Annelies Kepper
email: a.s.kepper@uu.nl

Karin Monshouwer

Saskia Van Dorsselaer

residential youth care institutions

HAL is a

Introduction

Adolescents who are in special education (SE) and/or live in residential youth care institutions (RYC) are characterized by learning disabilities, behavioural and/or emotional problems and appear to be at higher risk for a number of risk behaviours, including problematic substance use [1;4;9;21]. However, empirical studies on the prevalence of substance use in these groups are scarce. Most large population based studies among adolescents are conducted at mainstream schools, thus targeting only the majority group of adolescents [6;11;15]. To our knowledge, only a few studies have been performed on substance use among adolescents in RYC and SE [1;4;9;17;21]. In general, the results of these studies indicated that levels of substance use in these special groups were indeed (much) higher than in the general population of adolescents. Findings showed that 70% of US adolescents in SE were currently using cigarettes compared to 36% in mainstream education [START_REF] Grunbaum | Prevalence of health-related behaviors among alternative high school students as compared with students attending regular high schools[END_REF], among New Zealand boys, 86% in SE had ever used cannabis compared to 34% in mainstream education [START_REF] Denny | The health of alternative education students compared to secondary school students[END_REF] and among Eastern European adolescents aged 14-17 who live in RYC institutions, 13% was a current cannabis user, compared to 2% use among adolescents in the same age category who live with their parents [START_REF] Bacovic | Differences in substance use patterns among youths living in foster care institutions and in birth families[END_REF]. More research on these subgroups is needed, as most of these studies are outdated [9;17], while more recent studies are quite limited, with very small samples (n=58) [START_REF] Bacovic | Differences in substance use patterns among youths living in foster care institutions and in birth families[END_REF] that are not representative for the subpopulation of youth in SE [START_REF] Denny | The health of alternative education students compared to secondary school students[END_REF].

Vaughn [START_REF] Vaughn | Substance use and abuse among older youth in foster care[END_REF] investigated substance use among 406 17-year old adolescents in RYC settings, but did not compare findings with the general population, thus being unable to determine the relative risk of youngsters in youth care settings. In conclusion, the current literature suggests that adolescents in SE and in RYC are at an increased risk for problematic substance use. However, up-to-date and representative information on the actual levels of substance use and on the specific impact of risk factors is lacking.

The aim of this study is to examine substance use among adolescents in SE and RYC in the Netherlands. SE schools provide education for adolescents who are not able to attend mainstream schools owing to a number of problems like emotional and behavioural disorders, learning disabilities and developmental disorders. Within SE, there are subtypes of schools, of which two are included in the present study: schools for adolescents with behavioural problems (SEB) and schools for adolescents with learning disabilities (SEL).

Adolescents in RYC are living in residential youth care institutions owing to a variety of family problems and/or mental health and behavioural problems and generally attend SE [2;10].

When analysing substance use among adolescents in SE and RYC, it is important to take account of a number of background factors which are related to substance use, i.e. gender, age, ethnic background [START_REF] Donovan | Adolescent alcohol initiation: A review of psychosocial risk factors[END_REF] and family situation [3;14]. Two effects of these background factors on substance use levels can be discerned. First, subgroups with elevated risks for substance use may be overrepresented in SE and RYC as compared to the general population. For example, in RYC it is likely that the percentage of children from broken families is substantially higher than among the general population. As children with divorced or separated parents are, in general, at increased risk for substance use [3;14], this may (partly) explain differences in substance use levels between adolescents in RYC and in the general population. Second, the relationship of these background factors with substance use levels may be different for adolescents in SE and RYC as compared to their peer groups in the general population. For example, while boys are generally found to be at higher risk for substance use [START_REF] Fisher | Predictors of Initiation of Alcohol Use Among US Adolescents: Findings From a Prospective Cohort Study[END_REF], there are indications that the relationship between gender and substance use may be reversed in the SE and RYC setting. Loeber and Keenan found for example that girls with conduct disorders appear to be at higher risk for substance use problems than boys who suffer from conduct disorder, whereas the risk of (heavy) substance use is generally higher for boys [START_REF] Loeber | Interaction between conduct disorder and its comorbid conditions: effects of age and gender[END_REF]. This phenomenon is known as the gender paradox; in some types of disorders or problem behaviours that have an unequal sex ratio, those with the lower prevalence tend to be more seriously affected by high risk factors [START_REF] Taylor | The nature of gender differences explored trough ontogenetic analysis of sex ratios in disease[END_REF]. It can be hypothesized that this paradox not only applies to gender, but might be a more general phenomenon in the interaction of risk factors (risk paradox). Those subgroups that are normally at lower risk for substance use in the general population, are at higher risk for substance use when they are subjected to high risk factors. The present study will investigate this 'risk paradox' for the background variables age, gender, ethnic background and family situation. To our knowledge, no former studies have investigated this issue among adolescents in SE and RYC.

For the present study, a dataset is available including information from adolescents in SE, RYC and mainstream education. The data for these groups were collected according to the same protocol and by using the same measurement instruments. Thus, the present study is well suited to comparing the levels of substance use between adolescents in SE, RYC and mainstream education and to investigating the role of risk and protective factors. In conclusion, the aim of this study is [START_REF] Bacovic | Differences in substance use patterns among youths living in foster care institutions and in birth families[END_REF] to generate up-to-date information on the prevalence of substance use among two types of high risk groups: adolescents in RYC and adolescents in SE and to compare these findings with the results of a sample of mainstream school students (2) hereby taking into account the demographic background characteristics in these groups and (3) to investigate whether the association between background factors and substance use differs between adolescents in SE/RYC and those in mainstream education.

Methods

Sample

Residential youth care institutions (RYC)

In 2008 there were 48 Dutch RYC institutions, providing 24-hour day care for adolescents aged 12 to18. The Regional Child Welfare Office decides, under the authority of the Dutch juvenile court, whether a minor should be placed in residential care. Approximately 11% of the adolescents under the supervision of the Child Welfare Office are placed in RYC institutions every year [START_REF] Harder | Residentiële jeugdzorg in beeld [Residential child and youth care in the picture[END_REF]. In this study, all 48 RYC institutions in the Netherlands were invited to participate in the study. More than half of the institutions (N=26) agreed to cooperate (54% response). Within each participating institution, all adolescents in the age group of 12-18 living in a community group were invited to participate and were informed that they would receive a gift voucher of 10 euro if they filled in a questionnaire. This procedure resulted in 673 respondents. The response rate on respondent level was 83%. Main reasons of non-response were 'refusing to participate' (30%) and 'not present due to other obligations' (28%). Only information regarding the 12-16 age group has been used in order to compare substance use of adolescents in RYC with adolescents in mainstream schools (in mainstream education 17 and 18-year old students are not representative of the Dutch adolescent population, since school is compulsory in the Netherlands only until the age of 16 years). This resulted in a final sample of 531 respondents (92 adolescents aged 12 to13, 262 adolescents aged 14 to15 and 177 adolescents aged 16 years).

Special education for those with behavioural problems (SEB)

Dutch SEB schools are for students who are not able to attend mainstream education due to behavioural problems or emotional disturbance. In 2008, 1.4% of all Dutch students attended SEB schools [START_REF]Jaarboek onderwijs in cijfers 2008[END_REF]. In this study, a multiple-stage random sampling procedure has been used in order to select the schools. First, 50 SEB schools were drawn from a list of all SEB schools in the Netherlands and invited to participate. Twenty two schools agreed to contribute (44% response). Second, within each participating school one class from every grade [START_REF] Bacovic | Differences in substance use patterns among youths living in foster care institutions and in birth families[END_REF][START_REF] Baker | Mental Health and Behavioral Problems of Youth in the Child Welfare System: Residential Treatment Centers Compared to Therapeutic Foster Care in the Odyssey Project Population[END_REF][START_REF] Th | Family structure and adolescent cigarette smoking in eleven European countries[END_REF][START_REF] Denny | The health of alternative education students compared to secondary school students[END_REF] was selected randomly from a list of all classes provided by the school. When the selected classes within a school comprised a sample smaller than 35 students, a fifth class was drawn randomly (9 schools). Third, within the classes all students were drawn as a single cluster, resulting in 653 students. The student response rate was 75%. Main reasons for non-response were illness (28%), refusal to participate (11%) and absence due to work experience (34%). To compare the adolescents in SEB with mainstream students, only those aged 12-16 were included in the analysis (603 adolescents: 194 adolescents in the 12-13 year age group, 321 adolescents aged 14-15 and 88 adolescents aged 16 years).

Special education for those with learning disabilities (SEL)

SEL schools provide education for students who have special needs or other learning problems. These students get extra support and are prepared for directly entering the job market after finishing school. In 2008, 14% of all Dutch students were attending SEL schools [START_REF]Jaarboek onderwijs in cijfers 2008[END_REF]. In this study, SEL students were selected through the same procedure as described for the SEB schools (a multiple-stage random sampling procedure). First, 85 schools that provide special education for students with learning problems were invited to participate in the study and 41 schools agreed to collaborate (48% response). Second, within each participating school one class from every grade (1-4) was selected randomly from a list of all classes provided by the school. This procedure resulted in 1,965 students (89% response). Main reasons for absence were illness (50%) and teaching practice (23%).

Again, only the information of adolescents aged 12 -16 was used which resulted in 1,905 students (652 adolescents aged 12-13, 966 adolescents aged 14-15 and 287 16 year olds).

For both residential youth care institutions and SE schools, only minor differences were found between the participating and non-participating institutions / schools with respect to region, urbanization, number of students / adolescents, percentage ethnic minorities and percentage adolescents with low SES.

Mainstream education

To compare the adolescents from RYC and SE with adolescents in mainstream education, data were used from the most recent Dutch National School Survey on Substance Use [START_REF] Monshouwer | Jeugd en riskant gedrag 2007 [Youth and risk-taking behaviour 2007[END_REF]. The data were collected in October/November 2007, i.e. one year prior to the data collection in RYC and SE. For the present study, data of adolescents in the age group of 12-16 have been used (7,041 adolescents of whom 2673 adolescents in the age group of 12-13, 3,161 in the age group of 14-15 and 1,112 aged 16).

Collection of the data

All data on adolescents in SE and RYC were collected in the fall of 2008. Trained interviewers administered anonymous questionnaires in classes (SE) or living rooms (RYC) and underlined the anonymity of the respondents when introducing and clarifying the questionnaire. The data on students in mainstream education were collected in a similar way in the fall of 2007 [START_REF] Monshouwer | Jeugd en riskant gedrag 2007 [Youth and risk-taking behaviour 2007[END_REF].

Measures

Substance use

Daily use of cigarettes was measured by asking 'Have you ever smoked cigarettes?' Respondents could answer with 'I smoke every day', 'I smoke once in a while, but not every day', 'I have smoked once or twice', 'I used to smoke, but I have quit smoking completely' or 'I have never smoked'. Those answering with 'I smoke every day' were classified as daily smokers.

The number of glasses of alcohol respondents drank every week (Monday to Sunday) was assessed by using the Quantity-Frequency measure [START_REF] Engels | Alcohol use and intimate relationships in adolescence: when love comes to town[END_REF]. Frequency was measured by asking the number of days the adolescent usually drank on weekdays (Monday to Thursday) and weekend days (Friday to Sunday). Quantity was measured by asking how many glasses of alcohol the respondent usually drank on a weekday and weekend day. The total average weekly consumption of alcohol was computed by calculating the product of the number of days and the number of glasses and then summing the two products for weekdays and weekend days, resulting in the average total number of glasses per week [START_REF] Koning | Preventing heavy alcohol use in adolescents (PAS): cluster randomized trial of a parent and student intervention offered separately and simultaneously[END_REF]. This variable was coded as 1 (heavy alcohol use) and 0 (no heavy alcohol use). In order to determine a cut off score for heavy alcohol use, we assessed the group representing the 15% highest scores among our sample of 14-15 year olds students in mainstream education, for both boys and girls separately. This showed to correspond for boys with drinking 10 or more glasses a week and for girls with drinking 8 or more glasses of alcohol a week. Hence, in this study boys who drank more than 10 alcoholic beverages per week and girls who drank more than 8 alcoholic beverages per week were identified as heavy weekly alcohol drinkers.

Life-time use of cannabis was measured by asking 'How many times did you use cannabis in your life?' Respondents could answer by marking the number of times they had used cannabis (0,1,2,3,4,5,6,7,8,9,10,11-19,20-29,30-39, 40 times or more). Those answering 1 time or more were classified as life-time users of cannabis.

Finally, life-time use of hard drugs (XTC, cocaine, amphetamine, hallucinogens, GHB, LSD, crack or heroin) was measured by asking 'Have you ever used one of the following drugs during your life'. The respondent could answer with 'yes' or 'no' for every hard drug. Those answering with 'yes' for at least one hard drug were identified as life-time users of hard drugs.

Background factors

The following demographic factors were included: gender, age, ethnicity (Dutch/ethnic minority) and family situation (table 1). Age was recoded into three categories (12-13, 14-15 and 16 years; in tables 3 and 4 age is analysed as a continuous variable). Ethnicity was assessed by asking respondent to state their country of birth as well as that of both parents. Those respondents reporting that either he/she, his/her father or mother was born in a non-western foreign country were classified as belonging to an ethnic minority. Family situation was assessed by asking the respondents if their parents were divorced/separated and if respondents were in contact with their biological mother or father.

Statistical analysis

The association between substance use and RYC, SEB and SEL respectively and the role of background factors was investigated by using logistic regression analysis. In the first model the association between substance use (daily use of tobacco, heavy weekly alcohol drinking, lifetime prevalence of cannabis use and lifetime prevalence of hard drug use) and group membership (RYC, SEB,SEL, with mainstream education being the reference group) was tested. In a next step, this association was adjusted for the background factors gender, ethnic background, age and marital status of parents. Finally, to investigate a possible interaction between these background characteristics and group membership, the following two-way interaction terms were included: gender (0=girls, 1=boys) by respectively RYC, SEB or SEL, age (0=adolescents in the age of 12-13, 1= adolescents aged 14-16) by RYC, SEB or SEL, ethnic background (0=Dutch, 1=ethnic minority) by RYC, SEB or SEL and marital status of parents (0=non-divorced parents, 1=divorced parents) by RYC, SEB and SEL. In each of these analyses RYC, SEB, or SEL was coded as 1 and mainstream education was coded as 0.

All analyses were performed taking into account the fact that the data are clustered, i.e. adolescents from the same class or RYC group were drawn as a single group which results in dependence between observations (adolescents from the same class/group share several characteristics, like having the same teacher or living in the same region). This will influence standard errors and p-values. Therefore, robust standard errors were calculated by using the Huber/White/sandwich estimator as implemented in STATA version 9.1 [START_REF]Stata Release 9[END_REF].

Results

Table 1 illustrates the characteristics of the adolescents in each of the three subgroups. In SEB, there were more boys than girls. In the other groups the gender distribution was almost equal, although in SEL there were slightly more boys. In RYC there were more adolescents aged sixteen than in the age group of twelve to thirteen. In the other groups it was exactly the opposite: more 12-13 year olds and fewer adolescents aged sixteen. The number of adolescents aged 14-15 was about the same in all the groups (approximately 50%). Moreover, compared to mainstream education, adolescents in RYC, SEB and SEL more often reported having an ethnic minority background, divorced or separated parents and not being in contact with their biological father or biological mother.

[TABLE 1 AROUND HERE]

The prevalence of the use of tobacco, alcohol, cannabis and hard drugs by age category is reported in table 2.

Adolescents in RYC, SEB and SEL more often reported daily tobacco use, lifetime cannabis use and lifetime hard drug use compared to adolescents in mainstream education. In RYC the rates were highest, although differences with SEB were sometimes small (e.g. the lifetime prevalence of hard drug use among the 12-13 year olds was around 13% in both RYC and SEB). The differences in substance use between RYC, SEB andto a lesser degree SEL -and mainstream education were particularly marked in the youngest group (12-13 years). For example, of the 12-13 year olds in RYC, 22% reported daily tobacco use and 33% lifetime cannabis use. The latter is eight times higher compared to adolescents of the same age in mainstream education. But also among the older adolescents the differences were large, with four times more hard drug users in RYC and SEB (both 31%) than in mainstream education (7%).

[TABLE 2 AROUND HERE]

Table 3 reports the results of the logistic regression analysis on the association between substance use and group membership (RYC, SEL, SEB and mainstream education (reference category). The unadjusted model illustrates that being in RYC involved a higher risk of daily tobacco use and -to a lesser degree -heavy weakly alcohol use, lifetime prevalence of cannabis and hard drug use. Being a student in SEB also increased the risk of substance use, but these associations were less strong. For example, being in RYC involved a higher risk of daily use of tobacco (OR=16.91, CI=13.14 -21.76) compared to mainstream education, than being in SEB (OR=6.74, CI=5.01 -9.06) . Finally, SEL students had a greater risk of being a daily tobacco user and a hard drug user compared to adolescents in mainstream education. There were no significant differences between SEL students and mainstream education students in being a heavy weekly alcohol user or a cannabis user. In the second model, adjusting the analyses for gender, ethnic background, age and marital status, all associations between substance use and group membership became somewhat weaker (except for daily use of tobacco in SEL), though remaining significant. For example, being in RYC still involved a high risk for daily smoking, also when background factors were taken into account (OR=10.40, CI=7.90 -13.71). The risk of daily tobacco use due to being in SEB declined only slightly when adjusting for the background factors (OR=6.70, CI=4.99 -9.01).

[TABLE 3 AROUND HERE]

Four interaction terms of being in RYC, SEB or SEL and background factors were found to be significant (p<.001): a marital status of parents by RYC interaction for daily use of tobacco (OR=.43, CI=.27 -.67), an ethnic background by RYC interaction for heavy weekly drinking (OR= 2.72, CI=1.47 -5.03) and a gender by SEB interaction for both heavy weekly drinking (OR=.29, CI=.16 -.53) and lifetime use of cannabis (OR=.36, CI=.24 -.54).

Post estimation analyses of these interaction effects are reported in table 4. Among adolescents in RYC whose parents were divorced or separated, the risk of being a daily smoker was higher compared to their counterparts in mainstream education (OR=7.81, CI=5.76 -10.59). However, among adolescents whose parents were not divorced or separated, the risk of being a daily smoker was even more elevated for those in RYC (OR=18.18, CI=11.87 -27.85). The risk of heavy weekly drinking among Dutch adolescents in RYC was somewhat higher as compared to those in mainstream education (OR=1.81, CI=1.33 -2.45), however, among ethnic minorities this risk was more elevated (OR=4.92, CI=2.72 -8.90). Boys in SEB were not significantly at higher risk for heavy weekly drinking as compared to boys in mainstream education, however, for girls in SEB the risk of heavy weekly drinking was much higher as compared to girls in mainstream education (OR=5.05, CI=2.95 -8.67). Finally, boys in SEB were at higher risk for cannabis use (OR=2.36, CI=1.83 -3.06) compared to those in mainstream education, but the risk of cannabis use was even more elevated among girls in SEB (OR=6.57, CI=4.28 -10.10).

[TABLE 4 AROUND HERE]

Discussion

Main findings

Substance use rates are particularly high among adolescents living in residential youth care institutions (RYC).

Especially daily tobacco use was highly prevalent, with more than twenty times as many daily smokers among adolescents in the age of 12 -13 in RYC as compared to those in mainstream education. Being in RYC also increased the risk of lifetime prevalence of cannabis use, lifetime prevalence of hard drug use andto a smaller degree -heavy weekly drinking in comparison with those in mainstream education. In addition, elevated substance use rates among adolescents who are in special education due to behavioural problems (SEB) were found for all investigated types of substance use. Daily tobacco use and life time prevalence of hard drug use, among adolescents in special education for learning disabilities (SEL) were also higher compared to those in mainstream education, although differences were less pronounced as for those in SEB or RYC. Differences in substance use rates between the three risk groups and mainstream education were particularly large in the youngest age category (12-13 year olds).

Background factors, including age, ethnic background and family situation, partly explained the differences in substance use between mainstream education on the one hand and SE and RYC on the other hand, but all differences between the groups remained substantial and significant. Four significant interactions were found in the relationship between SE/RYC and substance use that were in line with the risk paradox: the elevated risk of substance use in SE/RYC was particularly noticeable among adolescents who are normally at lower risk for substance use (i.e. in SEB, girls for heavy alcohol drinking and cannabis use, and in RYC, ethnic minority adolescents for heavy weekly alcohol use and adolescents with a stable family situation for daily tobacco use).

Prevalence of substance use in the high risk groups

Our results are consistent with findings of previous studies that showed high prevalences of substance use among adolescents that attend SE schools [4;9]. As in our study, SE students in both New Zealand and the US reported elevated rates of substance use compared to students in mainstream education. In addition, previous studies that found high substance use rates in RYC settings [1;21] are confirmed by our results. For example, Vaughn and colleagues investigated substance use among 17 year olds who live in a RYC setting and found that almost half of these adolescents (46%) had ever used marijuana [START_REF] Vaughn | Substance use and abuse among older youth in foster care[END_REF]. Although the present study investigated a younger group, the number of adolescents that used cannabis was even higher (almost 70% of the 16 year olds in RYC had used cannabis at least once). In addition, the percentage of heavy alcohol users (consuming more than 10 alcoholic drinks per week for boys and more than 8 alcoholic drinks per week for girls), was much higher in SEB and RYC compared to mainstream education. About 40% of the adolescents in the age of sixteen in SEB and RYC and SEL reported to drink more than 10 or 8 glasses of alcohol per week. This is comparable to the average amount of alcoholic beverages Dutch adults drink [START_REF] Verdurmen | Bovenmatig drinken in Nederland [Excessive drinking in the Netherlands[END_REF].

Though the present study found an increased risk for substance use among adolescents with learning disabilities, this risk was not as elevated compared to adolescents in RYC or SEB. Adolescents in SEL are not able to attend mainstream education because they have difficulties with their learning abilities, whereas adolescents in SEB attend SE due to a whole range of problems: emotional disturbance, behavioural problems and, or family problems. Apparently, learning problems as such are less strongly associated with substance use.

The role of background factors

In line with previous studies on substance use, this study found that ethnic minorities [START_REF] Vaughn | Substance use and abuse among older youth in foster care[END_REF], younger adolescents [START_REF] Bacovic | Differences in substance use patterns among youths living in foster care institutions and in birth families[END_REF] and adolescents in intact families [3;14] were less likely to use substances. As shown in table 1, the percentage of broken families, was much higher in RYC and, to a lesser extent in SE, as compared to mainstream education. Therefore the high prevalence of substance use in RYC and SE can partially be attributed to the high number of adolescents with an unstable family situation.

Risk paradox.

Several interaction effects between risk factors were found, that are in line with the assumptions of the risk paradox : the elevated risk of substance use among adolescents in RYC/SE was in some cases particularly marked for those who would normally be at lower risk for substance use (i.e. girls, ethnic minorities and adolescents with a stable family situation). We can think of two explanations for these findings. First, those normally at lower risk (e.g. children from intact families) are generally living in more protective environments.

The threshold for entering the RYC-setting might accordingly be higherdue to more resources in these families for coping with their child's problems without professional help -and therefore, higher levels of problem behaviour in the children who are finally brought into these RYC-settings may prevail. Thus, elevated levels of problem behaviour in children in RYC originating from these settings are to be expected. Moreover, Loeber and Keenan noted in their review report that despite the lower prevalence of disruptive disorders in girls compared to boys, girls who do suffer from conduct disorder are at higher risk for developing comorbid conditions like substance use [START_REF] Loeber | Interaction between conduct disorder and its comorbid conditions: effects of age and gender[END_REF]. They hypothesized that since girls have a higher threshold for problematic behaviour or physical disorders than boys, they consequently require more harmful risk factors in order to develop problem behaviour than boys. Following this hypotheses it can be theorized that girls have a higher threshold for attending SEB schools and therefore have more complex and serious behavioural problems including the use of cannabis and excessive use of alcohol if they do attend SE schools.

Second, we can hypothesize that being in SE-settings has a levelling effect on the problem behaviour of all children entering these settingsdue to the effects of peer influence within these groups, and due to being in these settings as such. These effects may be particularly elevated in the groups normally at lower risk as well, because entering these settings brings them into contact with higher risk groups. Ethnic minority children may serve as an example in this respect. It is a well known fact that ethnic minority children are usually at (far) lower risk for drinking alcoholdue to the protective effect of restrictive parenting and cultural values forbidding alcohol in these groups [START_REF] Wallace | The Influence of Race and Religion on Abstinence from Alcohol, Cigarettes and Marijuana among Adolescents[END_REF]. Entry to a group of high risk peerswho tend to drink heavilymay have a particularly strong effect on children originating from an ethnic minority background. Future studies analysing peer processes in these settings are needed to test these assumptions.

Strength and limitations

A major strength of the current study is that the methodology was the same for RYC, SE and mainstream education. This makes the comparison of high risk groups with adolescents in mainstream education more valid.

Possible limitations have to be taken into account. First, analyses were based on self-report data which could result in over-or under-reporting of substance use. But as the administration of the questionnaires in school classes and living rooms occurred under the supervision of a trained interviewer and anonymity was assured, self-report is assumed to be reliable [START_REF] Smit | Monitoring Substance Use in Adolescents: school survey or household survey?[END_REF]. Second, there is a possible under-representation of substance users due to the number of adolescentsin particular in RYC and SEB -that refused to cooperate. It can be expected that the number of substance users is especially high among these refusers. Third, since the study is based on a cross sectional design, inferences on causal relationships cannot be made. Fourth, other non-assessed factors can possible also account for much of the elevated risk (family problems, deviant peers, delinquency, school problems) and therefore have to be taken in consideration in future studies.

In conclusion, this study found high rates of substance users among adolescents in residential youth care institutions and special education, in particular SEB, compared to adolescents in mainstream education. These results confirm that adolescents in these settings are not only at higher risk for psychosocial problems, but also at higher risk for developing substance use disorders. These differences are only partially explained by other risk factors that are elevated in this group (e.g. coming from broken families). Acknowledging these risks is of major importance for prevention and treatment programmes directed at these groups, as high levels of substance use may have a considerable impact on the success of these programmes. Results of this study have important implications for health education and intervention programs for adolescents in RYC institutions and in adolescents in SE. It is important to develop substance use prevention programmes that target high risk adolescents in SE and RYC.

Table 1

 1 Background factors of adolescents in residential youth care institutions (RYC), special education for behavioural problems (SEB) and special education for learning disabilities (SEL) compared to adolescents in regular education, %

	Background factors	RYC % (95% CI)	SEB % (95% CI)	SEL % (95% CI)	Mainstream % (95% CI)
		N=531	N=603	N=1,905	N=6,946
	Gender
	Girl	50.3 (43.3 -56.8) 18.9 (15.9 -22.4) 43.7 (40.3 -47.1) 51.1 (49.2 -52.9)
	Boy	49.7 (47.1 -50.3) 81.1 (77.7 -84.1) 56.3 (43.3 -56.8) 48.9 (47.1 -51.8)
	Age
	12-13	17.3 (13.7 -21.6) 32.2 (25.0 -40.3) 34.2 (28.3 -40.7) 38.5 (33.8 -43.3)
	14-15	49.3 (44.2 -54.5) 53.2 (46.7 -59.6) 50.7 (45.3 -56.1) 45.5 (41.4 -49.7)
	16	33.3 (28.3 -38.8) 15.6 (10.7 -19.6) 15.1 (11.5 -19.5) 16.0 (13.5 -18.9)
	Ethnic background
	Dutch	73.9 (69.6 -77.7) 74.2 (69.8 -78.2) 77.7 (74.0 -81.1) 87.5 (85.9 -88.9)
	Ethnic minority a	26.2 (22.3 -30.4) 25.8 (21.8 -30.2) 22.3 (19.0 -26.0) 12.5 (11.1 -14.1)
	Family situation
	Divorced / separated parents	79.7 (75.0 -83.7) 51.7 (47.5 -55.9) 28.1 (25.8 -30.6) 18.3 (17.3 -19.4)
	No contact with biological	10.7 (8.6 -13.4)	4.8 (3.3 -6.9)	3.3 (2.6 -4.2)	1.2 (1.0 -1.5)
	mother
	No contact with biological	35.2 (31.1 -39.5) 15.9 (13.2 -19.1) 9.8 (8.5 -11.4)	4.6 (4.1 -5.2)
	father

a In this study immigrants from western countries were categorized as Dutch Note: If the confidence intervals of RYC / SEB / SEL and mainstream education do not overlap, the differences are statistically significant and reported in bold

Table 2

 2 Substance use among adolescents in residential youth care institutions (RYC), special education for behavioural problems (SEB) and special education for learning disabilities (SEL), compared to adolescents in mainstream education, by age, %

				Aged 12-13			Aged 14-15			Aged 16
		RYC	SEB	SEL	Mainstream	RYC	SEB	SEL	Mainstream	RYC	SEB	SEL	Mainstream
		(N=92)	(N=94)	(N=652)	(N=2673)	(N=262)	(N=321)	(N=966)	(N=3161)	(N=177)	(N=88)	(N=287)	(N=1112)
	Daily use of tobacco	21.7	16.0	4.2	1.3	58.6	38.4	17.8	9.3	65.5	46.6	23.1	11.7
	Heavy weekly alcohol drinking 2	12.5	8.1	4.2	1.9	33.3	23.4	15.3	13.9	40.4	40.0	28.2	29.7
	Lifetime prevalence of	32.6	20.4	6.0	4.0	66.7	42.0	16.0	17.2	68.6	54.1	22.2	30.0
	cannabis use
	Lifetime prevalence of hard drug use 1	13.5	13.0	9.0	2.3	29.2	19.4	10.6	5.4	30.5	31.0	14.3	6.6

Table 3

 3 Association between daily use of tobacco, life time prevalence of cannabis use and hard drugs use and risk groups, adjusted for background factors, odds ratios OR Daily use of tobacco b Model 2: adjusted for all other variables in the model1 Reference group is mainstream education 2 reference group is female 3 reference group is Dutch 4 reference group is non divorced parents *p<.001

	Heavy weekly alcohol	Lifetime prevalence of	Lifetime prevalence of
	drinking	cannabis use	hard drug use

a Model 1: unadjusted

Table 4

 4 Association between substance use and risk groups among girls and boys, Dutch adolescents and ethnic minorities and adolescents with divorced parents and non-divorced parents, odds ratios (OR)

		OR	95% CI	OR	95% CI
	Daily use of tobacco
		Divorced parents	Non-divorced parents
	Mainstream education (reference)	1	-	1	-
	RYC	7.81*	5.76 -10.59	18.18*	11.87 -27.85
	Heavy weekly alcohol drinking
		Dutch adolescents	Ethnic minorities
	Mainstream education (reference)	1	-	1	-
	RYC	1.81*	1.33 -2.45	4.92 *	2.72 -8.90
			Boys	Girls
	Mainstream education (reference)	1	-	1	-
	SEB	1.48	1.04 -2.08	5.05 *	2.95 -8.67
	Lifetime prevalence of cannabis use
			Boys	Girls
	Mainstream education (reference)	1	-	1	-
	SEB	2.36*	1.83 -3.06	6.57 *	4.28 -10.10
	*p<.001

lifetime prevalence of at least one of the following substances: XTC, cocaine, amphetamine, hallucinogens, GHB, LSD, crack and/or heroin


Ï C Lo

F R Éd

Éric Patras

NATURAL ENDOMORPHISMS OF SHUFFLE ALGEBRAS

Introduction

Shuffles have a long history, starting with the probabilistic study of card shufflings in the first part of the 20th century by Borel, Hadamard, Poincaré and others. Their theory was revived in the 50's, for various reasons. In topology, the combinatorics of (non commutative) shuffle products was the key to the definition of topological products such as the ones existing on cochain algebras and the cohomology groups of topological spaces. Commutative shuffle products were the key to the study of the homology of abelian groups and commutative algebras. In combinatorics and for the theory of iterated integrals, commutative shuffle products played a key role resulting in the global picture of the modern theory of free Lie algebras given in C. Reutenauer's seminal Free Lie algebras [START_REF] Reutenauer | Free Lie algebras[END_REF].

The classical approach to shuffle algebras, as featured for example in Reutenauer's book, focussed on Lie theoretical properties, that is on the enveloping algebra structure of tensor algebras: the shuffle product arises naturally in this framework by dualizing the Hopf algebra structure of the tensor algebra and many properties of shuffles can be derived from that particular approach.

However, one can try to follow a different path, namely start directly from the combinatorics of shuffles, following the ideas originally developed by M.-P. Schützenberger [START_REF] Schützenberger | Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathématiques appliquées, Séminaire Dubreil-Jacotin Pisot (Algèbre et théorie des nombres)[END_REF]. A series of recent works by F. Chapoton, C. Malvenuto, C. Reutenauer, the second author of the present article, and others, provides many new tools to revisit the theory of shuffles. This is the purpose of the present article to put these tools to use.

Concretely, we focus on the adaptation to the study of shuffles of the main combinatorial tool in the theory of free Lie algebras, namely the existence of a universal algebra of endomorphisms for tensor and other cocommutative Hopf algebras: the family of Solomon's descent algebras of type A [START_REF] Reutenauer | Free Lie algebras[END_REF][START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF]. We show that there exists similarly a natural endomorphism algebra for commutative shuffle algebras, which is a natural extension of the Malvenuto-Reutenauer Hopf algebra of permutations, or algebra of free quasi-symmetric functions. We study this new algebra for its own, establish freeness properties, study its generators, bases, and also feature its relations to the internal structure of shuffle algebras.

As mentioned in the introduction, shuffle products can be understood in the commutative and noncommutative frameworks. The two uses still coexist (topological shuffles are noncommutative, whereas the use in combinatorics is to refer to shuffle products as the commutative ones of the theory of free Lie algebras). We survey briefly the historical foundations of the theory since it will appear later that the combinatorics of the objects that were first considered to study shuffle products (geometrical simplices and tensors) is closely related to the new algebraic structures to be introduced in the present paper.

According to [START_REF] Maclane | The Homology Products in K(II, n)[END_REF], the algebraic theory of these products was first established in [START_REF] Eilenberg | On the Groups H(π, n)[END_REF] together with the introduction of the notion of half-shuffles, which was to become the classical way to define recursively shuffle products. In view of later developments, the fundamental observation [START_REF] Eilenberg | On the Groups H(π, n)[END_REF]Fla 5.7] is that the topological (cartesian) product of simplices * decomposes into two half-products ≺, ≻:

x * y = x ≺ y + x ≻ y.
The associativity of the product follows then formally from the distributivity of left (≺) and right (≻) half-shuffles with respect to the * product [START_REF] Eilenberg | On the Groups H(π, n)[END_REF]Thm 5.2].

Whereas the topological product is associative but not commutative (the associativity holds automatically, but the commutativity holds only up to orientation and homotopy), Eilenberg and MacLane were the first to consider also the purely commutative case when dealing with the bar construction (a topological object which combinatorial structure is the one of the tensor algebra), see [START_REF] Eilenberg | On the Groups H(π, n)[END_REF]Sect. 18] and [START_REF] Eilenberg | Cohomology theory of abelian groups and homotopy theory III[END_REF]. These ideas were rediscovered independently by M.P. Schützenberger [START_REF] Schützenberger | Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathématiques appliquées, Séminaire Dubreil-Jacotin Pisot (Algèbre et théorie des nombres)[END_REF][START_REF] Aguiar | Infinitesimal bialgebras, pre-Lie and dendriform algebras, in "Hopf Algebras[END_REF][START_REF] Baumann | A Solomon-type epimorphism for Mantaci-Reutenauers algebra of a wreath product G Sn[END_REF][START_REF] Brouder | Hyperoctahedral Chen calculus for effective Hamiltonians[END_REF][START_REF] Chapoton | Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces[END_REF][START_REF] Chapoton | An operational calculus for the Mould operad[END_REF][START_REF] Cuvier | Homologie des algèbres de Leibnitz[END_REF][START_REF] Cuvier | Homologie de Leibniz et homologie de Hochschild[END_REF][START_REF] Duchamp | Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras[END_REF][START_REF] Ebrahimi-Fard | New identities in dendriform algebras[END_REF][START_REF] Ebrahimi-Fard | Dendriform Equations[END_REF][START_REF] Eilenberg | Cohomology theory of abelian groups and homotopy theory III[END_REF][START_REF] Eilenberg | On the Groups H(π, n)[END_REF][START_REF] Fisher | Cozinbiel Hopf algebras in combinatorics[END_REF][START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF][START_REF] Ginzburg | Koszul duality for operads[END_REF][START_REF] Loday | Dialgebras[END_REF][START_REF] Maclane | The Homology Products in K(II, n)[END_REF][START_REF] Malvenuto | Reutenauer Duality between Solomon's algebra and quasisymmetric functions[END_REF], who also clarified the set of relations necessary to prove the associativity relation (as was realized later, his proof of the associativity relation does not require the commutativity asumption and, up to a rewriting, coincides in the end essentially with the one given by Eilenberg-MacLane in a topological framework). With our previous notation, the half-shuffles associativity relations read:

x ≺ (y * z) = (x ≺ y) ≺ z; (1) (x * y) ≻ z = x ≻ (y ≻ z); (2) (x ≻ y) ≺ z = x ≻ (y ≺ z). (3)
In the commutative case, the commutativity property translates into x ≺ y = y ≻ x and these relations simplify to

(4) (a ≺ b) ≺ c = a ≺ (b ≺ c + c ≺ b),
see [12, (18.7)], [32, (S0)], [START_REF] Loday | Dialgebras[END_REF].

For simplicity, we stick from now on to the current terminology and call shuffle algebra a commutative shuffle algebra, that is an algebra with a non associative "half-product" ≺ satisfying the relation ( 4) and dendriform algebra a noncommutative shuffle algebra, that is an associative algebra with two half-products satisfying the associativity relations (1-3) (but not the commutativity relation x ≺ y = y ≻ x; the half-shuffles relations have also been attributed to Rota, see [START_REF] Loday | Dialgebras[END_REF]). See also [START_REF] Chapoton | Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces[END_REF][START_REF] Aguiar | Infinitesimal bialgebras, pre-Lie and dendriform algebras, in "Hopf Algebras[END_REF][START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF][START_REF] Ebrahimi-Fard | Dendriform Equations[END_REF][START_REF] Novelli | Construction of dendriform trialgebras[END_REF] for further general informations on the subject and applications of dendriform structures to various problems in algebra and combinatorics related to the ones we consider in the present article.

Shuffle algebras are sometimes refered to as Zinbiel algebras as a follow up of Cuvier's Jan. 1991 Thesis where Leibniz algebras were first introduced and studied (Zinbiel is the word Leibniz inverted, a successful joke suggested by the topologist J.M. Lemaire). Cuvier proved indeed that the cochain complex computing the homology of Leibniz algebras is the tensor algebra [START_REF] Cuvier | Homologie des algèbres de Leibnitz[END_REF][START_REF] Cuvier | Homologie de Leibniz et homologie de Hochschild[END_REF] -from which one can deduce by standard procedures that the notions of Leibniz algebras and shuffle algebras are Koszul dual [START_REF] Ginzburg | Koszul duality for operads[END_REF]. Since the original name "algèbres de shuffle" is better known and accepted we prefer to stick to the usual terminology.

The classical shuffle bialgebra over an alphabet fits into this picture. Let X be a graded, connected alphabet, that is to say X = n≥1 X n . For all x ∈ X n , we put |x| = n: this is the weight of x. Let T (X) be the tensor algebra generated by X over Q. For all n ∈ N, let T n (X) be the subspace of T (X) generated by the words y 1 ...y n , y i ∈ X of length n, and T n (X) be the subspace generated by the words of weight n, the weight of a word being the sum of the weights of its letters: |y 1 ...y n | := |y 1 | + ... + |y n |. The product in the tensor algebra (the concatenation product) is written •:

y 1 ...y n • z 1 ...z p := y 1 ...y n z 1 ...z p . Definition 1. The shuffle bialgebra Sh(X) = n∈N Sh n (X) is the graded connected (i.e. Sh 0 (X) = Q) commutative Hopf algebra such that
• The component of degree n of Sh(X), Sh n (X) is the linear span of the words of weight n over X (so that as vector spaces Sh n (X) = T n (X)). We write similarly Sh n (X) for the linear span of the words of length n; • The product is defined recursively as the sum of the two halfshuffle products ≺, ≻:

y 1 ...y n ≺ z 1 ...z p := y 1 • (y 2 ...y n ≺ z 1 ...z p )
with =≺ + ≻ and y 1 ...y n ≻ z 1 ...z p := z 1 ...z p ≺ y 1 ...y n . • The coalgebra structure is defined by the deconcatenation coproduct:

∆(y 1 ...y n ) := 0≤k≤n y 1 ...y k ⊗ y k+1 ...y n .
Recall that the notions of connected commutative Hopf algebra and connected commutative bialgebra are equivalent since a graded connected commutative bialgebra always has an antipode. The (graded) dual bialgebra of Sh(X) is the tensor algebra T (X) over X, we refer to [START_REF] Reutenauer | Free Lie algebras[END_REF] for details and proofs.

Equivalently, for all x 1 , . . . , x k+l ∈ X:

x 1 . . . x k ≺ x k+1 . . . x k+l = α∈Des ⊆{k} , α -1 (1)=1 x α -1 (1) . . . x α -1 (k+l) , x 1 . . . x k ≻ x k+1 . . . x k+l = α∈Des ⊆{k} , α -1 (1)=k+1 x α -1 (1) . . . x α -1 (k+l) ,
where the α are permutations of [k + l] = {1, ..., k + l}.

The notation α ∈ Des ⊆{k} means that α has at most one descent in position k. Recall that a permutation σ of [n] is said to have a descent in position i < n if σ(i) > σ(i + 1). The descent set of σ, desc(α) is the set of all descents of α, desc(σ) := {i < n, σ(i) > σ(i + 1)}. Proposition 2. As a commutative algebra, Sh(X) is the free algebra over X for the relations (4).

The result goes back to [START_REF] Schützenberger | Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathématiques appliquées, Séminaire Dubreil-Jacotin Pisot (Algèbre et théorie des nombres)[END_REF], where the reader can also find a discussion of the role of the unit in shuffle algebras (there is a subtelty to make sense of the half-products with 1, however this problem is easily settled and doesn't need to be discussed here: we will only use the fact that 1 is a unit for and, for half-shuffle products of 1 with words w use Schützenberger's conventions w ≺ 1 = w, 1 ≺ w = 0).

Whereas most studies focussed on shuffle algebras over sets, shuffle algebras over graded sets are equally important objects. Two classical examples are provided by the iterated bar construction (a key to the computation of the homology of K(Π, n) spaces [START_REF] Eilenberg | Cohomology theory of abelian groups and homotopy theory III[END_REF][START_REF] Eilenberg | On the Groups H(π, n)[END_REF]) and mould calculus , which focusses on problems such as the study and classification of differential equations by algebraic means [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF][START_REF] Menous | Mould calculus, polyhedral cones, and characters of combinatorial Hopf algebras[END_REF]. In the first framework, one constructs the shuffle algebra over a graded commutative algebra (this is actually one of the reasons for Eilenberg and MacLane works on shuffles), in the second case, the shuffle algebra over graded derivations (e.g., in dimension 1, the family of the x n ∂ x with degree n -1).

The remaining part of the present article is devoted to the internal study of Sh(X), where X is a graded alphabet. We insist on the action of natural endomorphisms, mimicking what is known for the shuffle algebra over a non graded set. We also recover as a byproduct Chapoton's rigidity theorem showing that an abstract shuffle bialgebra (an abstract shuffle algebra with a suitable coproduct) can always be realized as the shuffle algebra over a graded set [START_REF] Chapoton | Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces[END_REF].

Graded permutations

We have mentioned the foundational connexion between shuffles and the geometry of simplices. This relationship can be encoded purely combinatorially by the existence of a noncommutative shuffle product on the direct sum of the symmetric group algebras, this is the "geometrical ring of the symmetric groups" of [24, p. 180], a construction that relates directly the classical geometrical approach to shuffle products with the combinatorial approach.

The direct sum of the symmetric group algebras S carries in fact a much richer structure than a mere dendriform product: Malvenuto and Reutenauer first showed that it carries actually a noncommutative noncocommutative Hopf algebra structure and proved that it generalizes naturally various fundamental algebraic structures in the theory of free Lie algebras such as Solomon's descent algebras or quasi-symmetric functions, two noncommutative generalizations of the ring of symmetric functions [START_REF] Malvenuto | Reutenauer Duality between Solomon's algebra and quasisymmetric functions[END_REF]. This Hopf algebra or Malvenuto-Reutenauer (MR) Hopf algebra can be furthermore realized as an algebra of generalized quasi-symmetric functions and is often refered to in the litterature as the Hopf algebra of free quasi-symmetric functions [START_REF] Duchamp | Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras[END_REF].

The MR Hopf algebra is closely related to various fundamental notions of noncommutative representation theory such as the descent algebra of type A or the algebra of quasi-symmetric functions. These later notions are known to generalize to other Coxeter groups than the symmetric groups and, up to a certain extent, to wreath-products of symmetric groups with cyclic groups [START_REF] Mantaci | A generalization of Solomons descent algebra for hyperoctahedral groups and wreath products[END_REF][START_REF] Baumann | A Solomon-type epimorphism for Mantaci-Reutenauers algebra of a wreath product G Sn[END_REF]. Colored permutations appear naturally in this framework (the finite set of colors corresponding to the elements of the cyclic groups) [START_REF] Novelli | Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions[END_REF]. Some of our results generalize further these results from the case of finite cyclic groups to the integers.

These new Hopf algebras are typical examples of "combinatorial quantum groups" (graded Hopf algebras which are neither commutative nor cocommutative but can be naturally interpreted as a "group of symmetries", e.g. through the natural action of permutations on tensors) and have originated many studies. The one we will focus on and generalize is due to the second Author of the present article, who introduced the notion of bidendriform bialgebra and showed that the MR Hopf algebra carries such a structure -with various consequences such as the proof of the Free Lie conjecture (according to which the primitive elements of the MR Hopf algebra form a free Lie algebra) [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF]. These results, as we show now, generalize to graded permutations, which are a natural extension of the notions of permutations and colored permutations when studying shuffle algebras over graded sets.

Recall first that, as for any Hopf algebra, End(Sh(X)) carries an associative convolution product ⋆ defined by

(f ⋆ g) = • (f ⊗ g) • ∆.
We define two other products on End(Sh(X)) by

f ≺ g =≺ •(f ⊗ g) • ∆ and f ≻ g =≻ •(f ⊗ g) • ∆. As ≺ + ≻= , ⋆ =≺ + ≻. Lemma 3. (End(Sh(X)), ≺, ≻) is a dendriform algebra. Proof. Let f, g, h ∈ End(Sh(X)). Then: (f ≺ g) ≺ h = ≺ •(≺ ⊗Id) • (f ⊗ g ⊗ h) • (∆ ⊗ Id) • ∆ = ≺ •(Id ⊗ ) • (f ⊗ g ⊗ h) • (Id ⊗ ∆) • ∆ = f ≺ (g ⋆ h).
The two other axioms are proved in the same way.

These products on End(Sh(X)) dualize to the tensor algebra, for this dual point of view we refer to [START_REF] Fisher | Cozinbiel Hopf algebras in combinatorics[END_REF], which contains various applications of dendriform structures to Hopf algebras of graphs and twisted Hopf algebras (Hopf algebras in the category of species).

The following Lemma, although a direct, straightforward, consequence of the recursive definition of the shuffle product on Sh(X) and of the dendriform structure on End(Sh(X)) will prove very useful.

Lemma 4 (Rewriting Lemma). For any y 1 , ..., y n in X, the word y 1 ...y n can be rewritten:

y 1 ...y n = y 1 ≺ (y 2 ≺ (... ≺ (y n-1 ≺ y n )...)).
The proof is left as an exercice. The Lemma can be used to prove the Prop.2, see [32, 1.19].

Corollary 5. If we write π = n∈N * π n the projection on Sh 1 (X) (where π n is the projection on Sh 1 (X) ∩ Sh n (X)) orthogonally to the Sh n (X), n = 1, we get:

Id = exp ≺ (π) := n∈N π ≺n = n∈N π ≺ (π ≺ (...(π ≺ π)...)),
where π ≺0 stands for the canonical projection on the scalars, Sh 0 (X), and

π ≺n := π ≺ π ≺n-1 .
The series exp ≺ (π) is the "time-ordered exponential" of physicists; it is often called in analyis and physics the Picard series or Dyson-Chen series of π (see e.g. [START_REF] Brouder | Hyperoctahedral Chen calculus for effective Hamiltonians[END_REF], where the link between these series and the Malvenuto-Reutenauer Hopf algebra is explained). Its structure was investigated recently in a series of articles, focussing mainly on the Magnus problem (find an expression for Ω := log(exp ≺ (π))), see [START_REF] Ebrahimi-Fard | New identities in dendriform algebras[END_REF][START_REF] Ebrahimi-Fard | Dendriform Equations[END_REF]). We will be interested here in different issues, related to the meaning of the π n and their products with respect to the internal structure of shuffle algebras. Notice that, by their very definition and due to the Rewriting Lemma 4, we have:

Lemma 6. For any n 1 , ..., n k ∈ N * , π n 1 ,...,n k := π n 1 ≺ (π n 2 ≺ ...(π n k-1 ≺ π n k )...) is the canonical projection on the linear span of words x 1 ...x k with |x i | = n i . In particular, π n 1 ,...,n k • π n 1 ,...,n k = π n 1 ,.
..,n k , and the π n 1 ,...,n k form a complete family (i.e. with total sum Id) of orthogonal idempotents in End(Sh(X)).

We are now in the position to define and study the algebra of graded permutations.

Definition 7. Let us fix k ∈ N. Let σ ∈ S k and d : [k] -→ N * .
We define a linear endomorphism of Sh(X) by:

Φ (σ,d) : x 1 . . . x l -→ x σ(1) . . . x σ(l) if k = l and |x σ(i) | = d(i) for all i, -→ 0 if not.
For example, π n = Φ (idn,n) .

Notations.

(1) We put S = Proof. Direct computation.

Remark. If for all n ≥ 1, X n is infinite, it is not difficult to show that the linear extension Φ : S -→ End(Sh(X)) is injective. Hence, we can define an associative internal product on S by:

(σ, d) • (τ, e) = (τ • σ, d) if k = l and d = e • τ, 0 if not.
We shall from now on identify S with a subspace of End(Sh(X)) via Φ.

Let us write p n for the canonical projection on Sh n (X), so that Id = n p n . A direct inspection shows that the p n belong to S: Lemma 9. For all n ≥ 0:

p n = n k=1 p:[k]-→N * p(1)+...+p(k)=n (Id k , p) = n k=1 d(1)+...+d(k)=n 1 . . . k d(1) . . . d(k) .
Notations.

(

) Let σ ∈ S k , τ ∈ S l . We define σ ⊗ τ ∈ S k+l by (σ ⊗ τ )(i) = σ(i) if 1 ≤ i ≤ k and (σ ⊗ τ )(i) = τ (i -k) + k if k + 1 ≤ i ≤ k + l. (2) Let d : [k] -→ N * and e : [l] -→ N * . We define d ⊗ e : [k + l] -→ N * by (d ⊗ e)(i) = d(i) if 1 ≤ i ≤ k and (d ⊗ e)(i) = e(i -k) if k + 1 ≤ i ≤ k + l. 1
Lemma 10. S is a dendriform subalgebra of End(Sh(X)).

Proof. Let (σ, d) and (τ, e) ∈ S. We assume that σ ∈ S k and τ ∈ S l . If

n = k + l, then ((σ, d) ≺ (τ, e))(x 1 . . . x n ) = 0. If n = k + l, then: ((σ, d) ≺ (τ, e))(x 1 . . . x k+l ) = (σ, d).(x 1 . . . x k ) ≺ (τ, e).(x k+1 . . . x k+l ) =        x σ⊗τ (1) . . . x σ⊗τ (k) ≺ x σ⊗τ (k+1) . . . x σ⊗τ (k+l) if |x σ⊗τ (i) | = (d ⊗ e)(i) for all i, 0 if not, =            α∈Des ⊆{k} , α -1 (1)=1 x (σ⊗τ )•α -1 (1) . . . x (σ⊗τ )•α -1 (k+l) if |x (σ⊗τ )•α -1 (i) | = (d ⊗ e) • α -1 (i) for all i, 0 if not.
Hence:

(

) (σ, d) ≺ (τ, e) = α∈Des ⊆{k} ,α -1 (1)=1 ((σ ⊗ τ ) • α -1 , (d ⊗ e) • α -1 ). 5
Similarly:

(6) (σ, d) ≻ (τ, e) = α∈Des ⊆{k} ,α -1 (1)=k+1 ((σ ⊗ τ ) • α -1 , (d ⊗ e) • α -1 ).
So S is a dendriform subalgebra of End(Sh(X)).

Lemma 11. The idempotents π n 1 ,...,n k belong to S and generate a commutative subalgebra thereof for the composition product.

Proof. The second part of the proposition being a straigthforward consequence of the idempotency property, let us show that π n belong to S; since S is a dendriform subalgebra of End(Sh(X)), the Lemma will follow.

Since π 1 = (id 1 , 1), let us assume that π i ∈ S for i < n. We get:

π n = p n - i 1 +...+i k =n, k>1 π i 1 ,...,i k
and the Lemma follows by induction.

Proposition 12. For all n ≥ 1, we have:

π n = 1 n = n k=1 (-1) k-1 a 1 +...+a k =n p a 1 ≺ (p a 2 ⋆ . . . ⋆ p a k ).
Proof. Indeed, we have, according to the Rewriting lemma (4): p = n∈N π ≺n or, equivalently:

(1 -π) ≺ p = π ≺0 = p 0 or: π ≺ p = n∈N * p n . Let us set p + := n∈N * p n and write z for the convolution inverse of p in End(Sh(X)) (recall that p 0 is the identity for the convolution product):

z = p -1 = k∈N (-1) k (p + ) ,
we get (recall that according to our conventions, p 0 is a right unit for ≺):

π = π ≺ p 0 = π ≺ (p ⋆ z) = (π ≺ p) ≺ z
where the third identity follows from the half-shuffle relations, so that:

π = ( n∈N * p n ) ≺ z = ( n∈N * p n ) ≺ n∈N (-1) n (p + ) n ,
from which the Proposition follows.

Notice that the same argument would prove the following Lemma, useful to study Magnus formulas and Picard/Dyson-Chen series: Lemma 13. For any formally invertible series q = 1 + n∈N * q i in a dendriform algebra, we have:

µ = ( n∈N * q n ) ≺ k∈N (-1) k ( n∈N * q n ) k ,
where µ is given by: q = exp ≺ µ.

Remark. From ( 5) and ( 6 (1) Let w = (i 1 , . . . , i k ) be a word with letters in N * , all distinct. There exists a unique increasing bijection

f from {i 1 , . . . , i k } into [k]. The standardization of w is std(w) = (f (i 1 ), . . . , f (i k )). It is an element of S k . (2) Let σ ∈ S k and d : [k] -→ N * . We put: ∆ ≺ ((σ, d)) = n-1 k=σ -1 (1) std(σ(1), . . . , σ(k)) d(1), . . . , d(k) ⊗ std(σ(k + 1), . . . , σ(n)) d(k + 1), . . . , d(n) ∆ ≻ ((σ, d)) = σ -1 (1)-1 k=1 std(σ(1), . . . , σ(k)) d(1), . . . , d(k) ⊗ std(σ(k + 1), . . . , σ(n)) d(k + 1), . . . , d(n)
This defines two coproducts on the augmentation ideal S + of the dendriform algebra S + .

Example. Let a 1 , a 2 , a 3 , a 4 ∈ N * . ∆ ≺ 3 1 4 2 a 1 a 2 a 3 a 4 = 2 1 a 1 a 2 ⊗ 2 1 a 3 a 4 + 2 1 3 a 1 a 2 a 3 ⊗ 1 a 4 , ∆ ≻ 3 1 4 2 a 1 a 2 a 3 a 4 = 1 a 1 ⊗ 1 3 2 a 2 a 3 a 4 .
In other words, the coproducts of a biword σ(1) . . . σ(n) d(1) . . . d(n) are given by the cuts of the biword into two parts and the standardization of the first lines of the two parts of the biword; in ∆ ≺ , the biletter

1 d • σ -1 (1)
is in the left part and in ∆ ≻ , it is in the right part.

Notations. For all x ∈ S + , we put:

∆ ≺ (x) = x ′ ≺ ⊗ x ′′ ≺ , ∆ ≻ (x) = x ′ ≻ ⊗ x ′′ ≻ , ∆(x) = ∆ ≺ (x) + ∆ ≻ (x) = x ′ ⊗ x ′′ . Proposition 15. For all x, y ∈ S + : ∆ ≺ (x ≺ y) = x ′ ≺ ≺ y ′ ⊗ x ′′ ≺ ⋆ y ′′ + x ⊗ y + x ≺ y ′ ⊗ y ′′ + (7) x ′ ≺ ⊗ x ′′ ≺ ⋆ y + x ′ ≺ ≺ y ⊗ x ′′ ≺ , ∆ ≻ (x ≺ y) = x ′ ≻ ≺ y ′ ⊗ x ′′ ≺ ⋆ y + x ′ ≻ ≺ y ⊗ x ′′ ≻ + x ′ ≻ ⊗ x ′′ ≻ ⋆ y, (8) ∆ ≺ (x ≻ y) = x ′ ≺ ≻ y ′ ⊗ x ′′ ≺ ⋆ y ′′ + x ′ ≺ ≻ y ⊗ x ′′ ≺ + x ≻ y ′ ⊗ y ′′ , (9) ∆ ≻ (x ≻ y) = x ′ ≻ ≻ y ′ ⊗ x ′′ ≻ ⋆ y ′′ + y ⊗ x + y ′ ⊗ x ⋆ y ′′ + (10) x ′ ≻ ≻ y ⊗ x ′′ ≻ . Consequently, (S, ≻ op , ≺ op , ∆ op ≻ , ∆ op ≺
) is a bidendriform bialgebra. Proof. These identities are the axioms for dendriform bialgebras, as introduced in [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF], to which we also refer for the structure results on dendriform bialgebras we will use further on. We restrict ourselves to the case where x = (σ, d) and y = (τ, e) are two biwords. Then ∆ ≺ (x ≺ y) is obtained by taking all the shufflings of x and y such the first letter of the result is the first letter of x, then cutting these words after the letter

1 d • σ -1 (1)
. As a consequence, there are biletters of x in the left part of the result. Hence, five case are possible:

(1) There are letters of x and y in both parts: this gives the term

x ′ ≺ ≺ y ′ ⊗ x ′′ ≺ ⋆ y ′′ . ( 2
) There are letters of x in both parts, and all the letters of y are in the left part: this gives the term x ′ ≺ ≺ y ⊗ x ′′ ≺ . (3) There are letters of x in both parts, and all the letters of y are in the right part: this gives the term x ′ ≺ ⊗ x ′′ ≺ ⋆ y. (4) All the letters of x are in the left part, and there are letters of y in both parts: this gives the term x ≺ y ′ ⊗ y ′′ . (5) All the letters of x are in the left part, and all the letters of y are in the right part: this gives the term x ⊗ y. Let us now consider ∆ ≻ (x ≺ y). It is obtained by taking all the shufflings of x and y such the first letter of the result is the first letter of x, then cutting these words before the letter

1 d • σ -1 (1)
. As a consequence, there are biletters of x in both parts of the result. Hence, three cases are possible:

(1) There are letters of y in both parts of the result: this gives the term

x ′ ≻ ≺ y ′ ⊗ x ′′ ≻ ⋆ y ′′ . ( 2
) All the letters of y are in the left part: this gives the term x ′ ≻ ≺ y ⊗ x ′′ ≻ . (3) All the letters of y are in the right part: this gives the term x ′ ≻ ⊗x ′′ ≻ ⋆y. We now consider ∆ ≺ (x ≻ y). It is obtained by taking all the shufflings of x and y such the first letter of the result is the first letter of y, then cutting these words after the letter

1 d • σ -1 (1)
. Consequently, there are letters of x and y in the left part. So there are three possibilities:

(1) There are letters of x and y in both parts of the result: this gives the term x ′ ≺ ≻ y ′ ⊗ x ′′ ≺ ⋆ y ′′ . (2) All the letters of y are in the left part: this gives the term x ′ ≺ ≻ y ⊗ x ′′ ≺ .

(3) All the letters of x are in the left part: this gives the term x ≻ y ′ ⊗y ′′ . We now consider ∆ ≻ (x ≻ y). t is obtained by taking all the shufflings of x and y such the first letter of the result is the first letter of y, then cutting these words before the letter

1 d • σ -1 (1)
. Consequently, there are letters of y in the left part, and letter of x in the right part. Consequently, four cases are possible.

(1) There are letters of x and y in both parts of the result: this gives the term x ′ ≻ ≻ y ′ ⊗ x ′′ ≻ ⋆ y ′′ . (2) There are letters of y in both parts and all the letters of x are in the right part: this gives the term y ′ ⊗ x ⋆ y ′′ . (3) There are letters of x in both parts and all the letters of y are in the left part: this gives the term x ′ ≻ ≻ y ⊗ x ′′ ≻ . (4) All the letters of x are in the right part and all the letters of y are in the left part: this gives the term y ⊗ x. We obtain in this way the four compatibilities ( 7)- [START_REF] Ebrahimi-Fard | Dendriform Equations[END_REF].

Remark. We define ∆ : S -→ S ⊗ S by ∆(x) = x ⊗ 1 + 1 ⊗ x + ∆ ≺ (x) + ∆ ≻ (x) for all x ∈ S + , and ∆(1) = 1 ⊗ 1. Then (S, ⋆, ∆) is a Hopf algebra.

By the bidendriform rigidity theorem (according to which a bidendriform bialgebra is a free dendriform algebra, see [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF] for details):

Corollary 16. (S, ≺, ≻) is, as a dendriform algebra, freely generated by the subspace of dendriform primitive elements P rim Dend (S) := Ker(∆ ≺ ) ∩ Ker(∆ ≻ ).

The generating series of S as a graded vector space is:

R(x) = ∞ k=0 k!x k • ∞ k=1 x k .
As a consequence of the bidendriform rigidity theorem, the formal series of bidendriform elements of S is:

P (x) = R(x) -1 R(x) 2 .
Here are the first coefficients of R(x) and P (x): Recall that the descent algebra D of a tensor algebra and, more generally, of any graded bialgebra H, is the convolution algebra generated by the projections on the graded components of H [START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF][START_REF] Reutenauer | Free Lie algebras[END_REF]. The descent algebra is a graded algebra, and its graded components can be equipped with an internal composition product. These components identify with the classical Solomon descent algebras of type A and form the main building block of noncommutative representation theory. Motivated by the structural properties of the descent algebra, Fisher's thesis (where the dual notion is introduced and studied in various particular cases) [START_REF] Fisher | Cozinbiel Hopf algebras in combinatorics[END_REF] and by the Proposition 12, which shows that the canonical projection in shuffle algebras belong to the dendrifrom subalgebra of End(Sh(X)) generated by the p n : Definition 17. We define Descd ⊆ S as the dendriform subalgebra of End(Sh(X)) and S generated by the graded projections p n : Sh(X) -→ Sh n (X).

The algebra Descd is naturally graded (the degree of p n is n): Descd = n∈N Descd n and has the completion Descd = n∈N Decsd n . For simplicity, we do not emphasize the distinction between Descd and its completion when dealing with formal power series such as p = n p n and will allow us for example to write abusively p ∈ Descd.

Recall that for all n:

p n = n k=1 d(1)+...+d(k)=n 1 . . . k d(1) . . . d(k) .
For the classical descent algebra D, a key property is the group-like behavior of the graded projections p n (∆(p n ) = i≤n p i ⊗ p n-i ). We show now that this property is inherited, although in a more sophisticated way, in Descd.

Proposition 18. For all n ≥ 1, ∆ ≻ (p n ) = 0 and ∆ ≺ (p n ) = n-1 i=1 p i ⊗ p n-i .
Proof. Clearly, ∆ ≻ (p n ) = 0. Moreover:

∆ ≺ (p n ) = n k=1 d(1)+...+d(k)=n k-1 i=1 1 . . . i d(1) . . . d(i) ⊗ 1 . . . k -i d(i + 1) . . . d(k) = n-1 i=1   i p=1 d ′ (1)+...+d ′ (p)=i 1 . . . p d ′ (1) . . . d ′ (p)   ⊗   n-i q=1 d ′′ (1)+...+d ′′ (q)=n-i 1 . . . q d ′′ (1) . . . d ′′ (q)   = n-1 i=1 p i ⊗ p n-i .
Corollary 19. The dendriform descent algebra Descd is a sub bidendriform bialgebra of S. In particular, it is a free dendriform algebra over its dendriform primitive elements.

Theorem 20. The family (π n ) n≥1 is a basis of the space of dendriform primitive elements of Descd, and these elements freely generate Descd as a dendriform algebra.

Proof. Recall the notations p

+ = Id -1 = n≥1 p n , p 0 = 1. Then ∆ ≻ (p + ) = 0 and ∆ ≺ (p + ) = ∆(p + ) = p + ⊗ p + , so that ∆(Id) = Id ⊗ Id.
We set t := p -1 = ∞ n=0 (-1) n (p + ) n and t + = t -1 and get:

∆(t) = ∆(p -1 ) = p -1 ⊗ p -1 = t ⊗ t or, equivalently, ∆(t + ) = t + ⊗ t + .
We also have π = p + ≺ t. Since ∆ ≻ (p + ) = 0, we get ∆ ≻ (π) = 0 and

∆ ≺ (π) = p + ⊗ p + + p + ≺ t + ⊗ p + t + + p + ⊗ t + + p + ≺ t + ⊗ t + + p + ⊗ p + t + +p + ≺ t + ⊗ p + = p + ⊗ p + + p + ≺ t + ⊗ (-t + -p + ) + p + ⊗ t + + p + ≺ t + ⊗ t + +p + ⊗ (-t + -p + ) + p + ≺ t + ⊗ p + = 0
where we used the identities pt = (1+ p + )(1+ t + ) = 1 and p + t + = -p + -t + . So π is primitive for both coproducts. Taking its homogeneous component of weight n, we obtain that π n is dendriform primitive for all n.

Recall now that, by the Corollary 19, Descd is freely generated as a dendriform algebra by the space of its dendriform primitive elements, and that, by definition, Descd is generated by at most one generator in each weight. So the homogeneous components of the space of dendriform primitive elements for the weight are at most one-dimensional. Finally, (π n ) n≥1 is a basis of P rim dend (Descd).

Corollary 21. The formal series of Descd is:

1 -x -(1 -x)(1 -5x) 2x .
Proof. The formal series of Descd is 1-

√ 1-4x 2x • x 1-x .
Examples. Remark. As a consequence, it is not difficult to prove that the following families are bases of Descd n :

(1) n = 1: (π 1 ) = 1 1 .

(

) n = 2: (π 2 , π 1 ≺ π 1 , π 1 ≻ π 1 ) = 1 2 , 1 2 1 1 , 2 2
(

) n = 3: (π 3 , π 1 ≺ π 2 , π 1 ≻ π 2 , π 2 ≺ π 1 , π 2 ≻ π 1 , π 1 ≺ (π 1 ≺ π 1 ), π 1 ≺ (π 1 ≻ π 1 ), π 1 ≻ (π 1 ≺ π 1 ), π 1 ≻ (π 1 ≻ π 1 ) -(π 1 ≻ π 1 ) ≻ π 1 , (π 1 ≻ π 1 ) ≻ π 1 ) =         1 3 , 1 2 1 2 , 2 1 2 1 , 1 2 2 1 , 2 1 1 2 , 1 2 3 1 1 1 , 1 3 2 1 1 1 , 2 1 3 1 1 1 + 2 3 1 1 1 1 , 3 1 2 1 1 1 , 3 2 1 1 1 1         . 3
So Descd is not stable under the internal product •: for example,

1 3 2 1 1 1 • 3 1 2 1 1 1 = 2 1 3 1 1 1 / ∈ Descd.

Abstract shuffle bialgebras and the rigidity theorem

The coproduct acting on the shuffle bialgebra Sh(X) satisfies the relation:

∆(x ≺ y) = x ′ ≺ y ′ ⊗ x ′′ y ′′ + 1 ⊗ x ≺ y (11)
where we used the Sweedler notation ∆(x) = x ′ ⊗ x ′′ and set 1 ≺ 1 = 0 in the formula (recall that 1 ≺ w = w ≻ 1 = 0 and w ≺ 1 = 1 ≻ w = w whenever w is a non empty word). The relation follows from the recursive definition of the shuffle product of words (or from the observation that for two words w and z, the first letter of w is the first letter of w ≺ z, so that the expansion of ∆(w ≺ z) always starts with the first letter of w).

These identities lead to the abstract definition of a shuffle bialgebra, a particular case of the notion of dendriform bialgebra [START_REF] Chapoton | Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces[END_REF][START_REF] Ronco | Primitive elements in a free dendriform algebra[END_REF][START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF].

Recall from the discussion at the begining of the article that a (non unital) shuffle algebra is, in general, a vector space V equipped with a bilinear map ≺ satisfying the axiom (4): (a ≺ b) ≺ c = a ≺ (b ≺ c + c ≺ b). A unital shuffle algebra A is then obtained by adding a unit to V : A = A + ⊕ C := V ⊕ C, with a ≺ 1 := a, 1 ≺ a := 0. The half-product 1 ≺ 1 is defined to be 0. The shuffle product is defined on V = A + by a b := a ≺ b + b ≺ a or =≺ + ≻ with a ≻ b := b ≺ a and extended to A by requiring 1 to be the unit. It provides A with the structure of a commutative (and associative) algebra with unit. The shuffle algebras over generating sets Sh(X) are the free algebras over X for the relation (4) [START_REF] Schützenberger | Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathématiques appliquées, Séminaire Dubreil-Jacotin Pisot (Algèbre et théorie des nombres)[END_REF]. A shuffle algebra A is graded and connected if its decomposition into graded components A = n A n is such A 0 = C and that the half-product is compatible with the grading (A n ≺ A m ⊂ A n+m ). Definition 22. Let A = V ⊕ C be a unital shuffle algebra and ∆ a coassociative counital coproduct on A. The coproduct defines a shuffle bialgebra structure on A if and only if the relation ( 11) is satisfied. A shuffle bialgebra is graded connected if it is a graded connected shuffle algebra and if the coproduct is compatible with the grading (∆

(A n ) ⊂ i+j=n A i ⊗ A j ).
As expected, shuffle bialgebras over finite sets Sh(X) provide examples for this abstract definition of graded connected shuffle bialgebras. Lemma 23. The set of linear endomorphisms of a shuffle bialgebra End(A) is equipped with the structure of a dendriform algebra by the products:

f ≺ g(a) := f (a ′ ) ≺ g(a ′′ ) f ≻ g(a) := f (a ′ ) ≻ g(a ′′ ) = g(a ′′ ) ≺ f (a ′ ).
The proof follows from the same arguments as for the Lemma 3. From now on, A will denote an arbitrary graded connected shuffle bialgebra.

Corollary 24. There is a unique map φ of dendriform algebras from Descd to End(A) such that φ(p n ) := a n , where we write a n for the canonical projection from A to A n . Indeed, as the π n to which they are related by triangular equations (π n and p n are equal up to dendriform products of lower degrees elements), the p n form a free family of dendriform generators of Descd. The Corollary follows by the universal properties of free algebras. Lemma 25. Let τ = n≥1 τ n := φ(π). For any x, y ∈ A + , we have τ (x ≺ y) = 0. In other terms, τ acts trivially on A + ≺ A + . Indeed, let a + = φ(p + ). Since π = p + ≺ (Id) -1 , and since the convolution inverse of Id in End(A) is the antipode, we get:

τ (x ≺ y) = a + (x ′ ≺ y ′ ) ≺ S(y ′′ ) S(x ′′ ).
Since 1 ≺ u = 0 for an arbitrary u ∈ A + , we get:

τ (x ≺ y) = (a + (x ′ ) ≺ y ′ ) ≺ S(y ′′ ) S(x ′′ ) = a + (x ′ ) ≺ (y ′ S(y ′′ )) S(x ′′ ) = τ (x)
a 0 (y) = 0, from which the Lemma follows.

Corollary 26. The operator τ is an idempotent: τ 2 = τ . Indeed, since p = exp ≺ (π), Id A =: a = exp ≺ (τ ) and:

τ = τ • a = τ (exp ≺ (τ )) = τ • τ
since the image of τ in contained in A + , and therefore τ • (τ ≺ (τ...(τ ≺ τ )...) = 0 for iterated products (τ ≺ (τ...(τ ≺ τ )...) of an arbitrary length.

Proposition 27. The idempotent map τ is a projection onto the primitive elements of A.

From the identity τ = a + ≺ S we get that for a primitive element x of A, τ (x) = x ≺ 1 = x. Now, for y ∈ A + , ∆(τ (y)) = ∆(a + (y ′ ) ≺ S(y ′′ )) = ∆(y ′ ≺ S(y ′′ )) = (y 1 ≺ S(y 4 )) ⊗ (y 2 S(y 3 )) + 1 ⊗ y ′ ≺ S(y ′′ ) where we used the coassociativity of the coproduct, the property of the antipode ∆(S(y)) = S(y ′′ )S(y ′ ), and an extended Sweedler notation to write somehow abusively y 1 ⊗ ... ⊗ y n for the iterated coproduct of order n of y (so that e.g. (∆ ⊗ Id) • ∆(y) = y 1 ⊗ y 2 ⊗ y 3 , and so on).

From the coassociativity of the coproduct, we get y 2 S(y 3 ) = 1 (the convolution product of the antipode with the identity is the null map on A + and the identity on the scalars). Finally, by cancellation of the non scalar terms in y 2 S(y 3 ), we get: ∆(τ (y)) = y ′ ≺ S(y ′′ ) ⊗ 1 + 1 ⊗ y ′ ≺ S(y ′′ ) = τ (y) ⊗ 1 + 1 ⊗ τ (y), from which the Proposition follows.

In 2000, F. Chapoton introduced the breaking new idea that the classical Cartier-Milnor-Moore theorem holds in fact for generalized bialgebras such as dendriform bialgebras, provided an analogue of the Poincaré-Birkhoff-Witt theorem holds [START_REF] Chapoton | Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces[END_REF]. His main Theorem ([4, Thm 1]) implies a "rigidity theorem" in the commutative case: the underlying algebras are then free shuffle algebras. Chapoton's proof follows by adapting the classical proof of the Cartier-Milnor-Moore theorem [START_REF] Milnor | On the structure of Hopf algebras[END_REF] to dendriform bialgebras. M. Ronco contributed by various remarks to the final preprint version of [START_REF] Chapoton | Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces[END_REF] and proposed soon after another proof by adapting the combinatorial proof [START_REF] Patras | Homothéties simpliciales[END_REF][START_REF] Patras | La décomposition en poids des algèbres de Hopf[END_REF][START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF] of the Cartier-Milnor-Moore theorem [START_REF] Ronco | Eulerian idempotents and Milnor-Moore theorem for certain noncocommutative Hopf algebras[END_REF].

As far as classical bialgebras are concerned, it is a well-known fact that the Leray theorem (which asserts that a graded connected commutative bialgebra over a field of characteristic zero is a free commutative algebra) is much simpler to prove than the Cartier-Milnor-Moore theorem, see e.g. [START_REF] Patras | A Leray theorem for the generalization to operads of Hopf algebras with divided powers[END_REF] for a modern general proof and further references on the subject. This observation also holds for dendriform and shuffle algebras: the various proofs of the classical Leray theorem can be adapted to shuffle bialgebras to get simple and direct proofs of Chapoton's rigidity theorem. We deduce here a proof of the Theorem from the Descd approach. Theorem 28. A graded connected shuffle bialgebra A is isomorphic, as a shuffle bialgebra, to the free shuffle algebra over the vector space of its primitive elements P rim(A).

From the Proposition [START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF] we know that τ projects to P := P rim(A). From the identity Id = exp ≺ (τ ), we deduce that the image of τ , P , generates A as a shuffle algebra and that an arbitrary element in A can be written as a linear combination of iterated half-shuffle products p 1 ≺ (p 2 ≺ ...(p n-1 ≺ p n )...), with p i ∈ P Let us choose a graded basis B of P (by graded we mean that any b ∈ B belongs to a graded component P n of P ). To prove the theorem, it is enough to prove that an arbitrary linear combination of iterated half-shuffle products n≤N p 1 ,...,pn λ p 1 ,...,pn p 1 ≺ (p 2 ≺ ...(p n-1 ≺ p n )...) with the p i in B vanishes if and only if all the coefficients λ p 1 ,...,pn are null.

Let us prove this property by induction on N . We assume therefore that the p 1 ≺ (p 2 ≺ ...(p n-1 ≺ p n )...), where n < N -1 and the p i run over B are linearly independent. Assume now that X = n≤N p 1 ,...,pn λ p 1 ,...,pn p 1 ≺ (p 2 ≺ ...(p n-1 ≺ p n )...) = 0 and that ∃(p 1 , ..., p N ), λ p 1 ,...,p N = 0. Then, according to eqn [START_REF] Eilenberg | Cohomology theory of abelian groups and homotopy theory III[END_REF], 0 = ∆(X) = λ p 1 ,...,p N p 1 ⊗ p 2 ≺ (...(p n-1 ≺ p n )...) + Z, where Z is a linear combination of elements that, by induction, are linearly independent of p 1 ⊗ p 2 ≺ (...(p n-1 ≺ p n )...). The Theorem follows.

Corollary 29. In particular, the dendriform algebra of graded permutations S acts naturally on an arbitrary graded connected shuffle bialgebra A.

This follows from the existence of an isomorphism A ∼ = Sh(P rim(A)).

For

  I ⊂ [n], we write Des I := {σ, desc(σ) = I} and Des ⊆I := {σ, desc(σ) ⊆ I}.

k≥0S

  k × Hom([k], N * ), and S = V ect(S).

( 2 )

 2 Let σ ∈ S k and d : [k] -→ N * . We shall represent (σ, d) by the biword σ(1) . . . σ(k) d(1) . . . d(k) . Lemma 8. For all (σ, d) ∈ S k ×Hom([k], N * ) and (τ, e) ∈ S l ×Hom([l], N * ), Φ(σ, d) • Φ(τ, e) = Φ(τ • σ, d) if k = l and d = e • σ, 0 if not.

1 4 .

 14 ), we obtain that (σ, d) ≺ (τ, e) is the sum of the shufflings of the biwords σ(1) . . . σ(k) d(1) . . . d(k) and τ (1) + k . . . τ (l) + k eσ, d) ≻ (τ, e) is the sum of the shufflings of these two biwords such that the first biletter is τ (Bidendriform structures on graded permutations Definition 14.

5 .

 5 The dendriform descent algebra

  is sequence A002212 of the On-Line Encyclopedia of Integer Sequences.

Acknowledgements

We thank warmly C. Reutenauer for several discussions in Montréal that originated the present work, and the UQAM (Université du Québec à Montréal) and the LIRCO (Laboratoire International Franco-Québécois de Recherche en Combinatoire) for their support.

Shuffles