T Herremans
email: tineke.herremans@rivm.nl

L Kortbeek

D W Notermans

A review of diagnostic tests for congenital syphilis in newborns

Congenital syphilis (CS) can occur when a mother is inadequately treated or not treated at all for an active Treponema pallidum infection. Symptoms of CS are often subtle and non-specific, and it is estimated that up to 60% of affected infants are asymptomatic at birth, making the diagnosis dependent on laboratory findings. Despite decades of experience with CS, problems still arise in its diagnosis because laboratory test results for children at risk for CS can be inconclusive and no single diagnostic test can be used to diagnose CS. The development of diagnostic tests such as enzyme immunoassays, immunoblotting and polymerase chain reaction (PCR) has increased the sensitivity and specificity of diagnoses, but the detection of specific IgM is currently the most sensitive serological method, and the presence of specific IgM should be considered as evidence of a congenital T. pallidum infection. Suspected cases can also be confirmed or excluded by serial post-partum tests of antibody kinetics. The authors note strongly that it is considered unethical not to treat a baby at risk of contracting CS, even without a definitive diagnosis. In this review, we describe the various microbiological methods-and their shortcomings-used in the laboratory diagnosis of CS.

Introduction

Congenital syphilis (CS) is caused by a Treponema pallidum infection acquired from a mother with untreated or inadequately treated syphilis. Most infants at risk for CS can easily be identified by a positive maternal serologic test. However, the World Health Organization (WHO) estimates that maternal syphilis is responsible for 460,000 abortions or stillbirths and 270,000 low birth weight and/or premature babies every year (Table 1).

Transmission to the foetus can occur during any stage of a pregnancy, and while it usually occurs via the placenta, it may also occur during delivery if maternal genital lesions are present. The risk of CS transmission is also recognised to be higher in conjunction with a recent maternal infection and lack of treatment [START_REF] Fiumara | The incidence of prenatal syphilis at the Boston City Hospital[END_REF][START_REF] Ingraham Nr | The value of penicillin alone in the prevention and treatment of congenital syphilis[END_REF][START_REF] Rawstron | Maternal and congenital syphilis in Brooklyn, NY. Epidemiology, transmission, and diagnosis[END_REF]. The estimated rates of vertical transmission in untreated women are 70-100% for primary syphilis, 67% for secondary syphilis and 40-83% for early latent syphilis [START_REF] Hollier | Fetal syphilis: clinical and laboratory characteristics[END_REF]. Transmission from mothers with late latent syphilis to their infants can also occur, but at an estimated transmission rate of only 10% [START_REF] Bont | Positive results from serologic screening for syphilis in pregnancy in the Amsterdam region, 1991-1995[END_REF][START_REF] Fiumara | Syphilis in newborn children[END_REF][START_REF] Singh | Syphilis: review with emphasis on clinical, epidemiologic, and some biologic features[END_REF]. CS is also more likely to occur when the mother has high nontreponemal test titres (Venereal Diseases Research Laboratory, VDRL) of ≥16 at treatment or at delivery, and if short intervals between treatment and delivery (<4 weeks) occur [START_REF] Saloojee | The prevention and management of congenital syphilis: an overview and recommendations[END_REF][START_REF] Sheffield | Congenital syphilis after maternal treatment for syphilis during pregnancy[END_REF].

The diagnosis of CS is problematic. More than half of all infected infants are asymptomatic at birth, and the signs in symptomatic infants may be subtle and non-specific.

Clinical disease

CS can be divided into two clinical syndromes: early and late CS. Early CS is diagnosed during the first two years of life, and includes stillbirths. Late CS refers to cases that present after 2 years of age, and usually manifest near puberty. It mainly affects the bones, teeth and nervous system [START_REF] Chakraborty | Syphilis is on the increase: the implications for child health[END_REF][START_REF] Doroshenko | Syphilis in pregnancy and the neonatal period (review)[END_REF][START_REF] Egglestone | Serological diagnosis of syphilis. PHLS Syphilis Serology Working Group[END_REF][START_REF] Genç | Syphilis in pregnancy (review)[END_REF].

Early CS can present at any time before 2 years of age, but a review of the literature indicates that it usually presents in the neonatal period, and seldom later than 3-4 months of life. Most affected infants are asymptomatic at birth, with two-thirds developing symptoms by 3-8 weeks.

CS is responsible for a variety of clinical symptoms, the most characteristic of which are prematurity and low birth weight (10-40% of infants), hepatomegaly with or without splenomegaly (33-100%), a blistering skin rash (40%) and bone changes seen on X-ray (75-100%) [START_REF] Brion | Long-bone radiographic abnormalities as a sign of active congenital syphilis in asymptomatic newborns[END_REF][START_REF] Chawla | Congenital syphilis in the newborn[END_REF]. Other early signs include pseudoparalysis, respiratory distress, bleeding and fever [START_REF] Davanzo | Neonatal and post-neonatal onset of early congenital syphilis: a report from Mozambique[END_REF]. Clinical examination is important. However, it is estimated that up to 60% of infants with CS are asymptomatic at birth [START_REF] Lewis | Congenital syphilis. Serologic diagnosis in the young infant[END_REF]. Therefore, the diagnosis of CS in a newborn can only be made through a review of the mother's testing and treatment history, and through the infant's clinical and laboratory findings.

No single diagnostic test can diagnose CS, and it is considered unethical not to treat a baby at risk of CS, even without a definitive diagnosis. Diagnostic tests such as enzyme immunoassays, immunoblotting detecting specific IgM and polymerase chain reaction (PCR) have made the diagnosis of foetal syphilis more sensitive and specific, and clinical features such as hepatomegaly, jaundice and bone changes can resolve within three months of birth if symptomatic infants are given the appropriate therapy.

Laboratory diagnosis

The detection of T. pallidum during pregnancy and in newborns

The most sensitive and reliable method for detecting T. pallidum in clinical specimens has been the rabbit infectivity test (RIT). RIT provides definitive evidence of viable T. pallidum, but the use of the RIT as a routine diagnostic procedure is impractical because it requires animal testing and is not widely available. Dark-field microscopy has also been used to study amniotic fluid to confirm foetal infection. This method relies on fresh, good-quality specimens. The findings of spirochetes in amniotic fluid may be a marker for more severe foetal disease [START_REF] Lucas | Doppler systolicdiastolic ratios in pregnancies complicated by syphilis[END_REF][START_REF] Wendel | Identification of Treponema pallidum in amniotic fluid and fetal blood from pregnancies complicated by congenital syphilis[END_REF].

The immunofluorescent staining of T. pallidum is an alternative to dark-field microscopy [START_REF] Cummings | Comparison of methods for the detection of Treponema pallidum in lesions of early syphilis[END_REF], but in both techniques, failure to find the organism by microscopy does not exclude a diagnosis of CS. The detection of T. pallidum by PCR in amniotic fluids, neonatal sera and neonatal cerebrospinal fluid (CSF) has proved to be a fairly sensitive alternative method (78-86%), with a reported specificity of 100% for diagnosing CS compared to RIT. This could prove to be an important additional tool to darkfield microscopy and treponemal serology [START_REF] Zoechling | Molecular detection of Treponema pallidum in secondary and tertiary syphilis[END_REF][START_REF] Sánchez | Evaluation of molecular methodologies and rabbit infectivity testing for the diagnosis of congenital syphilis and neonatal central nervous system invasion by Treponema pallidum[END_REF][START_REF] Grimprel | Use of polymerase chain reaction and rabbit infectivity testing to detect Treponema pallidum in amniotic fluid, fetal and neonatal sera, and cerebrospinal fluid[END_REF][START_REF] Noordhoek | Detection by polymerase chain reaction of Treponema pallidum DNA in cerebrospinal fluid from neurosyphilis patients before and after antibiotic treatment[END_REF]. While one study found that RIT was more sensitive than PCR analysis, this was probably caused by inhibitory factors in the clinical specimens [START_REF] Grimprel | Use of polymerase chain reaction and rabbit infectivity testing to detect Treponema pallidum in amniotic fluid, fetal and neonatal sera, and cerebrospinal fluid[END_REF]. The use of non-T. pallidum tests in newborns

The first antibody test for syphilis, named after the bacteriologist August von Wassermann [START_REF] Von Wassermann | Eine serodiagnostische reaktion bei syphilis[END_REF], was described in 1906 and was based on complement-fixation and reactivity to cardiolipin. His assay was the basis of the still widely applied non-treponemal tests, the Venereal Diseases Research Laboratory (VDRL) test and the rapid plasma reagin (RPR) test, both based on this same principle. Between 56 and 66% of infants born to mothers with positive syphilis serologic findings are positive for nontreponemal syphilis tests [START_REF] Miller | A study of the biologic falsely positive reactions for syphilis in children[END_REF][START_REF] Rawstron | Comparison of maternal and newborn serologic tests for syphilis[END_REF][START_REF] Roberts | Congenital syphilis[END_REF]. One of the diagnostic criteria sometimes used to diagnose CS is the detection of non-treponemal antibodies by the VDRL or the RPR test. This diagnostic criterion requires that the infant's VDRL/ RPR titre be at least four-fold (two or more dilutions) greater than that of the mother's serum at birth, indicating foetal antibody synthesis. This suggests CS, but should not be the only marker used to make the diagnosis [START_REF] Zenker | New case definition for congenital syphilis reporting[END_REF][START_REF] Mascola | Congenital syphilis revisited[END_REF]. The estimated sensitivity of this method is only 4-13% [START_REF] Rawstron | Comparison of maternal and newborn serologic tests for syphilis[END_REF][START_REF] Mascola | Congenital syphilis revisited[END_REF][START_REF] Srinivasan | Congenital syphilis: a diagnostic and therapeutic dilemma[END_REF][START_REF] Taber | Congenital syphilis[END_REF], with a specificity of 99% [START_REF] Rawstron | Comparison of maternal and newborn serologic tests for syphilis[END_REF].

Most newborns with a four-fold titre ratio had obvious symptoms, and all had abnormal findings that enabled the diagnosis of CS to be made without the use of this criterion [START_REF] Rawstron | Comparison of maternal and newborn serologic tests for syphilis[END_REF]. In addition, no asymptomatic CS infections were detected with this method. It has also been suggested that the height of the titre of non-treponemal assays may be more useful than the ratio of newborn and maternal titres, since all newborns with RPR titres of 1,024 or greater had CS, while all newborns with titres lower than 1,024 did not [START_REF] Rawstron | Comparison of maternal and newborn serologic tests for syphilis[END_REF].

For the above reasons-and given their low sensitivitythe clinical application of non-treponemal tests seems to be very limited. However, the diagnosis of CS can be excluded if the VDRL/RPR tests become non-reactive before the age of six months in an infant who has not received treatment [START_REF] Sánchez | Congenital syphilis associated with negative results of maternal serologic tests at delivery[END_REF].

The detection of T. pallidum-specific IgG in infants

The T. pallidum haemagglutination assay (TPHA), the T. pallidum particle agglutination assay (TPPA) and the fluorescent treponemal antibody absorption test (FTA-ABS) are more specific serological tests for syphilis, but serology cannot differentiate between the other treponematoses (yaws, pinta and bejel). While they demonstrate IgG as well as IgA and IgM antibodies to T. pallidum, they do not distinguish among them. Since IgG from the mother is passively transferred to the newborn, a reactive serologic test in the neonate does not necessarily indicate that the infant is infected. It can be due to maternally derived IgG. Although foetuses are capable of producing IgG as early as 12.5 weeks of gestation, most immunoglobulin of this class at birth is of maternal origin [START_REF] Galatius-Jensen | On the inheritance of the Gm serum group[END_REF]. Thus, it is not surprising that the IgG reactivity of an infant's sera mirrors that of its mother in most instances, making IgG detection in newborns of limited value for diagnosing CS. In general, it is not used in routine diagnostics as a method of confirming infection in newborns. However, as in the case of non-treponemal tests, the TPPA/TPHA and FTA-ABS tests can be used to exclude CS if the tests are non-reactive before the age of one year in an infant who has not received treatment, or if these serologic tests are non-reactive in the mother and the infant [START_REF] Sánchez | Congenital syphilis associated with negative results of maternal serologic tests at delivery[END_REF].

The detection of T. pallidum-specific IgM in infants

The detection of specific IgM in a newborn is a very reliable serological method for the diagnosis of CS because maternal IgM, in contrast to IgG, does not cross the placenta. The foetus is capable of mounting an immune response by 24 weeks of gestation and can produce specific IgM antibodies [START_REF] Daffos | Prenatal diagnosis of congenital rubella[END_REF][START_REF] Eichenwald | Antibody production by the human fetus[END_REF]. While a negative IgM result cannot exclude the diagnosis of CS, a positive result should be considered as evidence of a congenital infection with T. pallidum. IgM tests can be negative, particularly in cases of early infection, and it should be recognised that the diagnosis of CS at birth is not always possible, no matter which serologic test is used. However, IgM testing can be used to identify women with a higher likelihood of active infection and who are, therefore, at a higher risk of infecting their babies. Performing the IgM test on mothers has sometimes been more effective at identifying infected babies than performing the IgM test on the babies themselves [START_REF] Rawstron | Evaluation of a Treponema pallidum-specific IgM enzyme immunoassay and Treponema pallidum western blot antibody detection in the diagnosis of maternal and congenital syphilis[END_REF]. Three different methods are currently used to detect treponemal-specific IgM in neonates for the diagnosis of CS: (1) 19S FTA-ABS tests, (2) IgM immunoblots and (3) IgM ELISAs.

(1) 19S FTA-ABS Because T. pallidum antigens are not all unique, and antibody reactivity with some of them may exist in normal serum, an absorption step was needed that used a non-pathogenic treponeme (T. reiteri, now called T. phagedenis). This gave rise to the more specific FTA-ABS test. A modification of the FTA-ABS test to detect IgM to T. pallidum was introduced in 1968 [START_REF] Scotti | A specific IgM antibody test in neonatal congenital syphilis[END_REF]. However, clinical testing in newborns showed a false-positive rate of 10% and a false-negative rate of up to 35% [START_REF] Kaufman | The FTA-ABS (IgM) test for neonatal congenital syphilis: a critical review[END_REF]. The protocol was, therefore, modified to improve the performance of the FTA-ABS test for IgM detection, with separation of the IgG fraction (maternal) from the IgM fraction (sedimentation coefficient 19S, foetal) prior to the FTA-ABS test. The 19S (IgM) FTA-ABS test is usually only available in specialised reference laboratories [START_REF] Lewis | Congenital syphilis. Serologic diagnosis in the young infant[END_REF]. The reported sensitivity of the 19S FTA-ABS test in clinical CS cases is 72-77% [START_REF] Larsen | Laboratory diagnosis and interpretation of tests for syphilis[END_REF][START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF][START_REF] Meyer | IgM rheumatoid factor removal and performance of the FTA-ABS (IgM) test in congenital syphilis[END_REF][START_REF] Stoll | Clinical and serologic evaluation of neonates for congenital syphilis: a continuing diagnostic dilemma[END_REF], with a specificity of 100% [START_REF] Meyer | IgM rheumatoid factor removal and performance of the FTA-ABS (IgM) test in congenital syphilis[END_REF][START_REF] Stoll | Clinical and serologic evaluation of neonates for congenital syphilis: a continuing diagnostic dilemma[END_REF][START_REF] Ozanne | Comparison of the fluorescent treponemal antibody absorption (FTA-ABS) test with the FTA-ABS double staining test for detection of antitreponemal immunoglobulin M in the 19S fraction of human serum[END_REF]. One to two percent (1-2%) of asymptomatic children at risk were found to be positive for IgM [START_REF] Larsen | Laboratory diagnosis and interpretation of tests for syphilis[END_REF][START_REF] Meyer | IgM rheumatoid factor removal and performance of the FTA-ABS (IgM) test in congenital syphilis[END_REF][START_REF] Stoll | Clinical and serologic evaluation of neonates for congenital syphilis: a continuing diagnostic dilemma[END_REF]. Following IgG and rheumatoid factor (RF) removal, there was an improvement in the specificity of the FTA-ABS IgM test, but this occurred at the expense of a loss of sensitivity, particularly in asymptomatic newborns [START_REF] Meyer | IgM rheumatoid factor removal and performance of the FTA-ABS (IgM) test in congenital syphilis[END_REF].

(2) IgM immunoblot Approximately ten different treponemal proteins are identified with IgM antibodies present in the sera of infected, symptomatic infants with immunoblot techniques. Reactions with 47, 17 and 15.5 kDa were found in most reactive blots, together with a variety of other reactions, including those corresponding to 72, 45, 42, 37, 34.5, 31, 24 and 7 kDa [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF][START_REF] Radolf | Identification and localization of integral membrane proteins of virulent Treponema pallidum subsp. Pallidum by phase partitioning with the nonionic detergent Triton X-114[END_REF]. Not all of the infected infants' sera reacted with all these antigens, but the 47-kDa band is one of the major immunogens in neonatal IgM antibody production in CS [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF][START_REF] Dobson | Recognition of Treponema pallidum antigens by IgM and IgG antibodies in congenitally infected newborns and their mothers[END_REF][START_REF] Sánchez | Molecular analysis of the fetal IgM response to Treponema pallidum antigens: implications for improved serodiagnosis of congenital syphilis[END_REF][START_REF] Herremans | Comparison of a Treponema pallidum IgM immunoblot with a 19S fluorescent treponemal antibody absorption test for the diagnosis of congenital syphilis[END_REF] and is also detected in the blood of foetuses [START_REF] Wendel | Identification of Treponema pallidum in amniotic fluid and fetal blood from pregnancies complicated by congenital syphilis[END_REF]. The 45-and 42-kDa antigens represent either breakdown products of the 47-kDa antigen or other related molecular weight derivates [START_REF] Chamberlain | Genetic and physicochemical characterization of the recombinant DNA-derived 47-kilodalton surface immunogen of Treponema pallidum subsp. Pallidum[END_REF].

IgM immunoblot results are considered to be reactive in neonates when one or more of the 47-, 45-, 30-and 15/17-kDa reactive bands are present [START_REF] Rawstron | Evaluation of a Treponema pallidum-specific IgM enzyme immunoassay and Treponema pallidum western blot antibody detection in the diagnosis of maternal and congenital syphilis[END_REF]. Lewis et al. used an even stricter interpretation in which the presence of at least five visual bands (including 47, 17 and 15.5 kDa) in serum samples was required to be considered as a reactive blot (the others were 42, 34.5, 31 and 24 kDa). In CSF, any IgM-reactive band in the 47 to 15.5 range was considered to be positive [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF].

The IgM immunoblot is very effective in confirming CS in symptomatic infants, with a reported sensitivity of 83-100% [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF][START_REF] Dobson | Recognition of Treponema pallidum antigens by IgM and IgG antibodies in congenitally infected newborns and their mothers[END_REF][START_REF] Sánchez | Molecular analysis of the fetal IgM response to Treponema pallidum antigens: implications for improved serodiagnosis of congenital syphilis[END_REF][START_REF] Wendel | Identification of Treponema pallidum in amniotic fluid and fetal blood from pregnancies complicated by congenital syphilis[END_REF]. In samples from asymptomatic infants at risk for CS, 4-22% were reported to be reactive in the IgM immunoblot [START_REF] Larsen | Laboratory diagnosis and interpretation of tests for syphilis[END_REF][START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF][START_REF] Sánchez | Molecular analysis of the fetal IgM response to Treponema pallidum antigens: implications for improved serodiagnosis of congenital syphilis[END_REF][START_REF] Sánchez | IgM antibody to Treponema pallidum in cerebrospinal fluid of infants with congenital syphilis[END_REF][START_REF] Sánchez | Evaluation of molecular methodologies and rabbit infectivity testing for the diagnosis of congenital syphilis and neonatal central nervous system invasion by Treponema pallidum[END_REF]. The patterns of IgM reactivity were similar for both asymptomatic and symptomatic infants [START_REF] Sánchez | Molecular analysis of the fetal IgM response to Treponema pallidum antigens: implications for improved serodiagnosis of congenital syphilis[END_REF]. The IgM immunoblot was more sensitive than the FTA-ABS IgM assay when used for the screening of newborns at risk for CS [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF]. No reactivity in an IgM immunoblot was detected in the control groups (infants born from adequately treated mothers or healthy controls or children with similar symptoms but with other causes [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF][START_REF] Dobson | Recognition of Treponema pallidum antigens by IgM and IgG antibodies in congenitally infected newborns and their mothers[END_REF][START_REF] Sánchez | Molecular analysis of the fetal IgM response to Treponema pallidum antigens: implications for improved serodiagnosis of congenital syphilis[END_REF][START_REF] Sánchez | IgM antibody to Treponema pallidum in cerebrospinal fluid of infants with congenital syphilis[END_REF]), leading to a specificity of 100%.

(3) IgM ELISAs Both in-house and commercially available IgM ELISAs have been evaluated for the detection of IgM responses in CS and asymptomatic cases [START_REF] Larsen | Laboratory diagnosis and interpretation of tests for syphilis[END_REF][START_REF] Stoll | Clinical and serologic evaluation of neonates for congenital syphilis: a continuing diagnostic dilemma[END_REF][START_REF] Bromberg | Diagnosis of congenital syphilis by combining Treponema pallidum-specific IgM detection with immunofluorescent antigen detection for T. pallidum[END_REF]. Most IgM ELISAs are based on sonic extracts of T. pallidum or a mix of cloned antigens [START_REF] Gerber | Recombinant Treponema pallidum antigens in syphilis serology[END_REF]. The sensitivity (88-100%) and specificity (100%) of the IgM ELISAs are high, and positive IgM results of 3-7% are detected in asymptomatic cases at risk for CS. Not many studies have compared the IgM ELISA to the IgM immunoblot, but the IgM ELISAs seem to be slightly less sensitive compared to the IgM immunoblot assays. A total of 21 out of 25 infants with a positive IgM blot were also positive in the IgM ELISA. Two of the falsenegative ELISA results were in patients with single bands on the immunoblot [START_REF] Bromberg | Diagnosis of congenital syphilis by combining Treponema pallidum-specific IgM detection with immunofluorescent antigen detection for T. pallidum[END_REF]. However, the IgM ELISA is an acceptable alternative for laboratories that have less experience with immunoblotting, and it is easier to perform.

The detection of T. pallidum-specific IgM in CSF

Neurosyphilis is believed to occur in 60% of infants with symptomatic CS, as judged by the presence of CSF abnormalities, such as reactivity in the VDRL test, pleocytosis and elevated protein content [START_REF] Platou | Treatment of congenital syphilis with penicillin[END_REF]. Others report that neurosyphilis is less common in infants with CS [START_REF] Sánchez | Evaluation of molecular methodologies and rabbit infectivity testing for the diagnosis of congenital syphilis and neonatal central nervous system invasion by Treponema pallidum[END_REF][START_REF] Srinivasan | Congenital syphilis: a diagnostic and therapeutic dilemma[END_REF], especially if the infants are asymptomatic. Guideline recommendations include a CSF examination in infants of untreated or inadequately treated mothers [START_REF]Sexually transmitted disease treatment guidelines, 2006[END_REF][START_REF] French | IUSTI: 2008 European Guidelines on the Management of syphilis[END_REF]. However, lumbar punctures in newborn infants are invasive and not entirely innocuous procedures.

A specific IgM response to T. pallidum antigensparticularly the 47-kDa antigen-was detected in the CSF of infants, showing clinical and laboratory evidence of CS, with fewer additional reactions than in the serum [START_REF] Srinivasan | Congenital syphilis: a diagnostic and therapeutic dilemma[END_REF]. However, the IgM immunoblotting of serum identified all infants who had a T. pallidum infection of the central nervous system [START_REF] Michelow | Central nervous system infection in congenital syphilis[END_REF]. The sensitivity of IgM immunoblotting in CSF was low, suggesting that this test is of limited use [START_REF] Michelow | Central nervous system infection in congenital syphilis[END_REF]. Comparison of IgM detection in CSF by immunoblot (2/7) and the detection of treponema in CSF by RIT (6/7) and PCR (5/7) revealed that the detection of Treponema in CSF was more sensitive than IgM detection [START_REF] Sánchez | Evaluation of molecular methodologies and rabbit infectivity testing for the diagnosis of congenital syphilis and neonatal central nervous system invasion by Treponema pallidum[END_REF].

None of the CSF samples from children with asymptomatic CS were IgM-positive [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF][START_REF] Sánchez | IgM antibody to Treponema pallidum in cerebrospinal fluid of infants with congenital syphilis[END_REF][START_REF] Srinivasan | Congenital syphilis: a diagnostic and therapeutic dilemma[END_REF] and no IgM was detected in the CSF of infants with late-onset disease [START_REF] Srinivasan | Congenital syphilis: a diagnostic and therapeutic dilemma[END_REF]. Therefore, no benefit from CSF examination can be expected in an asymptomatic infant.

Follow-up serology

All seroreactive infants (or infants whose mothers were seroreactive at delivery) should receive careful follow-up examinations and serologic testing at one month and at least every two to three months thereafter until the maternal antibodies have disappeared and the serological tests become non-reactive. In the case of an IgM-positive response at birth, treatment should be started and follow-up tests arranged at one, two, three, six and 12 months in order to assess response to treatment. After treatment in adults, the IgM antibodies usually disappear within one year, but can persist in a minority of patients [START_REF] Wilkinson | IgM-FTA test in syphilis in adults. Its relation to clinical findings[END_REF]. In the absence of detectable IgM in the newborn, or if the IgM test is inconclusive, the suspected diagnosis can be confirmed or excluded by serial postpartum tests of antibody kinetics only.

During follow-up, non-treponemal test reactivity should decline by three months of age, and should be non-reactive by six months of age if the infant was not infected. The serological response after therapy may be slower for infants treated after the neonatal period (>1 month). Treponemal tests are less effective for evaluating treatment responses because the results for an infected child can remain positive, despite effective therapy. IgG antibodies are eliminated in non-infected infants, with a half-life of 20.5 days, so passively transferred maternal treponemal antibodies can be present in an infant until 15 months of age [START_REF] Hagedorn | Prophylaxe und diagnostik der syphilis connata aus immunologischer sicht[END_REF].

A reactive treponemal test after 18 months is diagnostic of CS. If the non-treponemal test is non-reactive at this time, no further evaluation or treatment is necessary, as long as there is documentation of adequate therapy in the neonatal period. If the non-treponemal test is still reactive at 18 months of age, the infant should be re-evaluated and treated for CS. Given the appropriate therapy, the serological markers (RPR/VDRL and IgM) in symptomatic infants usually disappear within six months [START_REF] Dobson | Recognition of Treponema pallidum antigens by IgM and IgG antibodies in congenitally infected newborns and their mothers[END_REF].

Cord blood versus serum samples

The Centers for Disease Control and Prevention (CDC) suggests serological testing for syphilis on a newborn's serum sample, rather than on the cord blood. This is because serum samples show fewer false-positive results, whereas cord blood samples can become contaminated by maternal blood [START_REF]Sexually transmitted disease treatment guidelines, 2006[END_REF]. Most cord and newborn serum samples (79/87, 91%) were within two tube dilutions of each other in the RPR test [START_REF] Rawstron | Comparison of maternal and newborn serologic tests for syphilis[END_REF]. If newborn serum samples were taken to be the true-positive samples, then the number of false-positive cord blood RPR tests was 17/171 (10%). The number of false-negative cord RPR tests was reported to be 9/171 (5%) [START_REF] Rawstron | Comparison of maternal and newborn serologic tests for syphilis[END_REF].

Rheumatoid factors in CS

The interference of RFs in the measurement of IgM antibodies in adults against T. pallidum was described as early as 1959 [START_REF] Peltier | The presence of the "rheumatoid factor" in sera from patients with syphilis[END_REF]. IgM RF has also been detected in CS, and over 90% of infected infants have circulating RF [START_REF] Meyer | IgM rheumatoid factor in congenital syphilis: associations with clinical and laboratory findings[END_REF][START_REF] Reimer | The specificity of fetal IgM: antibody or anti-antibody?[END_REF]. Elevated RF levels were found to correlate with liver and renal involvement, as well as the extent of the disease. Following treatment, both RF levels and VDRL titres in the infants declined at a similar rate. These findings indicate a close relationship between the disease process and IgM RF levels [START_REF] Meyer | IgM rheumatoid factor in congenital syphilis: associations with clinical and laboratory findings[END_REF].

Infant sera that contained RF maintained its reactivity even after purification of the IgM and removal of the maternal IgG. This suggest that rigorous fractionation of serum immunoglobulins may not be necessary for IgM immunoblot analysis in neonates [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF][START_REF] Dobson | Recognition of Treponema pallidum antigens by IgM and IgG antibodies in congenitally infected newborns and their mothers[END_REF][START_REF] Sánchez | Molecular analysis of the fetal IgM response to Treponema pallidum antigens: implications for improved serodiagnosis of congenital syphilis[END_REF]. However, all of the serum sample tests from asymptomatic infants at risk, characterised by very the faint staining of only a few IgM reactions, yielded non-reactive blots when they were fractionated and the IgM fractions were retested [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF].

Conclusion

All infants born to mothers who have reactive nontreponemal and treponemal test results should be examined thoroughly for clinical evidence of congenital syphilis (CS). Furthermore, symptoms of CS are often subtle and nonspecific, and up to 60% of affected infants are asymptomatic at birth, so all infants at risk should be evaluated by serological assays.

Because maternal IgM does not cross the placenta, an IgM-positive neonatal serum should be considered as evidence of congenital infection. The IgM immunoblot currently seems to be the best available method to detect Treponema pallidum-specific IgM in neonates [START_REF] Herremans | Comparison of a Treponema pallidum IgM immunoblot with a 19S fluorescent treponemal antibody absorption test for the diagnosis of congenital syphilis[END_REF]. The IgM immunoblot is able to identify infants with clinical CS with high sensitivity, and IgM reactivity is sometimes the only evidence of an active infection in asymptomatic infants [START_REF] Lewis | Evaluation of immunoglobulin M western blot analysis in the diagnosis of congenital syphilis[END_REF][START_REF] Sánchez | Evaluation of molecular methodologies and rabbit infectivity testing for the diagnosis of congenital syphilis and neonatal central nervous system invasion by Treponema pallidum[END_REF]. Suspected cases can also be confirmed or excluded by serial post-partum tests of antibody kinetics.

Although tremendous progress has been made over the years in the new serologic techniques applicable to CS, there is still no single serologic assay that will reliably diagnose or exclude CS in young infants. Since it is considered unethical not to treat a baby at risk of CS, even without a definitive diagnosis, treatment decisions should not be based on definitive evidence of CS only, but also on the likelihood of infection.

Table 1

 1 Overview of the reported sensitivity and specificity of IgM assays in newborns at risk or with clinical or asymptomatic congenital syphilis (CS) and healthy newborns

	Test	Sensitivity	Specificity	Reference
	FTA-ABS IgM assay
	Clinical CS	88-92%		[1, 2]
	Asymptomatic CS	65-83%		[1, 2]
	At risk for CS	ND
	Healthy newborns		91-98%	[1, 2]
	(19S)FTA-ABS IgM assay
	Clinical CS	72-78%		[3-6]
	Asymptomatic CS	25%		[5]
	At risk for CS	2%		[3, 6]
	Healthy newborns		100%	[5-7]
	IgM immunoblot
	Clinical CS	83-100%		[4, 8-13]
	Asymptomatic CS	100%		[8]
	At risk for CS	4-42%		[4, 8, 10, 11, 14, 16]
	Healthy newborns		67-100%	[4, 9-11]
	IgM ELISA
	Clinical CS	88-100%		[3, 6, 8, 16]
	Asymptomatic CS	86%		[8]
	At risk for CS	2-66%		[3, 6, 8, 16]
	Healthy newborns		100%	[6]


Dominique Foata

Guo-Niu Han

Doubloons and q-secant numbers

Based on the evaluation at t = -1 of the generating polynomial for the hyperoctahedral group by the number of descents, an observation recently made by Hirzebruch, a new q-secant number is derived by working with the Chow-Gessel q-polynomial involving the flag major index. Using the doubloon combinatorial model we show that this new q-secant number is a polynomial with positive integral coefficients, a property apparently hard to prove by analytical methods.

Introduction

This paper, in harmony with our previous two papers on doubloons [START_REF] Foata | Doubloons and new q-tangent numbers[END_REF][START_REF] Foata | The doubloon polynomial triangle[END_REF], is motivated by our intention of finding a combinatorial connection between the Eulerian polynomials, on the one hand, and the trigonometric functions, tangent and secant, on the other hand, when the connection is further carried over to a q-analog environment.

Let (t; q) n := (1t)(1tq) • • • (1tq n-1 ) if n ≥ 1 and (t; q) 0 := 1 be the traditional q-ascending factorial and [ j ] q := 1 + q + • • • + q j-1 be the q-analog of the positive integer j. The q-analogs A n (t, q), introduced by Carlitz ( [START_REF] Carlitz | q-Bernoulli and Eulerian numbers[END_REF], [START_REF] Carlitz | A combinatorial property of q-Eulerian numbers[END_REF]), of the Eulerian polynomials, may be defined by the identity (1.1)

A n (t, q) (t; q) n+1 = j≥0 t j ([j + 1] q ) n (n ≥ 0).

For each n ≥ 0 the q-analog A n (t, q) is a polynomial with positive integral coefficients [in short, a PIC polynomial], such that A n (t, 1) is equal to the traditional Eulerian polynomial A n (t) introduced by Euler himself [Eul1755], who also derived the exponential generating function:

(1.2)

n≥0 u n n! A n (t) = 1 -t -t + exp(u(t -1

))

.

As A n (1, 1) = A n (1) = n!, each PIC polynomial A n (t) (resp. A n (t, q)) has been regarded as a generating function for the symmetric group S n by several integral-valued statistics (resp. pairs of such statistics) ( [START_REF] Riordan | An Introduction to Combinatorial Analysis[END_REF], [START_REF] Foata | Théorie géométrique des polynômes eulériens[END_REF], [START_REF] Carlitz | A combinatorial property of q-Eulerian numbers[END_REF]). Note that (1.2) is easily derived from (1.1).

In the same manner, the next two identities B n (t, q) (t; q 2 ) n+1 = j≥0 t j ([2j + 1] q ) n (n ≥ 0); (1.3)

n≥0 u n n! B n (t) =
(1t) exp(u(t -1)) -t + exp(2u(t -1)) ; (1.4) may serve to define two families of polynomials (B n (t)), (B n (t, q)) (n ≥ 0).

Again, both B n (t) and B n (t, q) are PIC polynomials and B n (t) = B n (t, 1). Moreover, (1.4) is easily derived from (1.3). The interpretation of B n (t) as a generating polynomial for the hyperoctahedral group B n , together with the derivations of (1.3) for q = 1 and (1.4), was first obtained by Reiner [START_REF] Reiner | Signed permutation statistics[END_REF], also by Cohen [START_REF] Arjeh | Eulerian polynomials of spherical type[END_REF] in the general context of the Coxeter groups of spherical type. Formula (1.3) was derived and fully interpreted by Chow and Gessel [CG07].

While studying the signatures of the toric varieties, Hirzebruch [Hi09] is led to calculate the values of both polynomials A n (t) and B n (t) at t = -1. He first quotes Euler's identities [Eul1755] (1.5)

A 2n (-1) = 0 (n ≥ 1); (-1) n A 2n+1 (-1) = T 2n+1 (n ≥ 0), where the coefficients T 2n+1 (n ≥ 0) are the tangent numbers occurring in the Taylor expansion of tan u: Then, he notes that B 2n+1 (-1) = 0 (n ≥ 0); (-1) n B 2n (-1) = 2 2n E 2n (n ≥ 0), (1.7) where the coefficients E 2n (n ≥ 0) are the secant numbers occurring in the Taylor expansion of sec u

tan u = n≥0 u 2n+1 (2n + 1)! T 2n+1 (1.6) = u 1! 1 + u 3 3! 2 +
sec u = 1 cos u = n≥0 u 2n (2n)! E 2n (1.8) = 1 + u 2 2! 1 + u 4 4! 5 + u 6 6! 61 + u 8 8! 1385 + u 10 10! 50521 + • • • since, by (1.4), n≥0 (iu) n 2 n n! B n (-1) = 2 e iu + e -iu = sec u = n≥0 u 2n (2n)! E 2n .
It so happens that (1.7) is just the relation needed to construct a new qanalog of the secant number, in parallel with what has been done already for the tangent number.

Theorem 1.1. Let (B n (t, q)) (n ≥ 0) be the sequence of polynomials defined by (1.3) and let

(1.9) E 2n (q) := (-1) n q n 2 B 2n (-q -2n , q) (n ≥ 1).

Then, (a) each E 2n (q) is a PIC polynomial;

(b) it admits the factorization

(1.10) E 2n (q) = (1 + q 2 )(1 + q 4 ) • • • (1 + q 2n )F 2n (q),
where

F 2n (q) is a PIC polynomial; (c) E 2n (1) = 2 n F 2n (1) = 2 2n E 2n (E 2n the secant number); (d) B 2n+1 (-q -(2n+1) , q) = 0 (n ≥ 0).
Property (c) follows from (1.7) and (1.9). Property (d) is proved in Section 5. As is often the case, it is much harder to derive the factorization shown in (b) and prove that the coefficients of E 2n (q) are positive. It requires a long combinatorial development, given in the next three Sections. We reproduce the first values of the polynomials B n (t, q) and E 2n (q) in Tables 1.1 and 1.2.

B 1 (t, q) = 1 + qt;
B 2 (t, q) = 1 + (2q + 2q 2 + 2q 3 )t + q 4 t 2 ; B 3 (t, q) = 1 + (3q + 5q 2 + 7q 3 + 5q 4 + 3q 5 )t + (3q 4 + 5q 5 + 7q 6 + 5q 7 + 3q 8 )t 2 + q 9 t 3 ; B 4 (t, q) = 1 + (4q + 9q 2 + 16q 3 + 18q 4 + 16q 5 + 9q 6 + 4q 7 )t + (6q 4 + 16q 5 + 30q 6 + 40q 7 + 46q 8 + 40q 9 + 30q 10 + 16q 11 + 6q 12 )t 2 + (4q 9 + 9q 10 + 16q 11 + 18q 12 + 16q 13 + 9q 14 + 4q 15 )t 3 + q 16 t 4 .

Table 1.1. The polynomials B n (t, q). E 2 (q) = (1 + q 2 )2; E 4 (q) = (1 + q 2 )(1 + q 4 )(6 + 8q + 6q 2 ); E 6 (q) = (1 + q 2 )(1 + q 4 )(1 + q 6 )(20 + 60q + 104q 2 + 120q 3 + 104q 4 + 60q 5 + 20q 6 ); E 8 (q) = (1 + q 2 )(1 + q 4 )(1 + q 6 )(1 + q 8 )(70 + 336q + 910q 2 + 1760q 3 +2702q 4 +3440q 5 +3724q 6 +3440q 7 +2702q 8 +1760q 9 +910q 10 + 336q 11 + 70q 12 ).

Table 1.2. The polynomials E 2n (q).

Following the method developed in [START_REF] Foata | Doubloons and new q-tangent numbers[END_REF] and [START_REF] Foata | The doubloon polynomial triangle[END_REF], the proof of Theorem 1.1 (a) and (b) will consist of making the polynomial E 2n+2 (q), defined in (1.9), appear as a generating function by an appropriate statistic "smaj," combined with a sign "sgn"

E 2n+2 (q) = w∈B 2n+2 sgn w q smaj w (n ≥ 1)
and constructing a sign-reversing involution on B 2n+2 , in such a way that after its application the remaining terms in the sum have positive signs. We leave out the banal case: E 2 (q) = 2(1 + q 2 ).

The final step is then to prove the identity

(1.11) E 2n+2 (q) = (1 + q 2 )(1 + q 4 ) • • • (1 + q 2n+2 ) w∈SN 2n+2 q smaj w ,
where the sum is over a specific class SN 2n+2 of signed permutations, called normalized signed doubloons (see Section 4). More importantly, the generating polynomial for SN 2n+2 occurring in (1.11) will be explicitly calculated by means of the doubloon polynomials (d n,j (q)) (n ≥ 1, 2 ≤ j ≤ 2n), which are defined by the recurrence (D1) d 0,j (q) = δ 1,j (Kronecker symbol); (D2) d n,j (q) = 0 for n ≥ 1 and j ≤ 1 or j ≥ 2n + 1; (D3) d n,2 (q) = j q j-1 d n-1,j (q) for n ≥ 1;

(D4) d n,j (q) -2 d n,j-1 (q) + d n,j-2 (q) = -(1 -q) j-3 i=1 q n+i+1-j d n-1,i (q) -(1 + q n-1 ) d n-1,j-2 (q) + (1 -q) 2n-1 i=j-1 q i-j+1 d n-1,i (q)
for n ≥ 2 and 3 ≤ j ≤ 2n; the first values being: d 1,2 (q) = 1; d 2,2 (q) = q; d 2,3 (q) = q + 1; d 2,4 (q) = 1; d 3,2 (q) = 2q 3 +2q 2 ; d 3,3 (q) = 2q 3 +4q 2 +2q; d 3,4 (q) = q 3 +4q 2 +4q+1; d 3,5 (q) = 2q 2 + 4q + 2; d 3,6 (q) = 2q + 2; d 4,2 (q) = 5q 6 +12q 5 +12q 4 +5q 3 ; d 4,3 (q) = 5q 6 +17q 5 +24q 4 +17q 3 +5q 2 ; d 4,4 (q) = 3q 6 + 15q 5 + 29q 4 + 29q 3 + 15q 2 + 3q; d 4,5 (q) = q 6 + 9q 5 + 25q 4 + 34q 3 + 25q 2 + 9q + 1; d 4,6 (q) = 3q 5 + 15q 4 + 29q 3 + 29q 2 + 15q + 3; d 4,7 (q) = 5q 4 + 17q 3 + 24q 2 + 17q + 5; d 4,8 (q) = 5q 3 + 12q 2 + 12q + 5.

Those polynomials were introduced and used in [START_REF] Foata | The doubloon polynomial triangle[END_REF] to evaluate a new q-analog (1.12) T 2n+1 (q) := (-1) n q ( n 2 ) A 2n+1 (-q -n , q)

of the tangent number based on the Carlitz q-Eulerian polynomial A n (t, q) defined in (1.1). It was shown that T 2n+1 (q) was a PIC polynomial equal to

(1.13) T 2n+1 (q) = (1 + q)(1 + q 2 ) • • • (1 + q n ) 2n+2 k=2 d n,k (q).
The parallel expression for the PIC polynomials E 2n+2 (q) is next stated.

Theorem 1.2. For each n ≥ 1 the polynomial E 2n+2 (q) has the following expression:

E 2n+2 (q) = (1 + q 2 )(1 + q 4 ) • • • (1 + q 2n+2 ) 2n k=2 d n,k (q 2 )P n,k (q), (1.14)
where the coefficients P n,k (q) (n ≥ 1, 2 ≤ j ≤ 2n) are defined by

P n,k (q) := 2n+1-k i=0 q n-1-2i i+k l=i+1 2n + 2 l q l . (1.15)
The quantities Q n,k (q) := q n+1-k P n,k (q) are PIC polynomials. Their first values are listed in Table 1.3. Q 1,2 (q) = 6 + 8q + 6q 2 ; Q 2,2 (q) = 15 + 26q + 30q 2 + 26q 3 + 15q 4 ; Q 2,3 (q) = 20 + 30q + 32q 2 + 30q 3 + 20q 4 ; Q 2,4 (q) = Q 2,2 (q) Q 3,2 (q) = 28 + 64q + 98q 2 + 112q 3 + 98q 4 + 64q 5 + 28q 6 ; Q 3,3 (q) = 56 + 98q + 120q 2 + 126q 3 + 120q 4 + 98q 5 + 56q 6 ; Q 3,4 (q) = 70 + 112q + 126q 2 + 128q 3 + 126q 4 + 112q 5 + 70q 6 ; Q 3,5 (q) = Q 3,3 (q); Q 3,6 (q) = Q 3,2 (q); Q 4,2 (q) = 45 +130q +255q 2 +372q 3 +420q 4 +372q 5 +255q 6 +130q 7 +45q 8 ; Q 4,3 (q) = 120+255q+382q 2 +465q 3 +492q 4 +465q 5 +382q 6 +255q 7 +120q 8 ; Q 4,4 (q) = 210+372q+465q 2 +502q 3 +510q 4 +502q 5 +465q 6 +372q 7 +210q 8 ; Q 4,5 (q) = 252+420q+492q 2 +510q 3 +512q 4 +510q 5 +492q 6 +420q 7 +252q 8 ; Q 4,6 (q) = Q 4,4 (q); Q 4,7 (q) = Q 4,3 (q); Q 4,8 (q) = Q 4,2 (q). Table 1.3. The polynomials Q n,k (q). The proofs of Theorems 1.1 and 1.2 are given in Sections 3 and 4. In the last Section we obtain a global expression for the generating polynomial for the group B n by a five-variable statistic, which takes the two classical descent definitions into account.

To end this introduction we point out that the identity

(1.16) T 2n+1 = 2 n 2n k=2 d n,k ,
which is the q = 1 version of (1.13), is originally due to Christiane Poupard [START_REF] Poupard | Deux propriétés des arbres binaires ordonnés stricts[END_REF], who worked out the recurrence for the now called Poupard triangle

d n,k := d n,k (1) (n ≥ 1, 2 ≤ k ≤ 2n), obtainable from (D1)-(D4) for q = 1.
We reproduce the first values of the Poupard triangle (d n,k ), together with the first values of (1.17) The q = 1 version of identity (1.14) reads:

Q n,k := Q n,k (1) = P n,k (1) = 2n+1-k i=0 i+k l=i+1 2n + 2 l . Both d n,k and Q n,k are displayed in triangles (2 ≤ k ≤ 2n, 1 ≤ n ≤ 4),
(1.18) 2 n+1 E 2n+2 = 2n k=2 d n,k Q n,k .
For instance, (1.18) for n = 2 yields:

2 3 E 6 = 8 × 61 = 488 = 1 × 112 + 2 × 132 + 1 × 112.
There exists a rich formulary of relations for tangent and secant numbers (see, e.g., the old monograph by Nielsen [START_REF] Nielsen | Traité élémentaire des nombres de Bernoulli[END_REF]). Identities (1.16) and (1.18) provide a new parametrization of those coefficients by means of the Poupard triangle (d n,k ).

Statistics on the hyperoctahedral group

The elements of the hyperoctahedral group B n , usually called signed permutations, may be viewed as words w = x 1 x 2 • • • x n , where each x i belongs to the set {-n, . . . , -1, 1, . . . , n} and |x

1 ||x 2 | • • • |x n | is a permu- tation of 12 . . . n.
The set (resp. the number) of negative letters among the x i 's is denoted by Neg w (resp. neg w). In the same manner, let Pos w (resp. pos w) be the set (resp. the number) of all positive letters in w. It is convenient to write i := -i for each integer i. There are 2 n n! signed permutations of order n. The symmetric group S n may be considered as the subset of all w from B n such that Neg w = ∅.

For each statement A let χ(A) = 1 or 0 depending on whether A is true or not. The usual number of descents and major index of each word

w = x 1 x 2 • • • x n are defined by des w := n-1 i=1 χ(x i > x i+1 ); (2.1) maj w := n-1 i=1 i χ(x i > x i+1 ). (2.2)
When B n is regarded as a Coxeter group, an extra descent is counted, when the first letter x 1 of the signed permutation w

= x 1 x 2 • • • x n is negative.
In the literature two definitions are then used:

des B w := χ(x 1 < 0) + des w; (2.3) fdes w := χ(x 1 < 0) + 2 des w. (2.4)
Furthermore, a flag major index "fmaj" defined by (2.5) fmaj w := 2 maj w + neg w, has been adopted for B n , because it is equidistributed with the Coxeter length "ℓ" for B n (see, e.g., [START_REF] Adin | Descent Numbers and Major Indices for the Hyperoctahedral Group[END_REF], [START_REF] Foata | Signed words and permutations, I: a fundamental transformation[END_REF]), a property that extends the corresponding property for the symmetric group S n , which says that the major index "maj" and the number of inversions "inv" (the Coxeter length for S n ) are equidistributed.

Proposition 2.1. The polynomial B n (t, q) defined by (1.3) has the following combinatorial interpretation:

(2.6) B n (t, q) = w∈B n t des B w q fmaj w .
In other words, B n (t, q) is the generating polynomial for the hyperoctahedral group B n by the pair (des B , fmaj).

The proof of the proposition can be found in [CG07]. This is also a consequence of Theorem 6.2, that takes both "des B " and "fdes" into account (see (6.15) and (6.16)).

From the definition of the polynomials E 2n+2 (q) given in (1.9) and (2.6) it follows that E 2n+2 (q) = (-1) n+1 q (n+1) 2 B 2n+2 (-q -2n+2 , q) may be expressed as

E 2n+2 (q) = (-1) n+1 w=x 1 •••x 2n+2 ∈B 2n+2 (-1) χ(x 1 <0)+des w q smaj w , (2.7)
where "smaj" is a new statistic -call it signed major index -defined for each signed permutation w

= x 1 x 2 • • • x 2n+2 ∈ B 2n+2 by (2.8) smaj w := (n + 1) 2 -2(n + 1) χ(x 1 < 0) + des w + 2 maj w + neg w.
A compressed major index "cmaj" was defined in [START_REF] Foata | Doubloons and new q-tangent numbers[END_REF], [START_REF] Foata | The doubloon polynomial triangle[END_REF] on the symmetric group S n . Extend its definition to each w ∈ B 2n+2 , as follows (2.9) cmaj w := maj w -(n + 1) des w + (n -1)n/2.

The next lemma only needs a straightforward calculation.

Lemma 2.2.

For each w = x 1 x 2 • • • x 2n+2 ∈ B 2n+2 we have: smaj w -2 cmaj w = 3n + 1 + neg w -2(n + 1)χ(x 1 < 0); (2.10) so that smaj w -2 cmaj w = n + neg w -1, if x 1 < 0. (2.11)
The mirror image of a signed permutation w

= x 1 x 2 • • • x 2n+2 is defined by r w := x 2n+2 • • • x 2 x 1 .
It is easily verified that des r w = (2n + 1)des w;

(2.12) maj r w = (2n + 2)(2n + 1)/2 -(2n + 2) des w + maj w.

(2.13) Those two relations suffice to prove the next lemma.

Lemma 2.3. For each w = x 1 x 2 • • • x 2n+2 ∈ B 2n+2 we have: (2.14) smaj r w -smaj w = 2(n + 1) χ(x 1 < 0) -χ(x 2n+2 < 0) ; (2.15) (-1) des r w+χ(x 2n+2 <0) × (-1) des w+χ(x 1 <0) = -(-1) χ(x 1 <0)+χ(x 2n+2 <0) .
The sum displayed in (2.7) may be decomposed into four subsums:

w=x 1 •••x 2n+2 ∈B 2n+2 = x 1 x 2n+2 >0, x 1 <x 2n+2 + x 1 x 2n+2 >0, x 1 >x 2n+2 + x 1 <0, x 2n+2 >0 + x 1 >0, x 2n+2 <0 = x 1 x 2n+2 >0, x 1 <x 2n+2 + r x 1 x 2n+2 >0, x 1 <x 2n+2 + x 1 <0, x 2n+2 >0 + r x 1 <0, x 2n+2 >0 .
It follows from Lemma 2.3 that the sum of the first two subsums vanishes, and the fourth one is equal to the product of the third one by q 2n+2 . Thus, (2.16)

E 2n+2 (q) = (-1) n+1 (1 + q 2n+2 ) w=x 1 •••x 2n+2 ∈B 2n+2 , x 1 <0<x 2n+2
(-1) des w+1 q smaj w since χ(x 1 < 0) = 1 for every w occurring in the sum. To pursue the calculation of E 2n+2 (q) we use the doubloon calculus, as developed in our previous two papers.

Doubloons

A doubloon of order (2n + 1) is defined to be a permutation of the word 012

• • • (2n+1), represented as a 2×(n+1)-matrix δ = a 0 ••• a n b 0 ••• b n . The word a 0 • • • a n b n • • • b 0 is called the reading ρ(δ) of δ. Define stat δ := stat ρ(δ),
whenever "stat" is equal to "des," "maj," "fmaj," "cmaj," or "smaj." Let F δ := a 0 , L δ := b 0 . The set of all doubloons of order (2n + 1) is denoted by D 2n+1 . The subset of all doubloons δ such that L δ = j (resp. F δ = i and L δ = j) is denoted by D 2n+1,j (resp.

D i 2n+1,j ). Each doubloon δ = a 0 ••• a n b 0 ••• b n from D 2n+1
is said to be interlaced (resp. normalized ), if for every k = 1, 2, . . . , n the sequence (a k-1 , a k , b k-1 , b k ) or one of its three cyclic rearrangements is monotonic increasing or decreasing (resp. decreasing). Let I i 2n+1 (resp. I i 2n+1,j , resp. N i 2n+1 , resp. N i 2n+1,j ) denote the set of all doubloons δ from D i 2n+1 , which are interlaced (resp. interlaced with L δ = j, resp. normalized, resp. normalized with L δ = j).

For instance, the doubloon δ = 0 4 3 2 1 5 is normalized, since both sequences (4, 2, 1, 0) and (5, 4, 3, 1), which are cyclic rearrangements of (0, 4, 2, 1) and (4, 3, 1, 5), respectively, are decreasing.

The geometry of interlaced and normalized doubloons has been studied in [START_REF] Foata | Doubloons and new q-tangent numbers[END_REF]. The connection between interlaced doubloons and split-pair arrangements, introduced by Graham and Zang [START_REF] Graham | Enumerating split-pair arrangements[END_REF], is explicitly made in [START_REF] Foata | The doubloon polynomial triangle[END_REF].

We now recall several properties on doubloons already proved in [START_REF] Foata | Doubloons and new q-tangent numbers[END_REF], [START_REF] Foata | The doubloon polynomial triangle[END_REF].

For each doubloon δ = a 0 a 1 ••• a n b 0 b 1 ••• b n from D 2n+1 and each integer h let δ + h be the doubloon (3.1) δ + h := a 0 + h a 1 + h • • • a n + h b 0 + h b 1 + h • • • b n + h ,
where each entry is expressed as a residue mod(2n + 2).

Property 3.1. The mapping δ → δ + h is a bijection of I i 2n+1,j (resp. N i 2n+1,j ) onto I i+h 2n+1,j+h (resp. N i+h 2n+1,j+h ) (superscript and subscript being taken mod(2n + 2)).

See [START_REF] Foata | The doubloon polynomial triangle[END_REF], Proposition 2.1.

Property 3.2. Let 0 ≤ i < j and δ = a 0 a 1 ••• a n b 0 b 1 ••• b n be a doubloon from D i 2n+1,j , so that δ -i = 0 a 1 -i ••• a n -i j-i b 1 -i ••• b n -i belongs to D 0 2n+1,j-i . Then, (3.2) des(δ -i) = des δ, cmaj(δ -i) = cmaj δ + i.
See [START_REF] Foata | The doubloon polynomial triangle[END_REF], Lemma 3.2.

Property 3.3. For each integer k there is a sign-reversing involution on D 0 2n+1,k \ I 0 2n+1,k having the property that

(3.3) δ∈D 0 2n+1,k (-1) n+des δ q cmaj δ = δ∈I 0 2n+1,k q cmaj δ .
Moreover,

δ∈I 0 2n+1,k q cmaj δ = (1 + q)(1 + q 2 ) • • • (1 + q n ) δ ′ ∈N 0 2n+1,k q cmaj δ ; (3.4)
Proof. Refer to the proofs of Theorems 4.2 and 1.6 in [START_REF] Foata | Doubloons and new q-tangent numbers[END_REF], and observe that the first column 0 k is left invariant under each macro flip.

Signed doubloons

Now, we extend the notion of doubloon to the group of signed permutations and speak of signed doubloons, but only for those signed permuta-

tions w = x 1 x 2 • • • x 2n+2 ∈ B 2n+2
occurring in the summation displayed in (2.16). They have the property that F w := x 1 < 0 < x 2n+2 =: L w.

We represent them as 2 × (n + 1)-matrices w =

x

1 x 2 • • • x n+1 x 2n+2 x 2n+1 • • • x n+2 .
The set of all those signed doubloons will be denoted by SD 2n+2 .

For

each w = x 1 x 2 • • • x n+1 x 2n+2 x 2n+1 • • • x n+2
from SD 2n+2 let φ w be the increasing bijection of {x 1 , x 2 , . . . , x 2n+2 } onto {0, 1, 2, . . . , 2n + 1} and form the (unnsigned) doubloon

δ w := φ w (x 1 ) φ w (x 2 ) • • • φ w (x n+1 ) φ w (x 2n+2 ) φ w (x 2n+1 ) • • • φ w (x n+2 ) .
The signed doubloon w is characterized by the pair (δ w , -Neg w). Moreover, stat w = stat δ w whenever "stat" is equal to "des," "maj," "fmaj," "cmaj," or "smaj." The signed doubloon w is said to be interlaced (resp. normalized), if δ w is interlaced (resp. normalized).

As F w < 0 < L w when w belongs to SD 2n+2 , the mapping

(4.1) w → (δ w , -Neg w)
is a bijection of the set SD 2n+2 onto the set of pairs (δ, J) such that δ ∈ D 2n+1 , and J a subset of {1, 2, . . . , 2n + 2} such that F δ + 1 ≤ #J ≤ L δ.

For instance, if δ = 042 315 ∈ D 5 , then F δ + 1 = 1 ≤ #J ≤ 3 = Lδ. Take J = {3}, {1, 3}, {2, 3, 5} for example, the three signed doubloons w ∈ SD 6 associated with those three subsets J are the following: We next make the composition product of the two mappings described in (3.1) and (4.1).

Theorem 4.1. For each pair (i, k) of integers such that 1 ≤ k ≤ 2n and 0 ≤ i ≤ 2n + 1k the mapping

(4.3) w → (δ w -i, -Neg w)
is a bijection of the set SD i 2n+2,i+k of the signed doubloons w satisfying F δ w = i, L δ w = i + k onto the set of pairs (δ, J) such that δ ∈ D 0 2n+1,k and J ⊂ [1, 2n + 2] with i + 1 ≤ #J ≤ i + k. Moreover, if w is interlaced (resp. normalized), so is δ wi, and conversely. Finally, if δ = δ wi, then (4.4) des w = des δ; smaj w = 2 cmaj δ -2i + #J + n -1.

Proof. The theorem is a consequence of Properties 3.1 and 3.2 and the properties of the bijection w → (δ w , -Neg w) given in (4.2). Identity (2.16) may be rewritten as

E 2n+2 (q) = (-1) n+1 (1 + q 2n+2 ) w∈SD 2n+2 (-1) des w+1 q smaj w = (1 + q 2n+2 ) 2n k=1 2n+1-k i=0 w∈SD i 2n+2,i+k (-1) n+des w q smaj w . Let (4.5) P n,i,k (q) := q n-1-2i i+k l=i+1 2n + 2 l q l .
Using the preceding theorem and Property 3.3 we evaluate the third sum as follows.

w∈SD i 2n+2,i+k (-1) n+des w q smaj w = δ∈D 0 2n+1,k i+1≤#J≤i+k (-1) n+des δ q 2 cmaj δ-2i+#J+n-1 = q n-1-2i δ∈D 0 2n+1,k (-1) n+des δ q 2 cmaj δ i+k l=i+1 2n + 2 l q l = P n,i,k (q) δ∈D 0 2n+1,k (-1) n+des δ q 2 cmaj δ = P n,i,k (q) δ∈I 0 2n+1,k q 2 cmaj δ = (1 + q 2 ) • • • (1 + q 2n )P n,i,k (q) δ∈N 0 2n+1,k q 2 cmaj δ (4.6) = (1 + q 2 ) • • • (1 + q 2n )P n,i,k (q) d n,k (q 2 ),
where the last equality follows from [START_REF] Foata | The doubloon polynomial triangle[END_REF], Theorem 1.2. By multiplying (4.6) by (1 +q 2n+2 ) and summing over all pairs (k, i) such that 1 ≤ k ≤ 2n and 0 ≤ i ≤ 2n + 1k we derive identity (1.14), keeping in mind that

P n,k (q) = 0≤i≤2n+1-k P n,i,k .
This achieves the proofs of both Theorems 1.1 and 1.2, except part (d).

Let SN i 2n+2,i+k be the set of the normalized signed doubloons w satisfying F δ w = i, L δ w = i + k. It also follows from Theorem 4.1 that

w∈SN i 2n+2,i+k q smaj w = P n,i,k (q) δ∈N 0 2n+1,k q 2 cmaj δ . (4.7)
From (4.6) it follows that

w∈SD i 2n+2,i+k (-1) n+des w q smaj w = (1 + q 2 ) • • • (1 + q 2n ) w∈SN i 2n+2,i+k q smaj w . (4.8)
By multiplying (4.8) by (1 + q 2n+2 ) and summing over all pairs (k, i) such that 1 ≤ k ≤ 2n and 0 ≤ i ≤ 2n + 1k we derive identity (1.11).

Proof of Theorem 1.1 (d)

Recall that for each w = x 1 x 2 • • • x 2n+1 ∈ B 2n+1 we have used the notations F w := x 1 and L w := x 2n+1 . As B 2n+1 (t, q) = w∈B 2n+1 t des B w q fmaj w , we may write (5.1) B 2n+1 (-q -(2n+1) , q) = w∈B 2n+1

(-1) sgn w q smaj w , where sgn w := (-1) des w+χ(F w<0) ; (5.2) smaj w := 2 maj w + neg w -(2n + 1)(des w + χ(F w < 0), (5.3) as there is no ambiguity to adopt this definition of "smaj" for signed permutations from B 2n+1 .

For proving the identity A 2n (-q -n , q) = 0 in [START_REF] Foata | Doubloons and new q-tangent numbers[END_REF] we had recourse to the classical properties of the dihedral group acting on S 2n . Actually, the mirror image r provided the sign-reversing involution that was needed. With the group B 2n+1 the supplementary descent to be counted, when the first letter is negative, makes it necessary to include another dihedral group involution, as well as a sign change operation.

In this section the elements of B 2n+1 will be regarded as two-row matrices w = |w|

ǫ := |x 1 | |x 2 | ••• |x 2n+1 | ǫ 1 ǫ 2 ••• ǫ 2n+1 , where |w| := |x 1 ||x 2 | • • • |x 2n+1 |
becomes an ordinary permutation and ǫ := ǫ 1 ǫ 2 • • • ǫ 2n+1 is the sign word defined by ǫ i := 1 or -1, dependeing on whether x i is positive or negative (1 ≤ i ≤ 2n + 1).

Three operations r, c, s are now introduced and further extended to all of B 2n+1 : first, the mirror image

r : y 1 y 2 • • • y 2n+1 → y 2n+1 • • • y 2 y 1 ,
defined for every arbitrary word; second, the complement to (2n + 2), defined for each permutation from S 2n+1 , by

c : y 1 y 2 • • • y 2n+1 → (2n + 2 -y 1 )(2n + 2 -y 2 ) • • • (2n + 2 -y 2n+1 );
third, the sign change s, defined for each binary word, such as ǫ = ǫ 1 ǫ 2 • • • ǫ 2n+1 , whose letters are equal to +1 or -1, by

s : ǫ 1 ǫ 2 • • • ǫ 2n+1 → ǫ 1 ǫ 2 • • • ǫ 2n+1 .
We use the same symbols for their extensions to B 2n+1 :

(5.4) r :

|w| ǫ = |x 1 | |x 2 | ••• |x 2n+1 | ǫ 1 ǫ 2 ••• ǫ 2n+1 → r |w| r ǫ = |x 2n+1 | ••• |x 2 ||x 1 | ǫ 2n+1 ••• ǫ 2 ǫ 1 ; (5.5) c : |w| ǫ = |x 1 | ••• |x 2n+1 | ǫ 1 ••• ǫ 2n+1 → c |w| ǫ = (2n+2-|x 1 |)•••(2n+2-|x 2n+1 |) ǫ 1 ••• ǫ 2n+1 ; (5.6) s : |w| ǫ = |x 1 | |x 2 |••• |x 2n+1 | ǫ 1 ǫ 2 ••• ǫ 2n+1 → |w| s ǫ = |x 1 | |x 2 | ••• |x 2n+1 | ǫ 1 ǫ 2 ••• ǫ 2n+1 .
Note that the three involutions r, c, s, defined on B 2n+1 by (5.4), (5.5) and (5.6) commute. The composition product b := c s r can also be written as

(5.7) b : |w| ǫ = |x 1 | ••• |x 2n+1 | ǫ 1 ••• ǫ 2n+1 → c r |w| r s ǫ = (n+2-|x 2n+1 |)•••(n+2-|x 1 |) ǫ 2n+1 ••• ǫ 1 .
Theorem 5.1. The composition product b defined in (5.7) is a signreversing involution of B 2n+1 , i.e., (5.12)

The third lemma requires a careful analysis.

Lemma 5.4. For each w

= |w| ǫ = |x 1 | |x 2 | ••• |x 2n+1 | ǫ 1 ǫ 2 ••• ǫ 2n+1 ∈ B 2n+1 we have: sgn c w = (-1) χ(F w<0)-χ(L w<0) sgn w; (5.13) smaj c w = -smaj w -(2n + 1) χ(F w < 0) -χ(L w < 0) . (5.14) Proof. If |x i | ǫ i > |x i+1 | ǫ i+1 (resp. |x i | ǫ i < |x i+1 | ǫ i+1 ) say that i is an interior descent (resp. rise), if |x i | > |x i+1 | (resp. |x i | < |x i+1 |) and ǫ i = ǫ i+1 .
Denote the set of all descents (resp. rises) of w by DES w (resp. RISE w), the set of all interior descents (resp. rises) being designated by DES i w (resp. RISE i w), so that DES w = DES i w + DES ǫ and RISE w = RISE i w + RISE ǫ.

First, DES i w = RISE i c w and RISE i w = DES i c w. Hence,

des w + des c w = (# DES ǫ + # DES i w) + (# DES ǫ + # DES i c w) = (# DES i w+# DES ǫ) + (# RISE i w+# RISE ǫ) + (# DES ǫ-# RISE ǫ) = 2n + (# DES ǫ -# RISE ǫ) := 2n + drise ǫ.
In the same way, let DRISE ǫ :

= i i χ(i ∈ DES ǫ) -χ(i ∈ RISE ǫ) . Then, maj w + maj c w = i i χ(i ∈ DES ǫ) + i χ(i ∈ DES i w) + i i χ(i ∈ DES ǫ) + i χ(i ∈ DES i c w) = i i χ(i ∈ DES w) + i χ(i ∈ RISE w) + i i χ(i ∈ DES ǫ) -χ(i ∈ RISE ǫ) = (1 + 2 + • • • + 2n) + DRISE ǫ = n(2n + 1) + DRISE ǫ. Let d 1 < d 2 < • • • (resp. r 1 < r 2 < • • •) denote
the sequence of the descents (resp. rises) of ǫ, when reading the word ǫ from left to right. Four cases are now considered.

(a) ǫ 1 = ǫ 2n+1 = -1; the rises and descents alternate in such a way that 1 ≤ r

1 < d 1 < r 2 < d 2 < • • • < r k < d k ≤ 2n and k ≥ 0. Hence, drise ǫ = 0 and DRISE ǫ = k i=1 (d i -r i ) = pos w. (b) ǫ 1 = +1, ǫ 2n+1 = -1; the alternation becomes: 1 ≤ d 1 < r 1 < d 2 < r 2 < • • • < d k < r k < d k+1 ≤ 2n (k ≥ 0). In this case, drise ǫ = 1 and DRISE ǫ = pos w. (c) ǫ 1 = ǫ 2n+1 = 1; the sequence is then: 1 ≤ d 1 < r 1 < d 2 < r 2 < • • • < d k < r k ≤ 2n (k ≥ 0). Hence, drise ǫ = 0 and DRISE ǫ = -neg w. (d) ǫ 1 = -1, ǫ 2n+1 = 1; then 1 ≤ r 1 < d 2 < r 2 < • • • < r k < d k < r r+1 ≤ 2n (k ≥ 0). Hence, drise ǫ = -1 and DRISE ǫ = -neg w.
Thus, sgn w +sgn c w = des w +χ(ǫ 1 < 0) +des c w +χ(ǫ 1 < 0) ≡ drise ǫ (mod 2), which is 0 when ǫ 1 and ǫ 2n+1 are of the same sign (cases (a) and (c)), equal to 1 when ǫ 1 = 1 , ǫ 2n+1 = -1 (case (b)) and -1 when ǫ 1 = -1 and ǫ 2n+1 = +1 (case (d)). Gathering in a common formula: sgn w + sgn c w ≡ χ(ǫ 1 = 1)χ(ǫ 2n+1 = 1). This implies (5.13). = sgn w(-1) χ(F w<0)-χ(L w<0) (-1) χ(L w<0)+χ(F w<0) = sgn w; smaj r c w = smaj c w + (2n + 1)(χ(F c w < 0)χ(L c w < 0)) = smaj c w + (2n + 1)(χ(F w < 0)χ(L w < 0)) =smaj w; sgn s r c w =sgn r c w =sgn w; smaj s r c w =smaj r c w = smaj w.

Which descent for the hyperoctahedral group?

The purpose of this Section is to work out a global expression for the generating polynomial for B n by the five-term statistic (neg, pos, Ξ, des, fmaj), where Ξ w is equal to 1 or 0, depending on whether the first letter of w is negative or positive, and to derive the specializations when the pair (Ξ, des) is replaced either by "des B ," or by "fdes," defined in (2.3) and (2.4). Our main result is the following. Theorem 6.1. Let (6.1) B n (X, Y, Z; t, q) = w∈B n X neg w Y pos w Z χ(x 1 <0) t des w q fmaj w .

Then,

(6.2) B n (X, Y, Z; t, q) (t; q 2 ) n+1 = t -Z t -1 s≥0 t s (qX + Y )[s + 1] q 2 n + Z -1 t -1 s≥0 t s (qX + Y )[s + 1] q 2 -Xq 2s+1 n .
When q = 1, write B n (X, Y, Z; t) := B n (X, Y, Z; t, 1). The exponential generating function for the latter polynomials can be derived in the following form.

Theorem 6.2. The following identity holds:

(6.3) n≥0 u n n! B n (X, Y, Z; t) = Z -t + (1 -Z) exp(uX(t -1)) -t + exp(u(X + Y )(t -1)) .
Proof of Theorem 6.1. Let w = x 1 x 2 • • • x n be a signed permutation from B n and φ be the unique increasing bijection of the set {x 1 , x 2 , . . . , x n } onto the interval [n] := {1, 2, . . . , n}. The word σ = σ(1)σ(2)

• • • σ(n) := φ(x 1 )φ(x 2 ) • • • φ(x n
) is then an (ordinary) permutation from S n and the map w → (Neg w, σ) a bijection of B n onto the Cartesian product 2 [n] ×S n having the following properties: χ(x 1 < 0) = χ(σ(1) ≤ neg w); des w = des σ; fmaj w = fmaj σ.

For convenience, introduce the polynomial

A k n (Z; t, q) := σ Z χ(σ(1)≤k) t des σ q maj σ (σ = σ(1) • • • σ(n) ∈ S n )
and express B n (X, Y, Z; t, q) in terms of the latter polynomials, to get:

B n (X, Y, Z; t, q) = n k=0 |E|=k Neg w=E (qX) neg w Y pos w Z χ(x 1 <0) t des w q 2 maj w = n k=0 (qX) k Y n-k |E|=k (E,σ) Z χ(σ(1)≤k) t des σ q 2 maj σ = n k=0 n k (qX) k Y n-k A k n (Z; t, q 2 ). Next, with each permutation σ = k σ(2) • • • σ(n) starting with k associate the permutation σ ′ = σ ′ (1) • • • σ ′ (n -1) := ψ(σ(2)) • • • ψ(σ(n)), where ψ is the unique increasing bijection of [n] \ {k} onto [n -1]. If σ(2) ≤ k -1, then des σ = des σ ′ +1, while maj σ = maj σ ′ +des σ ′ +1 and σ ′ (1) ≤ k -1. If σ(2) ≥ k + 1, then des σ = des σ ′ , while maj σ = maj σ ′ + des σ ′ and σ ′ (1) ≥ k. Hence, σ(1)=k, σ(2)≤k-1 Z χ(σ(1)≤k) t des σ q maj σ = Z σ ′ (1)≤k-1 t des σ ′ +1 q maj σ ′ +des σ ′ +1 = Z σ ′ (1)≤k-1 (tq) χ(σ(1)≤k-1) (tq) des σ ′ q maj σ ′ ; while σ(1)=k, σ (2)≥k+1
Z χ(σ(1)≤k) t des σ q maj σ = Z σ ′ (1)≥k

t des σ ′ q maj σ ′ +des σ ′ = Z σ ′ (1)≥k (tq) χ(σ(1)≤k-1) (tq) des σ ′ q maj σ ′ . Altogether σ(1)=k Z χ(σ(1)≤k) t des σ q maj σ = Z A k-1 n-1 (tq; tq, q).
In the same manner, σ(1)=k Z χ(σ(1)≤k-1) t des σ q maj σ = A k-1 n-1 (tq; tq, q).

Consequently, we have the relation:

(6.4) A k n (Z; t, q) = A k-1 n (Z; t, q) + (Z -1)A k-1 n-1 (tq; tq, q). By iteration we are led to:

(6.5) A k n (Z; t, q) = A 0 n (Z; t, q) + Z -1 t -1 k j=1 k j (t -1)(tq -1) • • • (tq j-1 )A 0 n-j (tq j , tq j , q).
But, the variable Z vanishes from A k n (Z; t, q) when k = 0 and then A 0 n (Z; t, q) = A n (t, q), which is the Carlitz q-analog of the Eulerian polynomial ([Ca54], [START_REF] Carlitz | A combinatorial property of q-Eulerian numbers[END_REF]) appearing in (1.1). Hence,

A k n (Z; t, q) = A n (t, q) + Z -1 t -1 k j=1 k j (-1) j (t; q) j A n-j (tq j , q) = t -Z t -1 A n (t, q) + Z -1 t -1 k j=0 k j (-1) j (t; q) j A n-j (tq j , q).
The next step is to report this new expression of A k n (Z; t, q) into the polynomial B n (X, Y, Z; t, q). We get:

B n (X, Y, Z; t, q) = n k=0 n k (qX) k Y n-k A k n (Z; t, q 2 ) = n k=0 n k (qX) k Y n-k t -Z t -1 A n (t, q 2 ) + Z -1 t -1 k j=0 k j (-1) j (t; q 2 ) j A n-j (tq 2j , q 2 ) = t -Z t -1 (qX + Y ) n A n (t, q 2 ) + Z -1 t -1 j,l,m≥0 j+l+m=n n! j! l! m! (qX) j+l Y m (-1) j (t; q 2 ) j A l+m (tq 2j , q 2 ),
where k = j + l.

Next, with r = l + m we get

B n (X, Y, Z; t, q) (t; q 2 ) n+1 = t -Z t -1 (qX + Y ) n A n (t, q 2 ) (t; q 2 ) n+1 + Z -1 t -1 j+r=n n! r! j! (-qX) j A r (tq 2j , q 2 ) (tq 2j ; q 2 ) r+1 l+m=r r! l! m! (qX) l Y m = t -Z t -1 (qX + Y ) n A n (t, q 2 ) (t; q 2 ) n+1 + Z -1 t -1 j+r=n n! r! j! (-qX) j A r (tq 2j , q 2 ) (tq 2j ; q 2 ) r+1 (qX + Y ) r .
Furthermore, n≥0

B n (X, Y, Z; t, q) (t; q 2 ) n+1

u n n! = t -Z t -1 n≥0
A n (t, q 2 ) (t; q 2 ) n+1

((qX + Y )u) n n! + Z -1 t -1 j≥0 (-qXu) j j! r≥0
A r (tq 2j , q 2 ) (tq 2j , q 2 ) r+1

((qX + Y )u) r r! .

Now, make use of the classical identity on the Carlitz q-Eulerian polynomials n≥0 u n n!

A n (t, q) (t; q) n+1 = s≥0 t s exp(u[s + 1] q ), to obtain (6.6) n≥0

B n (X, Y, Z; t, q) (t; q 2 ) n+1

u n n! = t -Z t -1 s≥0 t s exp((qX + Y )u [s + 1] q 2 ) + Z -1 t -1 j≥0 (-qXu) j j! s≥0
(tq 2j ) s exp((qX + Y )u [s + 1] q 2 ).

There remains to extract the coefficient of u n on both sides. This leads to:

(6.7) B n (X, Y, Z; t, q) n! (t; q 2 ) n+1 = t -Z t -

1 s≥0 t s (qX + Y ) [s + 1] q 2 n + Z -1 t -1 C,
where C is the coefficient of u n in j≥0 (-qXu) j j! s≥0 (tq 2j ) s m≥0

((qX + Y )u [s + 1] q 2 ) m m! , that is, C = s≥0 t s j≥0
(-qX) j j! q 2js ((qX + Y ) [s + 1] q 2 ) n-j (nj)! Reporting the last expression in (6.7) yields identity (6.2).

Proof of Theorem 6.2. When q = 1 in (6.2), we obtain (6.8) B n (X, Y, Z; t)

(1 -t) n+1 = t -Z t -1 s≥0 t s (X + Y )(s + 1) n + Z -1 t -1 s≥0 t s (X + Y )(s + 1) -X n .
Hence, which is identity (6.3) by replacing u by u(1t).

n≥0 u n (1 -t) n B n (X, Y, Z; t) = (Z -t)
Next, we derive specializations of Theorems 6.1 and 6.2 when the pair (Ξ, des) is replaced by "des B " and "fdes" (see (2.3) and (2.4)). We get:

w∈B n X neg w Y pos w t des B w q fmaj w = B n (X, Y, t; t, q); (6.9) w∈B n X neg w Y pos w t fdes w q fmaj w = B n (X, Y, t; t 2 , q). (6.10) Also, note that B n (0, 1, 1; t, q) is the Carlitz q-Eulerian polynomial A n (t, q). First,

B n (X, Y, t; t, q) (t; q 2 ) n+1 = s≥0 t s (qX + Y )[s + 1] q 2 -Xq 2s+1 n ; (6.11) B n (1, 1, t; t, q) (t; q 2 ) n+1 = w∈B n t des B w q fmaj w = s≥0 t s [2s + 1] q n . (6.12) Second, B n (X, Y, t; t 2 , q) (t 2 ; q 2 ) n+1 = t 2t t -1 s≥0 t 2s (qX + Y ) [s + 1] q 2 n + t -1 t 2 -1 s≥0 t 2s (qX + Y )[s + 1] q 2 -Xq 2s+1 n , so that (1 + t)B n (X, Y, t; t 2 , q) (t 2 ; q 2 ) n+1 = s≥0 t 2s+1 (qX + Y ) [s + 1] q 2 n (6.13) + s≥0 t 2s (qX + Y )[s] q 2 + Y q 2s n .

In particular, (1 + t)B n (1, 1, t; t, q) (t 2 ; q 2 ) n+1 = w∈B n t fdes w q fmaj w = s≥0 t s [s + 1] q n . (6.14)

  w , -Neg w) = (δ, J), then (see (2.11)) (4.2) des w = des δ; smaj w = 2 cmaj δ + #J + n -1.

  (sgn, smaj) |w| ǫ = (-sgn, smaj)b |w| ǫ . The proof of the theorem is based on the next three lemmas. The first two ones being easy to verify are given without proofs. Lemma 5.2. For each w = |w| ǫ ∈ B 2n+1 we have sgn r w = sgn w • (-1) χ(L w<0)+χ(F w<0) ; (5.9) smaj r w = smaj w + (2n + 1) χ(F w < 0)χ(L w < 0) . (5.10) Lemma 5.3. For each w = |w| ǫ ∈ B 2n+1 we have: sgn s w =sgn w; (5.11) smaj s w =smaj w.

  Finally, smaj c w + smaj w = 2(maj c w + maj w)+ neg c w + neg w -(2n + 1)(des c w + des w + 2χ(ǫ 1 < 0)) = 2n(2n + 1) + 2 DRISE ǫ + 2 neg w -(2n + 1)(2n + drise ǫ + 2χ(ǫ 1 < 0)) = 2 DRISE ǫ + 2 neg w -(2n + 1)(drise ǫ + 2χ(ǫ 1 < 0)) w + 2 neg w -(2n + 1)2 = 0, in case (a); 2 pos w + 2 neg w -(2n + 1) = 2n + 1, in case (b); -2 neg w + 2 neg w -(2n + 1)0 = 0, in case (c); -2 neg w + 2 neg w -(2n + 1) = -(2n + 1), in case (d).Altogether, smaj c w =smaj w -(2n + 1) χ(ǫ 1 = -1)χ(ǫ 2n+1 = -1) . This proves (5.14) and also Lemma 5.4.Proof of Theorem 5.1. Let w ∈ B 2n+1 . By the previous three lemmas sgn r c w = sgn c w • (-1) χ(L c w<0)+χ(F c w<0)

  2s ) j ((qX + Y ) [s + 1] q 2 ) n-j qX + Y )[s + 1] q 2 -Xq 2s+1 n

  s≥0

  (X + Y )(s + 1) -X) = (Z -t)(exp(u(X +Y ))+(1-Z) exp(uY ) s≥0 t s exp(u(X +Y )s) = (Zt) exp(u(X + Y )(1 -Z)) exp(uY ) 1t exp(u(X + Y )) ,

The specializations of (6.12) and (6.14) for q = 1 are banal and not reproduced. However, it is worth writing the exponential generating functions for the polynomials B n (1, 1, t; t) and B n (1, 1, t; t 2 ) directly obtained from (6.3):

. (6.16)

The statistics "fdes" and "fmaj" were introduced by Adin and Roichman [START_REF] Adin | The flag major index and group actions on polynomial rings[END_REF]. Identity (6.14) with their equivalent adaptations were derived by Brenti et al. [START_REF] Adin | Descent Numbers and Major Indices for the Hyperoctahedral Group[END_REF], Haglund et al. [START_REF] Haglund | Statistics on wreath products, perfect matchings, and signed words[END_REF] and reproved by the authors ( [START_REF] Foata | Signed words and permutations, III: the MacMahon Verfahren[END_REF], [START_REF] Foata | Signed words and permutations, V; a sextuple distribution[END_REF]) as specializations of identities involving severalvariable statistics. Note that (6.16) implies that w∈B n (-1) fdes w is null for every n ≥ 1. Accordingly, the statistic "fdes" would have been a wrong choice for obtaining a q-extension!