|g|) q fmaj(g) = [r] q [2r] ǫq [3r] ǫ 2 q • • • [(n -1)r] ǫ n-2 q • [nr/s] ǫ n-1 q , (16)
where ǫ = ±1. We now want to describe the polynomial g∈G(r,p,s,n) ǫ inv(|g|) q fmaj(g) . To this end we observe that an element g ∈ G(r, 1, s, n) belongs to G(r, p, s, n) if and only if fmaj(g) ≡ 0 mod p (see [START_REF] Caselli | Projective reflection groups[END_REF]Lemma 5.2]). Therefore we have g∈G(r,p,s,n)

ǫ inv(|g|) q fmaj(g) = g∈G(r,1,s,n) ǫ inv(|g|) q fmaj(g) q p . We observe that [r] q [2r] ǫq [3r] ǫ 2 q • • • [(n -1)r] ǫ n-2 q • [nr/s] ǫ n-1 q = = 1 -q r 1 -q • 1 -(ǫq) 2r 1 -ǫq • • • 1 -(ǫ n-2 q) (n-1)r 1 -ǫ n-2 q • • • 1 -(ǫ n-1 q) nr/s 1 -ǫ n-1 q = (1 -q r )(1 -q 2r ) • • • (1 -q (n-1)r )(1 -q nr/s ) (1 -q)(1 -ǫq) • • • (1 -ǫ n-1 q) . (17)
Notice that F (q) q p = 1 p p-1 j=0 F (ζ j p q). Hence from ( 16) and ( 17) we have g∈G(r,p,s,n)

ǫ inv(|g|) q fmaj(g) = [r] q [2r] ǫq [3r] ǫ 2 q • • • [(n -1)r] ǫ n-2 q • [nr/s] ǫ n-1 q q p = 1 p p-1 j=0 (1 -(ζ j p q) r )(1 -(ζ j p q) 2r ) • • • (1 -(ζ j p q) (n-1)r )(1 -(ζ j p q) nr/s ) (1 -ζ j p q)(1 -ǫζ j p q) • • • (1 -ǫ n-1 ζ j p q) = 1 p p-1 j=0 (1 -q r )(1 -q 2r ) • • • (1 -q (n-1)r )(1 -q nr/s ) (1 -ζ j p q)(1 -ǫζ j p q) • • • (1 -ǫ n-1 ζ j p q) = (1 -q r )(1 -q 2r ) • • • (1 -q (n-1)r )(1 -q nr/s ) • 1 p p-1 j=0 1 (1 -ζ j p q)(1 -ǫζ j p q) • • • (1 -ǫ n-1 ζ j p q) .
The last factor can be manipulated as follows:

1 p p-1 j=0 1 (1 -ζ j p q)(1 -ǫζ j p q) • • • (1 -ǫ n-1 ζ j p q) = 1 p p-1 j=0 [p] ζ j p q (1 -q p ) [p] ǫζ j p q (1 -(ǫq) p ) • • • [p] ǫ n-1 ζ j p q (1 -(ǫ n-1 q) p ) = [p] n-m q [p] m ǫq q p (1 -q p )(1 -(ǫq) p ) • • • (1 -(ǫ n-1 q) p ) ,
where m = n/2 if n is even and m = (n -1)/2 if n is odd. Hence we obtain that g∈G(r,p,s,n)

ǫ inv(|g|) q fmaj(g) = (1 -q r )(1 -q 2r ) • • • (1 -q (n-1)r )(1 -q nr/s ) (1 -q p )(1 -(ǫq) p ) • • • (1 -(ǫ n-1 q) p ) [p] n-m q [p] m ǫq q p = 1 -q r 1 -q p • • • 1 -q (n-1)r 1 -(ǫ n-2 q) p 1 -q nr/s 1 -(ǫ n-1 q) p [p] n-m q [p] m ǫq q p = r p q p 2r p (ǫq) p • • • (n -1)r p (ǫ n-2 q) p nr ps (ǫ n-1 q) p [p] n-m q [p] m ǫq q p
. Now we are ready to prove our main result.

Proof of Theorem 4.1. Now Theorem 4.1 is a direct consequence of Theorem 4.7 since k • fmaj(g) ≡ k • c(g) mod r for all g ∈ G(r, p, s, n) and so g∈G(r,p,s,n)

ǫ inv(|g|) ζ k•c(g) r q fmaj(g) = g∈G(r,p,s,n) ǫ inv(|g|) (ζ k r q) fmaj(g)

Carlitz's Identities

In this section we give a general method to compute the trivariate distribution of des (or fdes), fmaj and col over G(r, p, s, n). This unifies and generalizes all related results cited in the introduction, and provides two different generalizations of Carlitz's identity for the group G(r, p, s, n).

For f = (f 1 , . . . , f n ) ∈ N n let |f | = f 1 + • • • + f n and N n (p) = {f ∈ N n | |f | ≡ 0 mod p}. Moreover, if f ∈ N n , for any k > 0 we let col k (f ) def = n i=1 R k (f i )
be the sum of the residues of the f i 's modulo k.

Lemma 5.1. There is a bijection

N n (p) ←→ G(r, p, s, n) × P n × [0, s -1], such that if f ↔ (g, λ, h) then (a) max f = λ 1 (g) + rλ 1 + hr/s; (b) |f | = fmaj(g) + r|λ| + hnr/s; (c) col r s (f ) = col(g).
Proof. This result is similar to several others appearing in the literature and in particular is a special case of the bijection appearing in [START_REF] Caselli | Projective reflection groups[END_REF]Theorem 8.3] and so we simply describe how the bijection is defined for the reader's convenience. If f = (f 1 , . . . , f n ) ∈ N n (p), then g is the unique element in G(r, p, s, n) having a lift g ∈ G(r, p, n) satisfying:

• f |g(i)| ≥ f |g(i+1)| for all i ∈ [n -1]; • if f |g(i)| = f |g(i+1)| then |g(i)| < |g(i + 1)|; • c i (g) ≡ f |g(i)| mod r for all i ∈ [n].
Letting µ be the partition obtained by reordering the entries in (f 1 , . . . , f n ) in nonincreasing order one can show that µλ(g) (componentwise difference) is still a partition whose parts are all congruent to the same multiple of r/s mod r (i.e. µ is g-compatible in the notation of [START_REF] Caselli | Projective reflection groups[END_REF]). The partition λ ∈ P n and the integer h ∈ [0, s -1] are therefore uniquely determined by the requirement that µ = λ(g) + r • λ + (hr/s, . . . , hr/s).

The inverse map of this bijection is much simpler. Let (g, λ, h) be a triple in G(r, p, s, n

) × P n × [0, s -1], then the corresponding element f ∈ N n (p) is given by f i = λ |g -1 (i)| (g) + rλ |g -1 |(i) + hr/s.
All the other statements are straightforward consequences, since λ i (g) ≡ c i (g) mod r/s for any lift g of g, and so also col(g) = n i=1 R r/s (λ i (g)).

We will make use of the following observation several times: if F is a set and m : F → N is such that the set {f ∈ F | m(f ) ≤ k} is finite for all k ∈ N then we always have

1 1 -t f ∈F t m(f ) φ(f ) = 1 1 -t k∈N t k f ∈F |m(f )=k φ(f ) = k∈N t k f ∈F |m(f )≤k φ(f ), ( 18
)
where φ is any map defined on F taking values in a (polynomial) ring.

We can now state the main result of this section.

Theorem 5.2.

k≥0 t k [k + 1] q r/s + aq[k] q r/s r s -1 aq n q p
= g∈G(r,p,s,n) t des(g) q fmaj(g) a col(g) (1t)(1t s q r ) • • • (1t s q (n-1)r )(1tq nr/s ) Proof. For f ∈ N n (p) we let m(f ) = ⌊ s max(f )+r-s r ⌋ and we observe that m(f ) ≤ k if and only if max(f ) ≤ r s k. In fact ⌊ s max(f )+r-s r ⌋ ≤ k is equivalent to s max(f ) + rs ≤ kr + r -1. This happens if and only if max(f ) ≤ kr/s + s-1 s which is equivalent to max(f ) ≤ kr/s since max(f ) is an integer. We want to make use of Equation [START_REF] Stanton | A Schensted algorithm for rim hook tableaux[END_REF] with φ(f ) = q |f | a col r s (f ) and so we compute

f ∈N n (p)| m(f )≤k q |f | a col r s (f ) = kr/s j=0 q j a R r/s (j) n q p = 1 + q r s + • • • + q kr s + (aq + a 2 q 2 + • • • + a r s -1 q r s -1 )(1 + q r s + • • • + q (k-1) r s ) n q p = [k + 1] q r/s + aq r s -1 aq [k] q r/s n q p .
Therefore, by Equation ( 18),

f ∈N n (p) t m(f ) q |f | a col r s (f ) = (1 -t) k≥0 t k [k + 1] q r/s + aq r s -1 aq [k] q r/s n q p
. Now we want to compute this polynomial using the bijection described in Lemma 5.1. Observe that if f → (g, λ, h) then m(f ) = des(g) + sλ 1 + h. Therefore we have

f ∈N n (p) t m(f ) q |f | a col r s (f ) = (g,λ,h)
t des(g)+sλ1+h q fmaj(g)+r|λ|+nhr/s a col(g) = g∈G(r,p,s,n)

t des(g) q fmaj(g) a col(g) λ∈Pn t sλ1 q r|λ| s-1 h=0 t h q nhr/s = g∈G(r,p,s,n) t des(g) q fmaj(g) a col(g) (1t s q r ) • • • (1t s q (n-1)r )(1tq nr/s ) ,

and the result follows.

Letting s = p = 1 in the previous result we obtain [START_REF] Biagioli | Enumerating wreath products via Garsia-Gessel bijections[END_REF]Equation (8.1)]. Moreover, for a = 1 we have

[k + 1] q r/s + q[k] q r/s [ r s -1] q = r s k + 1
q and hence we obtain the following result.

Corollary 5.3 (Carlitz's identity for G(r, p, s, n) with des).

k≥0 t k r s k + 1 n q p = g∈G(r,p,s,n) t des(g) q fmaj(g) (1 -t)(1 -t s q r ) • • • (1 -t s q (n-1)r )(1 -tq nr/s ) .
The special case with r = 2, p = s = 1 of Corollary 5.3 is the main result of [START_REF] Chow | On the descent numbers and major indices for the hyperoctahedral group[END_REF], and for p = s = 1 we obtain [START_REF] Chow | A Carlitz identity for the wreath product Cr ≀ Sn[END_REF]Theorem 10 (iv)].

Now we see how a simple modification of the same ideas lead to the generalization of other identities that use different flag-descents.

Theorem 5.4.

k≥0 t k [Q r/s (k) + 1] q r/s + aq[r/s -1] aq • [Q r/s (k)] q r/s + aq mr/s+1) [R r/s (k)] aq n (19)
= g∈G(r,p,s,n) t fdes(g) q fmaj(g) a col(g)

(1 -t)(1 -t s q r ) • • • (1 -t s q (n-1)r/s ) • • • (1 -tq nr/s ) , (20)
where

Q r/s (k) is determined by k = r/s • Q r/s (k) + R r/s (k).
A proof of this result can be done by paralleling that of Theorem 5.2. In this case one should simply consider the statistics m(f ) = max(f ) and follow the rest mutatis mutandis. By letting a = 1 in Equation 19, one obtain a second Carlitz's identity type, with flag-descents. Theorem 5.5 (Carlitz's identity of G(r, p, s, n) with fdes).

k≥0 t k [k + 1] n q q p = g∈G(r,p,s,n) t fdes(g) q fmaj(g) (1 -t)(1 -t r q r )(1 -t r q 2r ) • • • (1 -t r q (n-1)r )(1 -t r s q nr s )
.

Multivariate generating functions

In this section we make further use of the bijection [START_REF] Caselli | Projective reflection groups[END_REF]Theorem 8.3] to compute new multivariate distributions for the groups G(r, p, s, n). We first concentrate on the case of the groups G(r, n) to make the used arguments more clear to the reader. We need to state the particular case of this bijection needed for our purposes, namely the special case of 2-partite partitions.

We recall that a 2-partite partition of length n (see [START_REF] Garsia | Permutation statistics and partitions[END_REF]) is a 2 × n matrix with non negative integer coefficients f = f

. . . f

(2) n satisfying the following conditions:

• f (1) 1 ≥ f (1) 2 ≥ . . . ≥ f (1) n ; • If f (1) i = f (1) i+1 then f (2) i ≥ f (2) i+1 .
One may think of a 2-partite partition as a generic multiset of pairs of nonnegative integers (the columns of f ) of cardinality n. If f is a 2-partite partition we denote by f (1) and f (2) the first and the second row of f respectively. To be consistent with the notation of [11, §8], we denote by B(n) the set of 2-partite partitions of length n and we let

B(r, n) def = {f ∈ B(n) | f (1) i + f (2) i ≡ 0 mod r for all i ∈ [n]} and B(r, s, 1, n) def = {f ∈ B(n) | there exists l ∈ [s -1] such that f (1) i + f (2) i ≡ lr/s mod r for all i ∈ [n]}. Note that B(1, n) = B(n) and that B(r, n) = B(r, 1, 1, n).
Proposition 6.1. There exists a bijection between B(r, s, 1, n) and the set of the 5-tuples (g, λ, µ, h, k), where g ∈ G(r, 1, s, n), λ, µ ∈ P n , and h, k

∈ [0, s -1]. In this bijection, if f ↔ (g, λ, µ, h, k) then (a) max(f (1) ) = λ 1 (g) + rλ 1 + hr/s; (b) max(f (2) ) = λ 1 (g -1 ) + rµ 1 + kr/s; (c) |f (1) | = fmaj(g) + r|λ| + hnr/s; (d) |f (2) | = fmaj(g -1 ) + r|µ| + knr/s; (e) col r s (f (1) ) = col(g); (f ) col r s (f (2) ) = col(g -1 )
. Proof. This is again a particular case of [START_REF] Caselli | Projective reflection groups[END_REF]Theorem 8.3]. In this case the bijection is defined as follows: if

(g, λ, µ, h, k) ↔ f then f (1) i = λ i (g) + rλ i + hr/s and f (2) i = λ |g(i)| (g -1 ) + rµ |g(i)| + kr/s,
and all the other statements follow immediately.

For the reader's convenience we first face the problem of the wreath products G(r, n). Theorem 6.2. Let r ∈ N. Then k1,k2≥0

t k1 1 t k2 2 i∈[0,rk1], j∈[0,rk2]: i+j≡0 mod r 1 1 -ua Rr(i) 1 a Rr(j) 2 q i 1 q j 2 = = n≥0 u n g∈G(r,n) t des(g) 1 t des(g -1 ) 2 q fmaj(g) 1 , q fmaj(g -1 ) 2 a col(g) 1 a col(g -1 ) 2 n j=0 1 (1 -t 1 q jr 1 )(1 -t 2 q jr 2 )
.

Proof. For i = 1, 2, we let m i (f ) = ⌊ max f (i) +r-1 r ⌋ and B k1,k2 = {f ∈ B(r, n) | m i (f ) ≤ k i , i = 1, 2}.
We can easily verify that m i (f ) ≤ k i if and only if max(f (i) ) ≤ rk i . Therefore, using an idea already appearing in [START_REF] Garsia | Permutation statistics and partitions[END_REF], one can show that

f ∈B k 1 ,k 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2 = i∈[0,rk1], j∈[0,rk2]: i+j≡0 mod r 1 1 -ua Rr(i) 1 a Rr(j) 2 q i 1 q j 2 u n
. Now we use a natural bivariate extension of Equation ( 18) to obtain k1,k2≥0

t k1 1 t k2 2 f ∈B k 1 ,k 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2 = 1 (1 -t 1 )(1 -t 2 ) f ∈B(r,n) t m1(f ) 1 t m2(f ) 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2
and hence

f ∈B(r,n) t m1(f ) 1 t m2(f ) 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2 = = (1 -t 1 )(1 -t 2 ) k1,k2≥0 t k1 1 t k2 2 f ∈B k 1 ,k 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2 = (1 -t 1 )(1 -t 2 ) k1,k2≥0 t k1 1 t k2 2 i∈[0,rk1], j∈[0,rk2]: i+j≡0 mod r 1 1 -ua Rr(i) 1 a Rr(j) 2 q i 1 q j 2 u n .
On the other hand, by Proposition 6.1 used with s = 1, we have

f ∈B(r,n) t m1(f ) 1 t m2(f ) 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2 = = (g,λ,µ) t des(g)+λ1 1 t des(g -1 )+µ1 2 q fmaj(g)+r|λ| 1 q fmaj(g -1 )+r|µ| 2 a col(g) 1 a col(g -1 ) 2 = g∈G(r,n) t des(g) 1 t des(g -1 ) 2 q fmaj(g) 1 q fmaj(g -1 ) 2 a col(g) 1 a col(g -1 ) 2 λ∈Pn t λ1 1 q r|λ| 1 µ∈Pn t µ1 2 q r|µ| 2 = g∈G(r,n) t des(g) 1 t des(g -1 ) 2 q fmaj(g) 1 q fmaj(g -1 ) 2 a col(g) 1 a col(g -1 ) 2 n j=1 1 (1 -t 1 q jr 1 )(1 -t 2 q jr 2 )
and the proof is complete. Now we want to extend this result to the quotients G(r, 1, s, n).

Theorem 6.3. Let r, s ∈ N, such that s|r. Then k1,k2≥0

t k1 1 t k2 2 s-1 l=0 i∈[0,k1 r s ], j∈[0,k2 r s ]: i+j≡l r s mod r 1 1 -ua R r/s (i) 1 a R r/s (j) 2 q i 1 q j 2 = = n≥0 u n g∈G(r,1,s,n) t des(g) 1 t des(g -1 ) 2 q fmaj(g) 1 q fmaj(g -1 ) 2 a col(g) 1 a col(g -1 ) 2 (1 -t 1 )(1 -t 2 )(1 -t 1 q n r s 1 )(1 -t 2 q n r s 2 ) n-1 j=1 (1 -t s 1 q jr 1 )(1 -t s 2 q jr 2 )
.

Proof. Say that f ∈ B(r, s, 1, n) is of type l, with l ∈ [0, s -1], if the column sums of f are all congruent to lr/s mod r. For i = 1, 2, we let

m i (f ) = ⌊ s max(f (i) )+r-s r ⌋ and B (l) k1,k2 = {f ∈ B(r, s, 1, n) of type l | m 1 (f ) ≤ k 1 , m 2 (f ) ≤ k 2 )}. We observe that m i (f ) ≤ k i if and only if max(f (i) ) ≤ r s k i and we compute f ∈B (l) k 1 ,k 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2 = i∈[0,k1 r s ], j∈[0,k2 r s ]: i+j≡l r s mod r 1 1 -ua R r/s (i) 1 a R r/s (j) 2 q i 1 q j 2 u n . Therefore 1 (1 -t 1 )(1 -t 2 ) f ∈B(r,s,1,n) t m1(f ) 1 t m2(f ) 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2 = k1,k2≥0 t k1 1 t k2 2 l∈[0,s-1] f ∈B (l) k 1 ,k 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2 = k1,k2≥0 t k1 1 t k2 2 l∈[0,s-1] i∈[0,k1 r s ], j∈[0,k2 r s ]: i+j≡l r s mod r 1 1 -ua R r/s (i) 1 a R r/s (j) 2 q i 1 q j 2 u n .
On the other hand, by Proposition 6.1, we have

f ∈B(r,s,1,n) t m1(f ) 1 t m2(f ) 2 q |f (1) | 1 q |f (2) | 2 a col r s (f (1) ) 1 a col r s (f (2) ) 2 = g,λ,µ,h,k t des(g)+sλ1+h 1 t des(g -1 )+sµ1+k 2 q fmaj(g)+r|λ|+kn r s 1 q fmaj(g -1 )+r|µ|+hn r s 2 a col(g) 1 a col(g -1 ) 2 = g∈G(r,1,s,n) t des(g) 1 t des(g -1 ) 2 q fmaj(g) 1 q fmaj(g -1 ) 2 a col(g) 1 a col(g -1 ) 2 s-1 h=0 t h 1 q hn r s 1 s-1 k=0 t k 2 q kn r s 2 λ t sλ1 1 q r|λ| 1 µ t sµ1 2 q r|µ| 2 = g∈G(r,1,s,n) t des(g) 1 t des(g -1 ) 2 q fmaj(g) 1 q fmaj(g -1 ) 2 a col(g) 1 a col(g -1 ) 2 1 -t s 1 q nr 1 1 -t 1 q n r s 1 1 -t s 2 q nr 2 1 -t 2 q n r s 2 n j=1 1 (1 -t s 1 q jr 1 )(1 -t s 2 q jr 2 )
.

The corresponding result for the the full class of projective reflection groups G(r, p, s, n) can now be easily deduced from Theorem 6.3. We remind that G(r, p, s, n) is defined if and only if p|r, s|r, and sp|rn. In particular if r, p, s are fixed then G(r, p, s, n) is defined only if sp/GCD(sp, r) divides n. Theorem 6.4. Let r, p, s ∈ N, such that s and p divide r. Let d = sp/GCD(sp, r). Then k1,k2≥0

t k1 1 t k2 2 s-1 l=0 i∈[0,k1 r s ], j∈[0,k2 r s ]: i+j≡l r s mod r 1 1 -ua R r/s (i) 1 a R r/s (j) 2 q i 1 q j 2 u d q p 1 = = n≥0 : d|n u n g∈G(r,p,s,n) t des(g) 1 t des(g -1 ) 2 q fmaj(g) 1 q fmaj(g -1 ) 2 a col(g) 1 a col(g -1 ) 2 (1 -t 1 )(1 -t 2 )(1 -t 1 q n r s 1 )(1 -t 2 q n r s 2 ) n-1 j=1 (1 -t s 1 q jr 1 )(1 -t s 2 q jr 2 )
.

6.1. Hilbert series of diagonal invariant algebras. We can now make a specialization of this result. We set a 1 = a 2 = 1 in Theorem 6.4

k1,k2≥0

t k1 1 t k2 2 s-1 l=0 i∈[0,k1 r s ], j∈[0,k2 r s ]: i+j≡l r s mod r 1 1 -uq i 1 q j 2 u d q p 1 = = n≥0 : d|n u n g∈G(r,p,s,n) t des(g) 1 t des(g -1 ) 2 q fmaj(g) 1 q fmaj(g -1 ) 2 (1 -t 1 )(1 -t 2 )(1 -t 1 q n r s 1 )(1 -t 2 q n r s 2 ) n-1 j=1 (1 -q jr 1 t s 1 )(1 -q jr 2 t s 2 )
.

If one considers the special case of Theorem 5.2 with p = s = 1, and relate it to [8, Theorem 5.1] one can deduce that the two polynomials g∈G(r,n) t des G (g) q fmaj(g) a col(g) and g∈G(r,n) t des ′ G (g) q fmaj ′ (g) a col (g) are equal. This can also be easily proved bijectively.

Proposition 7.1. There exists an explicit involution φ : G(r, n) → G(r, n) such that Des G (φ(g)) = Des ′ G (g) and col(φ(g)) = col(g), and in particular fmaj(φ(g)) = fmaj ′ (g).

Proof. If g ∈ G(r, n) let S(g) = {g(1), . . . , g(n)}. The set S(g) is totally ordered by both < and < ′ . We let ι : S(g) → S(g) be the unique involution such that ι(x) < ι(y) if and only if x < ′ y, for all x, y ∈ S(g). We define φ(g) = [ι(g(1)), . . . , ι(g(n))]: it is clear that the map φ satisfies the conditions of the statement.

It follows from Proposition 7.1 that the two statistics fmaj and fmaj ′ are equidistributed on G(r, n). Nevertheless the two polynomials g∈G(r,n) χ(g)q fmaj(g) and g∈G(r,n) χ(g)q fmaj ′ (g) , where χ is any linear character of G(r, n), do not coincide in general. So, in order to obtain the results appearing in §4 we must consider the order <. We should also mention that the polynomial g∈G(r,n) χ(g)q fmaj ′ (g) does not factor nicely at all in general.

There is a more subtle difference if one considers the multivariate distributions g∈G(r,n) t desG(g) 1 t desG(g -1 ) 2 q fmaj(g) 1 q fmaj(g -1 ) 2 a col(g) 1 a col(g -1 ) 2

, and the analogue computed with the order < ′ . In fact, by comparing Theorem 6.2 and [9, Theorem 7.1] we deduce that these two polynomials coincide for r = 1, 2, and are distinct if r > 2. The case r = 1 being trivial we can justify this coincidence for r = 2 with a bijective proof. We split the set Des G (g) into four subsets.

Des G (g) = HDes 0 (g) ∪ HDes 1 (g) ∪ Des ± (g) ∪ D 0 (g), where

• HDes 0 (g) = {i ∈ [n -1] | g(i) > g(i + 1) > 0};

• HDes 1 (g) = {i ∈ [n -1] | 0 > g(i) > g(i + 1)};

• Des ± = {i ∈ [n -1] | g(i) > 0 > g(i + 1)};

• D 0 (g) = {0} if g(1) < 0 and D 0 (g) = ∅ otherwise.

If we let NN(g) = {i ∈ [n -1] | g(i) < 0 and g(i + 1) < 0} we have Des ′ G (g) = HDes 0 (g) ∪ (NN(g) \ HDes 1 (g)) ∪ Des ± (g) ∪ D 0 (g). By the Robinson-Schensted correspondence for B n (see [START_REF] Stanton | A Schensted algorithm for rim hook tableaux[END_REF][START_REF] Caselli | Projective reflection groups[END_REF]) we have a bijection g → [(P 0 , P 1 ), (Q 0 , Q 1 )],

where (P 0 , P 1 ) and (Q 0 , Q 1 ) are bitableaux of the same shape. Given a tableau P let Des(P ) def = {i | both i and i + 1 belong to P with i strictly above i + 1}.

If we let Neg(g) = {i ∈ [n] | c i (g) = 1}, in the Robinson-Schensted correspondence we have:

• Neg(g) = content of Q 1 , Neg(g -1 ) = content of P 1 .

• HDes 0 (g) = Des(Q 0 ), HDes 1 (g) = Des(Q 1 )

• HDes 0 (g -1 ) = Des(P 0 ), HDes 1 (g -1 ) = Des(P 1 )

• Des ± (g) = {i | i belongs to Q 0 and i + 1 belongs to Q 1 },

• Des ± (g -1 ) = {i | i belongs to P 0 and i + 1 belongs to P 1 } • D 0 (g) = {0} if and only if 1 belongs to Q 1 ;

• D 0 (g -1 ) = {0} if and only if 1 belongs to P 1 ;

Let ϕ : B n → B n be the bijection defined by the requirement that if g → [(P 0 , P 1 ), (Q 0 , Q 1 )], then ϕ(g) → [(P 0 , P ′ 1 ), (Q 0 , Q ′ 1 )], where T ′ denotes the transposed tableau. Proof. It follows from the previous facts with the further observation that if g → [(P 0 , P 1 ), (Q 0 , Q 1 )] then Des(Q ′ 1 ) = NN(g) \ Des(Q 1 ) and Des(P ′ 1 ) = NN(g -1 ) \ Des(P 1 ).

  Moreover, we denote by ζ r the primitive r-th root of unity ζ r def = e 2πi r

For r = 2

 2 and p = s = 1 we obtain [2, Theorem 4.2], for r = s = 2 and p = 1 [7, Theorem 4.3], and for p = 1 [5, Theorem 11.2].

Proposition 7 . 2 .

 72 The bijection ϕ satisfies the following properties:(a) Neg(g) = Neg(ϕ(g)); (b) Des G (g) = Des ′ G (ϕ(g)); (c) Des G (g -1 ) = Des ′ G (ϕ(g) -1).

Example 7 . 3 .

 73 If g = [5, -2, -1, -4, 6, -3, -7] ∈ B 7 , then One then can check that ϕ(g) = [5, -3, -7, -1, 6, -4, -2]. Hence Neg(g) = Neg(ϕ(g)) = {2, 3, 4, 6, 7}, Des G (g) = Des ′ G (ϕ(g)) = {1, 2, 5}, and Des G (g -1 ) = Des ′ G (ϕ(g) -1 ) = {0, 1, 3, 6}.

Multiplying both sides of the previous identity by (1t 1 )(1t 2 ) and then taking the limit as t 1 , t 2 → 1 we obtain s-1 l=0 i+j≡lr/s mod r 1 1uq i 1 q j 2 u d q p 1 = n≥0 : d|n u n g∈G(r,p,s,n) q fmaj(g) 1 q fmaj(g -1 ) 2

(1 -

There is a nice algebraic interpretation of the previous identity that goes back to the work of Adin and Roichman [START_REF] Adin | The Flag Major Index and Group Actions on Polynomial Rings[END_REF]. We let S p [X, Y ] be the subalgebra of the algebra of polynomials in 2n variables x 1 , . . . , x n , y 1 , . . . , y n generated by 1 and the monomials whose degrees in both the x's and the y's variables is divided by p. Then we can observe (see [11, §8]) that the factor

is the bivariate Hilbert series of the invariant algebra of the "tensorial action" of the group G(r, s, p, n

where ∆G(r, s, p, n) denotes the diagonal embedding of G(r, s, p, n) in G(r, s, p, n) 2 . We can now conclude that Equation ( 21) provides the following interpretation for the generating function of the Hilbert series of the diagonal invariant algebras of the groups G(r, p, s, n).

Corollary 6.5. Let r, p, s ∈ N, such that s and p divide r. Let d = sp/GCD(sp, r). Then n≥0 : d|n

This identity was apparently known only in the case of the symmetric group [START_REF] Garsia | Permutation statistics and partitions[END_REF][START_REF] Roselle | Coefficients associated with the expansion of certain products[END_REF].

Orderings

In the literature, when considering statistics based on descents and/or inversions on the groups G(r, n) at least two distinct orderings have been considered on the set of colored integers. In this last section we would like to clarify some of the relationships between two of these orderings concerning the results shown in this paper.

We recall that we have been using the order defined in [START_REF] Adin | Descent Numbers and Major Indices for the Hyperoctahedral Group[END_REF]. One can consider also the order

The order < ′ is the "good" one to give a combinatorial interpretation of the length function in G(r, n) in terms of inversions (see e.g. [START_REF] Reiner | Signed permutation statistics[END_REF], [START_REF] Bagno | Euler-Mahonian parameter on colored permutation groups[END_REF]), while the order < is often used in the study of some algebraic aspects such as the invariant theory of B n and G(r, n) (see [START_REF] Adin | The Flag Major Index and Group Actions on Polynomial Rings[END_REF], [START_REF] Caselli | Projective reflection groups[END_REF]). Here we give a motivation of these choices. So we denote