Marie Jaspard
email: marie.jaspard@coral.alima.ngo

Sabue Mulangu

Sylvain Juchet
email: sylvain.juchet@coral.alima.ngo

Beatrice Serra
email: beatrice.serra@coral.alima.ngo

Ibrahim Dicko
email: barahimadicko@gmail.com

Baweye Hans-Joeg Lang
email: hansjoerg.lang@alima.ngo

Mayoum Baka

Gaston Musemakweli Komanda
email: drkomanda.gas@gmail.com

Jeremie Muhindo Katsavara
email: jeremiemkats@gmail.com

Patricia Kabuni
email: pkabuni@gmail.com

Baweye Mayoum Baka

Fabrice Mbika Mambu

Margaux Isnard
email: margauxisnard@gmail.com

Christophe Vanhecke
email: christophevanhecke@yahoo.fr

Alexia Letord
email: alexia.letord@gmail.com

Ibrahima Dieye
email: ibrahimadieye@g.harvard.edu

Oscar Patterson-Lomba
email: oscar.patterson-lomba@analysisgroup.com

Olivier Tshiani Mbaya

Fiston Isekusu

Donatien Mangala
email: donatienmangala@gmail.com

Jean Luc Biampata
email: jlbiampata@gmail.com

Richard Kitenge
email: richardkitenge2@gmail.com

Moumouni Kinda
email: moumouni.kinda@alima.ngo

Xavier Anglaret
email: xavier.anglaret@u-bordeaux.fr

Jean Jacques Muyembe
email: jjmuyembet@gmail.com

Richard Kojan
email: richard.kojan@alima.ngo

Khaled Ezzedine
email: khaled.ezzedine@aphp.fr

Denis Malvy
email: denis.malvy@chu-bordeaux.fr

Development of the PREDS score to predict in-hospital mortality of patients with Ebola virus disease under advanced supportive care: Results from the EVISTA cohort in the Democratic Republic of the Congo

Keywords: Ebola virus, Sub-Saharan Africa, Supportive care, Outcome, Predictive score, In-hospital mortality Research

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Evidence before this study Mortality from Ebola virus disease (EVD) remains a major issue, despite the development of new specific treatments. Advanced supportive care and early detection of severe cases are levers that could further reduce mortality. We search Pubmed from January 1 st 2014 to July 1 st 2022 for cohorts related to intensive care in EVD patients on low-resource settings. The search was done with no language restriction and using the search terms "Ebola virus", "West Africa", "Guinea", "Sierra Leone", "Liberia", "Democratic Republic of the Congo" and "intensive care" in titles and abstracts. Advance supportive care were already provided to EVD patients in Sierra Leone in an Ebola treatment center (ETC) equipped with intensive care unit on a small number of patients in comparison with "regular ETC". Access to intensive care was then associated with lower mortality. In addition, several article highlight the urgent need to scale up the level of supportive care to improve mortality.

We did a similar search for articles related to the development or validation of prognostic models for in-hospital death (search terms "prognostic", "death", "outcome", "score"). Two scores were developed using clinical, socio-demographics, or virological data from the Sierra Leone 2014-2015 outbreak data but were not internally or externally validated and did not include biological variables. Furthermore, the clinical relevance and feasibility of these two scores is debatable. One score includes a lot of clinical variables, including socio-demographic variables whose association with mortality is not obvious, with no biological variables, while the other includes very few variables, with no biological considerations either. The scoring of EVD severity remains then a key question in order to orientate rapidly patients in in advanced care unit.

Added value of this study

We collected clinical, biological, and virological data on all patients attending to the three ETCs providing advanced supportive care and investigational treatments during the 10 th outbreak in the DRC. With these data, we were able to develop a prognostic score model that predicts the day 28 in-hospital risk of death of EVD patients.

Introduction

The Ebola virus disease (EVD) outbreak in the Democratic Republique of the Congo (DRC) that occurred between August 2018 and June 2020 involved 3,470 cases and resulted in 2287 deaths. It affected three provinces in the east of the country (Ituri, North Kivu and South Kivu). 1 After being validated in the 2014-2016 West African outbreak, the specific Ebola Virus (EBOV) vaccine (r-VSV-ZEBOV, ERVEBO, Merck) was used for ring vaccination of contacts and contacts of contacts. [START_REF] Henao-Restrepo | Efficacy and effectiveness of an rVSV-vectored vaccine in preventing Ebola virus disease: final results from the Guinea ring vaccination, open-label, cluster-randomised trial (Ebola Ça Suffit!)[END_REF][START_REF] Coller | Clinical development of a recombinant Ebola vaccine in the midst of an unprecedented epidemic[END_REF] In addition, during this DRC 10 th outbreak, a specific treatment was identified. The MEURI (Monitored Emergency Used for Unregistered Intervention) and the PALM trials evaluated 3 investigational drugs, two of them showed efficacy in term of mortality improvement: REGN-EB3and MAb114 (monoclonal antibody products). [START_REF] Mulangu | A randomized, controlled trial of Ebola virus disease therapeutics[END_REF] Even using these treatment strategies, the mortality remains high for severe cases. [START_REF] Mulangu | A randomized, controlled trial of Ebola virus disease therapeutics[END_REF] Thus, any strategy that can help reduce mortality should be explored. We provide in this paper two leads that could help in future outbreaks: advanced supportive care and early triage of at-risk patients.

Supportive care is an important therapeutic strategy. [START_REF] Fischer | Shifting the paradigmapplying universal standards of care to ebola virus disease[END_REF][START_REF] Lamontagne | Evidence-based guidelines for supportive care of patients with Ebola virus disease[END_REF][START_REF] Leligdowicz | Ebola virus disease and critical illness[END_REF][START_REF] Langer | Intensive care support and clinical outcomes of patients with Ebola virus disease (EVD) in West Africa[END_REF][START_REF] Fowler | Caring for critically ill patients with ebola virus disease. Perspectives from West Africa[END_REF][START_REF] Lamontagne | Doing today's work superbly well-treating Ebola with current tools[END_REF] However, the precise spectrum of supportive care is wide and there is no clear indication as to its application, as it depends on a combination of several complicated actions. [START_REF]Optimized supportive care for Ebola virus disease[END_REF] The spectrum of supportive care therefore requires definition in order to optimize its feasibility in the field.

Early triage of patients also remains of prime importance in reducing mortality rates. Understanding which patients are more at risk of death and have the highest chance of being saved is a crucial issue to address.

The present study set out to describe the supportive care provided to patients during the 10 th Ebola outbreak in the DRC, and to develop an individual death risk score model, herein referred to as PREDS (PRedicting Ebola Death risk Score).

Methods

Participants and setting

Within the context of the operational response to Ebola outbreaks in the DRC, the national Ministry of Health, with the support of the French NGO ALIMA (Alliance for International Medical Action), has established three Ebola Treatment Centers (ETCs) in the North Kivu and Ituri provinces, at the Beni, Katwa and Mambassa urban centers.

EVISTA (Ebola Virus STAndard of care) is an observational cohort set up for the duration of the outbreak, with the aim of describing the standard of care provided to Ebola patients, and the clinical and biological course of the disease. All patients with a positive EBOV RT-PCR test admitted to an ETC during the inclusion period were included. Patients were followed from admission until discharge or death in ETC.

Care, treatment and follow-up

All EVD patients received a standard of care in accordance with the DRC Ministry of Health recommendations and WHO guidelines. [START_REF]Optimized supportive care for Ebola virus disease[END_REF] Clinical evaluation was carried out using a modified version of the NEWS2 (mNEWS2) score without the respiratory rate, for practical reasons (Figure S4). [START_REF]National Early Warning Score (NEWS) 2[END_REF] Advanced supportive care tools were used to improve clinical management. Four investigational drugs were available through PALM or MEURI (Remdesivir, REGN-EB3, Zmapp and MAb11). Follow-up was similar during PALM and MEURI. An innovative measure, the Biosecure Emergency Room (BER), was developed to treat cases in individual spaces within the units. [START_REF] Fischer | Shifting the paradigmapplying universal standards of care to ebola virus disease[END_REF][START_REF] Lamontagne | The evolution of supportive care for Ebola virus disease[END_REF][START_REF] Malvy | Ebola virus disease[END_REF] Haemotological, biochemical, and virological analysis were done in each ETC (more detail on the devices available in annex 4).

Data collection

Data regarding the demographic and clinical situations, biological parameters and medication received were collected for all patients at admission. The outcome and a summary of the hospitalization were collected at discharge or death. Data regarding vaccination status were based on patient declaration.

Statistical analysis

Baseline characteristics, follow-up supportive care, treatment, and observed outcomes were summarized using median and interquartile range (IQR) for continuous variables, and count and percentage for categorical variables. Statistical comparisons were carried out using the Wilcoxon rank-sum test for continuous variables and chi-squared test for categorical variables. Time-to-event was estimated as the period from date of admission to date of death for the patients who died. Patients who survived the whole study period ceased to be monitored at the end of their follow-up period, i.e. day 28. Univariate Cox proportional hazard regression models were used to assess the association of key baseline characteristics with risk of death for all patients.

Prognostic model development

The selection of candidate predictors included in the model was based on the literature, the results from the univariate Cox models, and input from clinical experts. Three multivariable Cox proportional hazard model were explored: one full model (i.e. with all candidate predictors), and two least absolute shrinkage and selection operator (LASSO) models. The outcome was death before day 28 after admission. To develop and validate the PREDS score, patients with non-missing data on all candidate predictors and the outcome were randomly allocated to either a training sample (2/3 of the data) or a validation sample (1/3 of the data). The training sample was used to develop and parametrize the prediction model, and the validation sample was used to test outof-sample prediction model performance. Predictive performance was assessed in the validation set by 1) model concordance, measured by the C-statistic, and 2) model calibration for risk of death at 28 days, quantified by the Hosmer-Lemeshow goodness-of-fit test. The Breslow estimator of the baseline hazard was combined with the HRs to obtain the predicted risk of death for each patient at 28 days from admission date.

Prognostic score development

PREDS was developed as a simple points-based system to characterize patient risk scores using the methods of Sullivan and D'Agostino. [START_REF] Sullivan | Presentation of multivariate data for clinical use: the Framingham study risk score functions[END_REF] Creatinine was categorized according to a clinically relevant cut-off, and a reference value operationalized as the midpoint was calculated using the 1 st and 99 th percentile values to minimize the influence of extreme values. A base risk profile was set to correspond to the lowest risk category for each variable. The constant for the points system (number of regression units corresponding to one point) was defined as the increase in risk associated with a 2-unit (mg/dL) increase in creatinine. Factor states associated with a higher risk of death were assigned more points, so a higher points total represented greater risk. The points for each predictor were totaled, and a table indicating risk (defined as the probability of dying within 28 days of admission) for each points total was established.

Prognostic score internal validation

To evaluate the robustness of the risk-scoring algorithm, the points system based on the training sample was used to generate risk scores for patients in the validation sample, and the correlation between the points-based risk scores and the Cox model-based risk scores was assessed. To evaluate the impact of missing data, a sensitivity analysis was carried out to assess the model's predictive performance in multiple imputed held-out (i. e., validation) datasets. This approach allowed the robustness of the prediction model to be evaluated across a range of imputed datasets to assess how well the model would perform in other datasets with no missing data. Multiple imputation was carried out using the multivariate imputation by chained equations (MICE) approach, [START_REF] Buuren | MICE: multivariate imputation by chained equations in R[END_REF] which has been the preferred approach for handling missing data based on published simulation studies. [START_REF] Marshall | Comparison of imputation methods for handling missing covariate data when fitting a Cox proportional hazards model: a resampling study[END_REF] In this analysis, 200 validation imputations were generated. Then, model concordance C-statistic, p-value for the Hosmer-Lemeshow goodness-of-fit test, and correlation coefficient between points-predicted risk and model-predicted risk were obtained for each imputed validation set. The mean values across all 200 validation sets were then calculated and compared with the respective values obtained in the non-imputed (empirical) validation set.

All statistical analyses were performed with the R statistical package version 4.0.3 or later (R Foundation for Statistical Computing). The results were reported following the TRIPOD statement (supplementary material).

Ethical consideration

The consent of patients included in the cohort was (i) extrapolated from their signed consent to participation in the RCT or MEURI studies (the data collected were part of the medical support we had provided for these patients) or (ii) collected through specific consent forms for EVISTA. For patients unable to sign the consent form, relatives were contacted to sign for them. The EVISTA protocol was approved by the DRC National Ethics Committee (151/CNES/BN/PMMFI2019) and registered with clinicaltrial.gov (NCT04815175).

Role of funding sources

The funding sources took no part in designing the study, collecting, analyzing or interpreting the data, writing the report or making the decision to submit the article for publication.

Results

Description of the cohort (Table 1 and S1)

Between August 1 st 2018 and December 31 th 2019, 711 patients were eligible for analysis (Figure 1). The mean ( § SD) age was 28 ( § 17) years. Patients were admitted at a median of 4 days from onset of symptoms. In terms of vaccination status, 175 (33%) declared they had been vaccinated with the rVSV-ZEBOV-GP vaccine. Of the 134 patients who reported their vaccination date, 93 had been vaccinated 10 days or less before admission (median 7 days, IQR 5-8), and 41 more than 10 days before admission (median 15 days, IQR 12-41). Biological parameters at baseline (Table S2) showed a low platelet count in 158 patients (54%), a high white blood cell count in 138 (46%), and an increased level of creatinine in 271 (49%). The CRP value was above 5 mg/l for 512 patients (99%). Serum sodium was low (<130mmol/ l) for 183 patients (33%) and 166 (35%) had an abnormal level of potassium. Values for AST and ALT were more than 5 times the upper limit for normal rang in 224 (59%) and 262 patients (48%) respectively. The albumin level was low (≤ 35 g/l) in 490 patients (88%). In terms of viral load measurement, the EBOV RT-PCR NP Ct value was ≤ 22 for 356 patients (54%), while the GP Ct value was ≤ 22 for only 83 patients (13%). S1) Almost all patients (94%) experienced viral-illness-like symptoms such as fatigue (91%), anorexia (70%) and headache (57%). Half of them showed respiratory symptoms and a critical oxygen saturation below 92%. In addition, 86% of patients had digestive symptoms, of which 75% diarrhea and 34% dysphagia. A total of 238 patients (35%) experienced an impaired level of consciousness (CVPU on the ACVPU scale) and 264 (37%) had no neurological symptoms. The most common neurological symptoms were agitation and coma (in 25% and 29% of patients respectively). 103 patients (15%) experienced seizure during follow-up. Overall, 275 patients (39%) had bleeding symptoms of some type, the most common being venous puncture point bleeding, melena, gingival bleeding and hematemesis (20%, 14%, 12% and 11% respectively).

Follow-up (Table

Investigational treatment and patient care

Overall, 623 patients (88%) were given an investigational treatment, 453 (73%) in the context of PALM and 170 (27%) in MEURI. Most patients were managed in the BER from the time of admission, mainly for biosafety reasons. All the supportive interventions provided to the patients are detailed in Table 2.

Outcomes

Of the 711 patients involved, 323 (45%) died after a median of 2 days (IQR 1-5) from admission and 9 days (IQR 6-12) from onset of symptoms (Table 2). In addition, lethality dropped to 36% and 29% in patients hospitalized for more than 24 h and 48 h respectively.

The case fatality rate (CFR) was 39% (243/623) among the patients who received an investigational treatment, and 91% (79/87) among those who did not (64 of them could not receive it because they died before having the possibility to receive the treatment, within the first 24 h of admission). The CFR of the 175 patients who declared being vaccinated was 30% (vs. 49% for unvaccinated). A total of 32 pregnant women and 85 children aged 5 or under were included in the study. Details of the outcomes for these specific groups are given in Figure S1 and table S2 respectively.

Prognostic model development

Six candidate predictors were evaluated: four from the univariate analysis and the literature (ALT, creatinine, mNEWS2 and EBOV RT-PCR), and two from clinical expertise (age and time from 1 st symptom to admission, widely used by clinicians on the field to evaluate severity at admission). Some variables were significant at the univariate analysis (table S2) but were not kept to propose a score that is not field dependant (e.g. socio-economic variables), that can be used on resource-limited terrains (e.g. biological variables), and with reliable data (e.g. vaccination). The training sample to develop the prediction model comprised 279 patients randomly selected (Figure 2). Compared to the two LASSO models, prediction performance was similar and calibration at 28 days was better for the full model (i.e. including all candidate predictors). Proportional Hazard was checked graphically and with scaled Schoenfeld residuals (annex 3). The score was developed using this model. 3). The final score ranged from 0 to 13 points, with lower score indicating a lower risk of the outcome. In the internally validated dataset, the estimated risk of death at day 28 ranged from 1.9% (0 point) to 80.8% (13 points) (annex 3). Three risk groups were defined: low (score 0-5, survival probability at D28 92.8% (95%CI 86.3-99.9)), medium (score 6-9, survival probability at D28 53.7% (95%CI 40.4-71.3)), high (score 10-13, survival probability at D28 11.9% (95%CI 5.2-27.1)) (Figure 3).

Prognostic score development

Prognostic score validation

The predictive performance of the multivariable Cox model showed good concordance (C = and calibration with the Hosmer-Lemeshow test (p = 0.12) for risk of death at 28 days in the validation sample S3).

The correlation between points-based scores and the Cox model-based scores in the validation sam-(N = 140) was 93.8%. A plot of the observed survival rates at different time points stratified by predicted risk quartiles S2) shows that the model-predicted risk scores discriminate well between patient groups with different survival profiles. The sensitivity analyses indicated that the model had a similar predictive value across the 200 multiple imputed held-out datasets. The mean C-statistic (SD) was 82.2% (0.96), compared to 86.2% in the empirical validation set. The mean p-value for the Hosmer-Lemeshow goodness-of-fit test p-value (SD) was 0.09 (0.11), while the corresponding value in the empirical dataset was 0.12. Finally, the mean (SD) correlation coefficient between points-predicted risk and model-predicted risk was 92.7% (0.67), compared to 93.8% in the empirical dataset.

Discussion

This study, characterizing more than 700 patients, represents to our knowledge the largest prospective observational cohort for EVD.

The initial clinical presentation of these patients was similar to that described during the 2014-2016 Ebola outbreak in West Africa. hyponatremia, renal dysfunction, thrombocytopenia, dyskaliemia and hepatic failure are factors previously described in EVD. [START_REF] Schieffelin | Clinical illness and outcomes in patients with Ebola in Sierra Leone[END_REF][START_REF] Van Griensven | Electrolyte and metabolic disturbances in Ebola patients during a clinical trial, Guinea, 2015[END_REF][START_REF] Hunt | Clinical presentation, biochemical, and haematological parameters and their association with outcome in patients with Ebola virus disease: an observational cohort study[END_REF] We also recognized that acute renal high viral load, and a high ALT level were indicators of a fatal outcome. [START_REF] Malvy | Ebola virus disease[END_REF][START_REF] Van Griensven | Electrolyte and metabolic disturbances in Ebola patients during a clinical trial, Guinea, 2015[END_REF][START_REF] Hunt | Clinical presentation, biochemical, and haematological parameters and their association with outcome in patients with Ebola virus disease: an observational cohort study[END_REF] The overall ity for our cohort was 45% -higher than that in the PALM trial in which patients with severe on admission were not included. 4 Because there have been strong calls for improvements in care standard of care by caregivers, [START_REF] Lamontagne | Evidence-based guidelines for supportive care of patients with Ebola virus disease[END_REF][START_REF] Leligdowicz | Ebola virus disease and critical illness[END_REF][START_REF] Langer | Intensive care support and clinical outcomes of patients with Ebola virus disease (EVD) in West Africa[END_REF][START_REF] Fowler | Caring for critically ill patients with ebola virus disease. Perspectives from West Africa[END_REF][START_REF] Lamontagne | Doing today's work superbly well-treating Ebola with current tools[END_REF] symptomatic treatment has improved since the West African epidemic. [START_REF] Fowler | Caring for critically ill patients with ebola virus disease. Perspectives from West Africa[END_REF][START_REF] Loignon | Barriers to supportive care during the Ebola virus disease outbreak in West Africa: results of a qualitative study[END_REF] Innovative options such as admission and management in BER, cardiopulmonary ultrasound to guide vascular filling, monitoring of biological parameters allowing the adaptation of electrolyte compensation and oxygen therapy are now available in the field. [START_REF] Fischer | Shifting the paradigmapplying universal standards of care to ebola virus disease[END_REF][START_REF] Lamontagne | The evolution of supportive care for Ebola virus disease[END_REF] Some patients have even benefited from the placement of a central venous or bone catheter, or treatment with a vasopressive drug to manage cardio-circulatory shock. Specific treatments only have a partial impact on fatality rates in severe patients on admission, [START_REF] Mulangu | A randomized, controlled trial of Ebola virus disease therapeutics[END_REF][START_REF] Iversen | Recent successes in therapeutics for Ebola virus disease: no time for complacency[END_REF] which is largely supported by the experience of EVD management in northern countries with high levels of supportive care and showing mortality rates of less than 20%, even without specific treatments. [START_REF] Uyeki | Clinical Management of Ebola virus disease in the United States and Europe[END_REF] The optimization of supportive care should be mandatory in future outbreak management and therefore always involve three elements: (i) very close monitoring of the patient's hydration level (vital constant, electrolytes and possibly cardiopulmonary ultrasound), (ii) early access to oxygen therapy, antibiotics and blood transfusion and (iii) the possibility of advanced resuscitation care such as central catheter placement, dialysis or mechanical ventilation, which could lower fatality rates. The availability of innovative measures such as BERs for severe patients could also help with close monitoring while maintaining biomedical safety for caregivers. [START_REF] Fischer | Shifting the paradigmapplying universal standards of care to ebola virus disease[END_REF][START_REF] Lamontagne | The evolution of supportive care for Ebola virus disease[END_REF] However, recourse to these advanced supportive care measures, even where they are available, remains limited in poor-resource settings. In this context, patient triage is of prime importance. The NEWS2 score appears to be a useful marker of severity that is compatible with operational constraints; however, it omits important risk factor variables that have been included in the PREDS scoring model developed here.

Two previously-published study attempted to generate a scoring system for evaluation of death. Our colleagues in Sierra Leone developed a score including age, level of education, occupation and symptom description. [START_REF] Kangbai | Severity score for predicting in-facility Ebola treatment outcome[END_REF] Similarly, Hartley et al proposed a score based on age, symptom at triage and day since first symptom). [START_REF] Hartley | Predicting Ebola severity: a clinical prioritization score for Ebola virus disease[END_REF] Nonetheless, none of those scores include biological and virological parameters, and the use of some socio-demographical data are not easily replicable across different field. Finally, the score developed by Kangbai et al is too complicated (5 groups and up to 9 sub groups of variables) to be used during outbreaks. PREDS, on the other hand, does include those biological and clinical data, which might help early distinction of patients requiring immediate advanced supportive care from those at a lower risk of negative outcome. Moreover, in view of the data currently available in case of epidemics, its operational feasibility is high. The NEWS2 has demonstrated its simplicity in large cohorts of Lassa fever patients, [START_REF] Duvignaud | Lassa fever outcomes and prognostic factors in Nigeria (LASCOPE): a prospective cohort study[END_REF] standard biological devices are now widely deployed in ETCs and viral load is systematically performed at patient admission. We have also developed a form to be filled in for the calculation of the PREDS score and the determination of the risk level at admission (annex 5). In the framework of our study we propose, with the aim to improve survival rates, that patients who score 6 or more (medium or high PREDS score) should benefit immediately from innovative advanced care. This aggressive strategy should be tested with external cohorts to confirm its general applicability and accuracy.

This study has some limitations. First, due to its observational nature, it did not allow the effects of advanced supportive care to be distinguished from those of specific investigational treatments on mortality rates. Second, PREDS was only internally validated and requires external validation in future epidemics in the field. Third, data collection was a major challenge in the field context (war in the region, extremely isolated sites), leading to issue in data quality and missingness. To address missing data, we conducted a sensitivity analysis using multiple imputed validation datasets to test the external validity of the prediction model and subsequent risk-scoring system, which indicated that the model performed well on these multiple imputed heldout datasets. Finally, our study population was heterogeneous, because some patients were vaccinated and some received specific treatment or benefit from advanced supportive care which might alter the interpretation of case fatality rate. However, it should be underlined that this study was set up in real life condition, and reflects the daily life of field teams.

In conclusion, despite the development of specific treatments for EVD, mortality remains high, especially in patients who present with critical symptoms on admission to the ETC. Improving the availability of advanced supportive care is therefore essential as a first step in strategic management of the disease. In addition, early triage after admission at the ETC, aimed at identifying patients at risk of death, is key to managing strategies. The PREDS score allows this early referral of patients to appropriate medical care for them. Considering the trends in EVD cases in recent outbreaks, it is likely that interventions will be carried out at multiple small ETCs quickly set up near to where cases of the disease develop, and that PREDS will be useful in this regard for identifying severe cases at these small intervention units.

Contributors

Marie Jaspard (MJ), Sabue Mulangu (SM), Sylvain Juchet (SJ), Beatrice Serra (BS), Richard Kojan (RK), Xavier Anglaret (XA) and Denis Malvy (DM) designed the study. MJ, SM, SJ, BS, RK, Hans-Joerg Lang (HJL), Ibrahim Dicko (ID), Baweye Mayoum Baka (BMB), Gaston Musemakweli Komanda (GMK), Jeremie Muhindo Katsavara (JMK), Jean Louis Muanza Nyengele (JLM),

The

  PREDS score was developed by attaching adding points according to the HR of the multivariable Cox model: ALT > 5N (aHR = 2.48 [95%CI 1.46-4.23], 3 points), creatinine > 1.3 mg/dl (aHR = 1.19 [95%CI 1.12 -1.26], 3 points), mNEWS2 score > 4 (aHR = 2.311 [95%CI 1.54-3.48], 2 points), EBOV RT-PCR NP Ct value ≤ 22 (aHR = 2.56 [95%CI 1.56-4.19], 3 points), age ≥ 50 years old (aHR = 1.50 [95%CI 0.89-2.54], 1 point) and time from 1 st symptom to admission ≥ 4 days (aHR = 1.44 [95%CI 0.97-2.16], 1 point) (Table

Figure 1 .

 1 Figure 1. Inclusion flowchart.

Figure 2 .

 2 Figure 2. Kaplan Meier probability of death for PREDS individual parameters in the training sample (N = 279).

Figure 3 .

 3 Figure 3. Estimated risk of death and survival probability for a (Score 0-5), Medium (Score 6-9), or High (Score 10-13) PREDS score.

Table 1 (

 1 Continued)

		Value	N		Value	N
	Age, n (%)		676	Amylase -U/liter	106 [72;177]	534
	≤ 5 yr	85 (13%)		Corrected calcemia-mmol/l f	2.3 [2.2;2.5]	554
	≤ 28 days	6 (1%)		Albumine -g/l	28 [23;33]	556
	6 to 18 yr	101 (15%)		Glycemia -mg/dl	97 [79;121]	556
	19 to 49 yr	406 (60%)		Liver function, median [IQR]
	≥ 50 yr	84 (12%)		Bilirubin -mg/dl	0.6 [0.5;1.1]	511
	Sex, n (%)		711	ALT -U/liter	212 [60;589]	544
	Male	302 (42%)		AST -U/liter	292 [93;1018]	378
	Female	409 (58%)		Virology, n (%)
	Positive result on pregnancy test, n (%)	32 (8%)		EBOV RT-PCR NP Ct value ≤ 22	356 (54%)	655
	Vaccination			EBOV RT-PCR GP Ct value ≤ 22	83 (13%)	644
	Patient-reported vaccination with rVSV,	175 (33%)	533
	n (%)
	Time from vaccination to admission	8 [5;11]	134
	(days), median [IQR]
	> 10 days before admission, n (%)	41 (31%)
	≤ 10 days before admission, n (%)	93 (69%)
	Time from 1st symptom to admission	4 [2;7]	697
	(days), median [IQR]
	Vital signs, n (%)
	Respiratory rate ≥ 24 breaths per	328 (51%)	641
	minute
	Heart rate ≥ 110 per minute	207 (31%)	666
	Diastolic Arterial Pressure < 60 mmHg	134 (23%)	579
	Systolic Arterial Pressure < 90 mmHg	89 (15%)	579
	SpO2 < 92%	63 (10%)	602
	Mean Arterial Pressure < 65 mmHg a	49 (8%)	579
	Symptoms, n (%)
	Any type of bleeding symptom b	123 (17%)	711
	Any type of neurologic symptom c	99 (14%)	711
	Level of consciousness, n (%)
	ACPVU classification: d		638
	Alert	577 (90%)
	Impaired	61 (10%)
	mNEWS2 score, n (%) e		496
	Total score ≤ 4	357 (72%)
	Total score > 4	139 (28%)
	Hematology, median [IQR]
	Hemoglobin -g/dl	14 [12;15]	301
	Hematocrit -%	40 [36;45]	298
	Platelets -G/l	145 [97;222]	294
	Leucocytes -G/l	10 [5;21]	300
	Lymphocytes -G/l	3 [1;6]	232
	Neutrophils -G/l	5 [3;12]	224
	Monocytes -G/l	0.4 [0.2;1.0]	222
	Renal function, median [IQR]
	Urea -mg/dl	18 [9;48]	553
	Creatinine -mg/dl	1.1 [0.7;3.6]	548
	Electrolytes and Biochemistry, median [IQR]
	CRP -mg/l	39 [11;109]	517
	CPK -U/liter	674 [267;2116]	519
	Sodium -mmol/l	131 [128;134]	551
	Potassium -mmol/l	4.2 [3.7;4.8]	469

Articles 6 www.thelancet.com Vol 54 December, 2022

Table 1 :

 1 Baseline characteristics (N=711).

		Value	N
	Supportive care
	Antibiotics, n (%) a	636 (100%)	638
	Duration (days), median [IQR]	8 [4;14]
	Inotropes, n (%) b	111 (16%)	711
	Received 1 dose	69 (62%)
	Received 2 doses or more	42 (38%)
	Oxygen therapy, n (%)	250 (38%)	661
	Duration (days), median [IQR]	2 [1;3]	188
	Highest flow, n (%)		218
	< 10 l/min	154 (71%)
	≥ 10 l/min	64 (29%)
	Transfusion, n (%)	101 (14%)	711
	Volume (ml), median [IQR]	450 [250;525]	99
	Urinary catheter, n (%)	199 (32%)	628
	Feeding tubes, n (%)	45 (7%)	654
	UltraSound, n (%)	15 (2%)	711
	Intraosseus catheter, n (%)	7 (1%)	659
	Hospitalized in cube, n (%)	560 (85%)	656
	Investigational treatment
	Investigational treatment received,	623 (88%)	710
	n (%)
	Program, n (%)
	MEURI c	170 (27%)
	PALM RCT d	453 (73%)
	Drugs, n (%)
	MAb114	197 (32%)

a Mean Arterial pressure = 2/3 Diastolic Arterial Pressure + 1/3 Systolic Arterial Pressure. b Epistaxis, hematemesis, hematuria, hemoptysis, melena, purpura, conjunctival bleeding, gingival bleeding, venous puncture point bleeding. c Headache, focal neurological deficit, disorientation/confusion, agitation, convulsions, coma/consciousness disorder, photophobia. d ACVPU classification: A: Alert; C: new confusion; V: responsive to voice; P: responsive to pain, U: Unresponsive. e NEWS2 score: National Early Warning Score 2nd version; mNEWS2: modified NEWS2 score without respiratory rate. f Corrected calcemia = measured calcemia -0.025*(albuminemia -40); measured calcemia in mmol/L, albuminemia in g/L.

Table 2 (

 2 Continued)

Table 2 :

 2 [START_REF] Malvy | Ebola virus disease[END_REF][START_REF] Jacob | Ebola virus disease[END_REF][START_REF] Bah | Clinical presentation of patients with Ebola virus disease in Conakry, Guinea[END_REF][START_REF] Mobula | Clinical manifestations and modes of death among patients with Ebola virus disease, Monrovia, Liberia, 2014[END_REF][START_REF] Schieffelin | Clinical illness and outcomes in patients with Ebola in Sierra Leone[END_REF][START_REF] Lado | Clinical features of patients isolated for suspected Ebola virus disease at Connaught Hospital, Freetown, Sierra Leone: a retrospective cohort study[END_REF] Likewise, Care, treatment and outcome (N=711).

		Value	N
	Regeneron	191 (31%)
	Remdesivir	122 (20%)
	Zmapp	107 (17%)
	Outcome
	Dead, n (%)	323 (45%)	711
	Time from admission to death (days),	2 [1;5]
	median [IQR]
	Alive, n (%)	388 (55%)	711
	Time from admission to discharge	17 [14;22]
	(days), median [IQR]

a Ceftriaxone, cefixim, metronidazole, gentamycin, amoxicillin, ciprofloxacin, cloxacillin b Adrenaline, noradrenaline. c Consultation on Monitored Emergency Use of. Unregistered and Investigational Interventions. d Mulangu et al, NEJM, 2019.

Table 3 :

 3 Multivariate cox proportional-hazards analysis of the training sample and PREDS scoring system (N=279) The sample size of 279 patients correspond to patients randomly selected from the 711 patients of the cohort with no missing data on all candidate risk factors (i.e. training sample). CI denotes confidence interval, Hazard Ratios for each variable is adjusted on the other variables shown in the table. Crude Hazard Ratios corresponding to the univariate analysis are shown in TableS4.b Assignment of points to risk factors was based on a linear transformation of the corresponding b regression coefficient. The constant for the points system (number of regression units corresponding to one point) was defined as the increase in risk associated with a 2-unit (mg/dL) increase in creatinine (B=0.34). (cf annex 3).

	Risk factors	Categories	Points
	ALT > 5N (U/L)	No	0
		Yes	3
	Age (years)	< 50	0
		≥ 50	1
	Creatinine (mg/dL)	≤ 1.3	0
		> 1.3	3
	Time from 1st symptom to	< 4	0
	admission (days)	≥ 4	1
	mNEWS2 score > 4	N o	0
		Yes	2
	EBOV RT-PCR NP Ct value ≤ 22	No	0
		Yes	3
	Score category a	Low	0-5 Points
		Medium	6-9 Points
		High	10-13 Points

a . a c In the Multivariate Cox Proportional-Hazards Analysis creatinine is used as a continuous variable, but to assign points for the PREDS score it was categorized according to a clinically relevant cut-off (cfTable 4).

Table 4 :

 4 Points associated with each of the categories of the PREDS score and risk categorization.

a The score is obtained by adding the points obtained by individual risk factors.

www.thelancet.com Vol 54 December, 2022

www.thelancet.com Vol 54 December, 2022

Acknowledgments

Our thanks to all the patients involved in the current study as well as their caregivers, the investigators and the research staff at the participating care centers.

Our thanks to the World Health Organization for technical advice.

Funding This study was funded by ALIMA.

Patricia Kabuni (PK), Fabrice Mbika Mambu (FMM), Margaux Isnard (MI), Christophe Vanhecke (CV), Alexia Letord (AL), Olivier Tshiani (OT, Fiston Isekusu (FE, Jean Luc Biampata (JB) and Jean Jacques Muyembe (JM) set up the study in DRC, enrolled and monitored the patients and recorded clinical data. MJ, SM, SJ, BS, Ibrahima Dieye (ID), Oscar Peterrson-Lomba (OPS) had access to the raw data. MJ, SM, SJ, BS, ID, OPL, Khaled Ezzedine (KE) carried out the analyses. MJ, SM, SJ, BS, RK, Moumouni Kinda (MK), KE and DM drafted the manuscript. All authors revised the manuscript critically for important intellectual content and approved the final version before submission.

Data sharing statement

The anonymized individual data and data dictionary for the study will be made available to other researchers by Professor Denis Malvy (denis.malvy@chu-bordeaux.fr) after a methodologically sound proposal has been approved and a data access agreement signed.

Declaration of interests

No conflict of interest is declared by any of the authors.

SM is listed as the inventor on the patent application for mAb 114, US Application No.62/087, 087 (PCT Application No.PCT/US2015/060733) related to anti-Ebola virus antibodies and their use.

Supplementary materials

Supplementary material associated with this article can be found in the online version at doi:10.1016/j. eclinm.2022.101699.


Salim Rostam

Limit shape for regularisation of large partitions under the Plancherel measure

A celebrated result of Kerov-Vershik and Logan-Shepp gives an asymptotic shape for large partitions under the Plancherel measure. We prove that when we consider e-regularisations of such partitions we still have a limit shape, which is given by a shaking of the Kerov-Vershik-Logan-Shepp curve. We deduce an explicit form for the first asymptotics of the length of the first rows and the first columns for the e-regularisation.

Introduction

Partitions of a given integer n are the different way to decompose n as an unordered sum of positive integers. In other words, a partition of n ∈ Z ≥0 is a non-increasing sequence λ = (λ 1 ≥ . . . ) of non-negative integers with sum n. This mathematical object appears for instance in the study of the symmetric group S n of permutations of {1, . . . , n}, since partitions of n index the conjugacy classes (via the cycle decomposition).

A representation of S n of dimension N over a field k is a group homomorphism ρ : S n → GL N (k). We focus for the moment at the case where k is the field C of complex numbers, in which case we say that we have a complex representation. As any integer decomposes into a product of primes, any complex representation decomposes into a sum of irreducible complex representations. It turns out that the fact that the set P n of partitions of n index the conjugacy classes implies that P n also index the set of irreducible complex representations. If we denote by ρ λ the irreducible complex representation associated with λ ∈ P n , a standard result of complex representation theory shows that:

#S n = n! = λ∈Pn (dim ρ λ ) 2 .
(1.1)

A remarkable fact is that we are able to explicitly compute the numbers dim ρ λ (namely with the famous hook length formula). We refer for instance to Sagan [Sa] or James-Kerber [JaKe] for more details on the complex representations of S n .

Now if we want to study the representations of S n over the field k = F p with p elements (with p a prime number; we will say that we have a p-representation), almost everything that is known for complex representations falls apart. A fundamental difference is that some representations may not decompose into a sum of irreducible ones. Nevertheless, the study of irreducible representations is still interesting since any representation can always be decomposed into irreducible constituents via a composition (or Jordan-Hölder) series.

A general theorem of Brauer says that the irreducible p-representations are in bijection with the p-regular conjugation classes, that is, with conjugation classes formed by elements whose order does not divide p. For the symmetric group, one can see that the set of p-regular conjugation classes is in one-to-one correspondence with the set of p-regular partitions, that is, partitions with no p (or more) consecutive equal parts. Note that the dual notion of p-restricted partitions is also present in the literature, where the difference between two consecutive parts of a p-restricted partition it at most p -1. If λ is a p-regular partition, we will denote by ρ p λ the associated irreducible p-representation. Contrary to the complex case, there is no formula expressing the dimension of ρ p λ (yet). We refer for instance to James-Kerber [JaKe] for more details on the p-representations of S n .

A way to understand p-irreducible representations is to study the p-irreducible representations that appear in the decomposition series of an irreducible complex representation reduced modulo p. More explicitly, it can be shown that any irreducible complex representation can be realised over Z, that is, we can assume that ρ λ : S n → GL N (Z). The reduction modulo p, that we denote by ρ p : S n → GL N (F p ), is then simply the reduction modulo p of the matrix entries. Now if we return to the problem of determining which irreducible p-representations appear in a decomposition series of ρ λ , James [Ja] gave an explicit combinatorial construction of a p-regular partition reg e (λ) such that ρ p λ appears in a decomposition series of ρ λ . This partition reg e (λ) is the p-regularisation of λ. The p-regularisation operation has in fact a meaning also when p is not prime, in the context of the Iwahori-Hecke algebra of S n (see, for instance, Mathas [Ma]). In particular, for an integer e ≥ 2 we will also use the terms e-regular and e-regularisation. Note that the e-regularisation map was recently generalised by Millan Berdasco [Mi] to an (e, i)-regularisation map on partitions: it is likely that our results generalise to this setting.

We now go back to (1.1). This equation shows that Pl n (λ) :

= (dim ρ λ ) 2 n!
is a probability measure on the set P n of partitions of n, called the Plancherel measure. Via some calculations involving the hook integral (defined in the spirit of the hook length formula), Kerov-Vershik [KeVe] and Logan-Shepp [LoSh] proved independently that there is a curve Ω so that the upper rim ω λ of the Young diagram (in the Russian convention) of a large partition λ converges uniformly in probability to Ω. An illustration of this convergence is given in Figure 1 (note 1 ). This limit shape theorem allowed to determine the first asymptotics of the length of the first row (and first column) of a Young diagram, taken under the Plancherel measure. Note that, via the Robinson-Schensted correspondence (which provides a bijective proof of (1.1)), this provides a solution to the Ulam problem on the length of a longest increasing subsequence of a word in S n chosen uniformly. We refer for instance to Romik [Rom] for more details on the Plancherel measure and related asymptotics.

In the previous context of regularisation of partitions, the following question in thus natural: for an integer e ≥ 2, what can be said about the partition reg e (λ) when λ is a large partition taken under the Plancherel measure? The aim of this paper is to give a first answer to this question. More precisely, if K is a compact convex subset of R 2 then its symmetrisation with respect to the line L is the set S L (K) that we obtain after sliding the different slices of K (with respect to the orthogonal direction of L) until the midpoint of the slice is on L. An example of symmetrisation with respect to a vertical line in given in Figure 2. The link with the isoperimetric problem is that S L (K) has the same area as K but a smaller perimeter. Note the following result, known as the sphericity theorem of Gross: from any compact convex subset of R 2 one can obtain the unit disk after a succession of (possibly infinite number of) symmetrisations. Note that this concept of symmetrisation can also be extended to maps, with for example Schwarz symmetric rearrangment, which is a powerful tool to study functional inequalities. We refer for instance to Gruber [Gr] or Krantz-Parks [KrPa] for more details on the Steiner symmetrisation.

A variation of Steiner symmetrisation is the notion of shaking. This notion was first introduced by Blaschke [Bl] to solve Sylvester's "four points problem". The difference with Steiner symmetrisation is that we slide the slices until we meet L. An example is given in Figure 2. Shaking and Steiner symmetrisation share many properties, for instance, Gross theorem holds with "unit disk" replaced by "simplex". We refer to [CCG] for fore details on the shaking operation. We will in fact use the shaking operation in a context where the slices are not orthogonal to the line L; such an operation is for instance used in [FrLu].

We can now state our main result (Theorems 4.7 and 6.1).

Theorem A. Let e ≥ 2. Under the Plancherel measure Pl n , the upper rim ω reg e (λ) of the Young diagram (in the Russian convention) of reg e (λ) converges uniformly in probability to the shape Ω α as n → +∞, in other words, for any > 0 we have

Pl n sup R ω reg e (λ) -Ω α > n→+∞ -----→ 0.
The shape Ω α is obtained by shaking the part Y(Ω) of the graph of Ω that is above the graph of the absolute value, with respect to the line of equation y = -x and angle α := 1 -2e -1 , and is given by:

Ω α (x) = Ω(x), for all x ≤ x - α , Ω α (x + δ x ) = Ω(x) + αδ x , for all x ∈ (x - α , x + α ), Ω α (x) = x,
for all x ≥ s α , where:

• x + α = Ω -1 (α) ∈ [0, a), • δ x = (1 -α) -1 F α (x) -F -1 α F α (x) , where F α : x → Ω(x) -αx and F -1 α denotes its inverse on (x + α , a), • x - α ∈ (-a, x + α ) (is the unique point that) satisfies F α (x - α ) = (1 -α)a, • (1 -α)s α = F α (x + α ) = Ω Ω -1 (α) -αΩ -1 (α) ∈ Ω(0) -αΩ(x + α )
, Ω(0) . We illustrate the convergence in Figure 3 and compare the limit shapes Ω 0 and Ω. The fact that Ω 0 is obtained via the horizontal shaking of Y(Ω) (with respect to the line of equation y = -x) is highlighted in Figure 4. In the particular case e = 2, the statement of Theorem A becomes explicit (Corollary 4.14 and Theorem 6.1).

Corollary B.

Under the Plancherel measure Pl n , the upper rim ω reg 2 (λ) of the Young diagram (in the Russian convention) of reg 2 (λ) converges uniformly in probability to the shape Ω 0 given by:

Ω 0 (x) = Ω(x),
for all x ≤ -2,

Ω 0 2x + Ω(x) = Ω(x), for all x ∈ (-2, 0), Ω 0 (x) = x,
for all x ≥ Ω(0). Figure 4: The maps Ω 0 (in blue) and Ω (in red). In green the tangent for Ω at 0. The left magenta segments are what is "added" to the red curve, and comes from the right magenta segments by shaking Y(Ω).

We now give the outline of the paper. In Section 2 we recall the definition of the e-regularisation map on the set of partitions of n and we state the limit shape result of Kerov-Vershik and Logan-Shepp for large partitions taken under the Plancherel measure. The short Section 3 is devoted to the (non-orthogonal) shaking operation on compact subsets of R2 . Section 4 contains the first main result of the paper, Theorem 4.7, which compute the shape associated with the shaking of the graph of a convex function (more precisely, the part of the graph that is above the graph of the absolute value). We give in Corollary 4.14 a particular case when the shaking is made by horizontal slices, where the result is explicit. In Section 5 we introduce two approximations ρ ± (λ) for the upper rim ω λ of the Young diagram (in the Russian convention) of a partition λ, in order to facilitate the shakings computations. More precisely, in §5.2 we define the outer regularisation ρ + (λ) of an e-regular partition λ (Definition 5.4) and in Proposition 5.5 we prove that ρ + (λ) is both close to ω λ and that its graph ρ + (λ) is stable under the shaking operation. A similar construction is made in §5.3 for the inner regularisation ρ -(λ). Finally, Section 6 is devoted to the proof of the second main result, Theorem 6.1, which states that the (rescaled) upper rim ω reg e (λ) of the Young diagram (in the Russian convention) of the e-regularisation of a partition taken under the Plancherel measure converges uniformly in probability to the shape defined in Theorem 4.7. The proof uses the results of Sections 4 and 5. We also prove the convergence of the support of ω λ -| • | (Theorem 6.4), and we deduce the asymptotic length of the first row and the first column of reg e (λ) (Corollary 6.5).

Regularisation

A partition is a finite non-increasing sequence λ = (λ 1 ≥ • • • ≥ λ h > 0) of positive integers. If |λ| := h i=1 λ i = n then we say that λ is a partition of n. We denote by P n the set of partitions of n. The Young diagram of a partition λ = (λ 1 ≥ • • • ≥ λ h > 0) is the subset of Z 2
≥1 given by:

Y(λ) = (a, b) ∈ Z 2 ≥1 : 1 ≤ a ≤ h and 1 ≤ b ≤ λ a .
Note that we will consider that the a-coordinate in the Young diagrams goes downwards. A node is an element of Z 2 ≥1 . Example 2.1. The Young diagram of the partition λ = (4, 4, 1) is . The node in blue has coordinates (1, 2).

Definition 2.2. Let e ≥ 2. A partition λ = (λ 1 , . . . , λ h ) is e-regular if no parts repeat e times or more, that is, if λ i > λ i+e-1 for any i ∈ {1, . . . , h -e}.
As we mentioned in the introduction, the notion of e-regular partitions appears for instance in the context of the representation theory of the Iwahori-Hecke algebra H q (S n ) of S n (a certain deformation of CS n ), with q ∈ C × having order e, see, for instance, [Ma] (note that in [Ma] the dual notion of e-restricted partition is used). The next two definitions are due to James [Ja].

Definition 2.3. Let e ≥ 2.
• The e-ladder number (or simply ladder number)

of a node γ = (a, b) ∈ Z 2 ≥1 is: lad e (γ) := a + (e -1)(b -1) ∈ Z ≥1 .
• Let ≥ 1. The (e, )-th ladder (or simply -th ladder) is the (finite) set of all nodes of Z 2 ≥1 with e-ladder . The e-ladder of a node γ is the (e, lad e (γ))-th ladder. Example 2.4. In the Young diagram of λ = ( 4,4,3,3,3,3,3,1 Note that reg e (λ) is an e-regular partition, and if λ is e-regular then reg e (λ) = λ. As we have mentioned in the introduction, the e-regularisation map has a significance in terms of modular representation theory of the symmetric group (or its associated Iwahori-Hecke algebra).

Example 2.6. The 4-regularisation of the partition of Example 2.4 is (5,4,4,3,3,2,2,1). The

• for each k ∈ Z we have ω λ | (k,k+1) = ±1, (2.7) • we have ω λ (x) = |x| for |x| 0 (more precisely, for x ≤ -λ 1 or x ≥ h),
• we have R ω λ (x) -|x| dx = 2n. An illustration of the construction of ω λ is given in Figure 5. (We warn the reader that in the literature the convention is sometimes reversed, that is, our ω λ is sometimes reflected with respect to the axis {0} × R.) We will use a particular rescaling ω λ : R → R of ω λ , given by ω λ (s) := 1

< t(a + 1, b) if (a + 1, b) ∈ Y(λ) and t(a, b) < t(a, b + 1) if (a, b + 1) ∈ Y(λ).
We denote by Std(λ) the set of standard Young tableaux of shape λ.

We have the following standard identity:

n! = λ∈Pn #Std(λ) 2 .
Definition 2.8. The Plancherel measure on the set P n of the partitions of n is given by

Pl n (λ) := #Std(λ) 2 n! ,
for all λ ∈ P n .

The next result describes the Plancherel measure Pl n for large n.

Theorem 2.9 ( [LoSh], [KeVe], [START_REF] Romik | The Surprising Mathematics of Longest Increasing Subsequences[END_REF]Theorem 1.26]). Let Ω : R → R be defined by

Ω(s) := 2 π s arcsin( s 2 ) + √ 4 -s 2 , if |s| ≤ 2, |s|, otherwise.
Then, under the Plancherel measure Pl n , the function ω λ converges uniformly in probability to Ω as n → +∞. In other words, for any > 0 we have

Pl n sup R ω λ -Ω > n→+∞ -----→ 0.
Moreover, we also have convergence of the supports, that is:

inf s ∈ R : ω λ (s) = |s| -→ -2,
and sup s ∈ R : ω λ (s) = |s| -→ 2,
in probability under Pl n as n → +∞.

An illustration of Theorem 2.9 for n = 1000 is given in Figure 1. Note that the limit shape Ω has a much simpler form after derivation.

Lemma 2.10. The map Ω is an antiderivative on R of:

s → arcsin s 2 , if |s| ≤ 2, sgn(s), if |s| > 2.
In particular, the minimum of

Ω on R is Ω(0) = 4 π .

Shakings

The aim of this paper is to put together the notions of §2.1 and §2.2. As a first step, we show in Figure 6 what do the 4-ladders of Example 2.4 look like in the Russian convention.

Proposition 3.1. In the Russian convention, the e-regularisation makes the nodes going as left as possible in the direction of y = 1 -2e -1 x.

Proof. In the Young diagram we know that the node (e, 1) ∈ Z 2 ≥1 goes to (1, 2). In the Russian convention it means that the box with top vertex at (e -1, e + 1) goes to the box with top vertex (-1, 3). The corresponding slope is thus e+1-3 e-1-(-1) = e-2 e = 1 -2e -1 as announced (note that the slope do not change when both axes are rescaled by a same constant).

We now define the shaking operation that will be of interest for us. As we mentioned in the introduction, this is a variation of Steiner symmetrisation first introduced by Blaschke [Bl]. If K is a compact subset of R 2 , its shaking with direction α (against y = -x) is the compact subset Sh α (K) of R 2 that we obtain by sliding the set K along the direction of (L α ) : y = αx until we meet the line (L) : y = -x, that is:

Sh α (K) := x∈L K v ,
where K v is the segment with extreme points x and x

+ K ∩ (x + L α ) v α with | • | denoting the one-dimensional Lebesgue measure.
In the usual definition of shaking the direction L α is orthogonal to the the line L against which we shake. This non-orthogonal variation is for instance considered in [START_REF] Freyer | Interpolating between volume and point enumerator with successive minima[END_REF]§5].

Example 3.3. Take α = 0 and let K be the unit square with bottom left corner at (1, 1). Then, as shown on Figure 7, the set Sh α (K) is the parallelogram with vertices (-2, 2), (-1, 2), (0, 1) and (-1, 1).

The following property is a standard property of Steiner symmetrisations and shakings. The proof is immediate from the definition.

Proposition 3.4. If K ⊆ K are compact subsets of R 2 then Sh α (K) ⊆ Sh α (K ).
Proof. See [START_REF] Freyer | Interpolating between volume and point enumerator with successive minima[END_REF]Lemma 5.3]. Note that the proof is in fact the same as for the usual (orthogonal) shakings.

Recalling Proposition 3.1, we are interested in special cases of shakings. Definition 3.5. For any e ≥ 2, we denote by sh e the shaking Sh α with direction α e := 1 -2e -1 .

Shaking the limit shape

Let a ∈ R >0 . We define Y a to be the set of all functions f : R → R of class C 1 such that:

f is even, (4.1a)
for any |s| ≥ a we have f (s) = |s|, (4.1b)

f is strictly convex on [-a, a]. (4.1c)
Note that by Lemma 2.10 we have Ω ∈ Y 2 .

Example 4.2. Let Σ : R → R be given by Σ

(x) = |x| if |x| ≥ 1 and Σ(x) = 1 2 (x 2 + 1) otherwise. Then Σ ∈ Y 1 .
Note that the last two equations in (4.1) imply that for all |s| < a we have:

f (s) > |s|, (4.3a)
since f is above its tangent at s = a, and:

|f (s)| < 1, (4.3b)
for all s ∈ (-a, a), since f is strictly increasing on [-a, a].

Before studying the shaking operation on the elements of Y a , we first define a map between some sets Y a via double scaling. Definition 4.4. Let f : R → R be any map and let ∈ (0, 1). We define the two maps f ± : R → R by, for any s ∈ R,

f ± (s) := (1 ± 2 ) f s 1 ± 2 .
For instance, the maps Ω ±1 are as follows:

Ω

Ω +1 Ω -1
We now gather some informations about f ± .

Lemma 4.5. Let f ∈ Y 2 and ∈ (0, 1).

(i) We have f ± ∈ Y 2± . (ii) For any s ∈ R we have (f ± ) (s) = f s 1± 2 .
(iii) The maps g + := f + -f and g -:= f -f -are decreasing on R ≥0 .

(iv) For any s ∈ R we have f -(s) ≤ f (s) ≤ f + (s), and:

(a) we have f -(s) = f (s) if and only if |s| ≥ 2, (b) we have f + (s) = f (s) if and only if |s| ≥ 2 + , (v) We have f -f ± ∞ = 2 f (0).
Proof. (i) Clear.

(ii) Clear.

(iii) Follows from the previous point and from the fact that f is increasing.

(iv) Since the functions are even, it suffices to prove the inequality on R ≥0 . We prove only the relations for f + , the ones for f -being similar. By the previous point, we know that g + is decreasing on R ≥0 , thus we deduce that g ≥ 0 since g(s) = |s| -|s| = 0 for s 0. For the equality case, first note that the necessary condition holds. For the sufficient condition, since f is strictly convex on [-2, 2] we have in fact g (s) < 0 for all s ∈ (0, 2] thus as before we have g(s) > 0 for all s ∈ [0, 2]. Now for s ∈ [2, 2 + 2 ) we have g (s) = f s 1+ -1 < 0 since 1 = f (2) and f is strictly increasing on [0, 2] and we conclude the proof as before.

(v) Again we only prove the result for f + . With the previous notation, we know that g = f + -f is non-negative and decreases on R ≥0 thus its maximum is reached at s = 0.

We need one more definition before giving the main result of this section.

Definition 4.6. Let f : R → R be a map. We denote by Y(f ) the part of the graph of f that is above the graph of the absolute value, that is:

Y(f ) := {(x, y) ∈ R 2 : |x| ≤ y ≤ f (x)}.
For instance, we picture here in blue the set Y(Ω):

Theorem 4.7. Let α ∈ [0, 1) and a ∈ R >0 . For any f ∈ Y a we have:

Sh α Y(f ) ∪ Y(id R ) = Y(f α ),
where f α : R → R is given by:

f α (x) = f (x), for all x ≤ x - α , f (x + δ x ) = f (x) + αδ x , for all x ∈ (x - α , x + α ), f α (x) = x,
for all x ≥ s α := x + α + δ x + α , where: Proof. As in Definition 3.2, let L α be the line of equation y = αx. For M ∈ R 2 , the line L M α := M + L α cuts the curve of f at most twice since f is strictly convex. Note that we can ignore the parts of Y(f ) outside (-a, a) since it has measure 0 there. Note also that there is only one intersection point if and only if Y(f ) ∩ L M α is stable under Sh α . Moreover, it suffices to choose M of the form (0, y M ), and if y M0 is the unique real number such that L M0 α contains (a, a) then: for any y ≤ y M0 then L M α intersects the curve of f in (-a, a) only once. Let x - α < a be such that L M0 α intersects the curve of f at x - α , f (x - α ) . This means that:

• x + α = f -1 (α) ∈ [0, a), • δ x = (1 -α) -1 F α (x) -F -1 α F α (x) , where F α : x → f (x) -αx and F -1 α denotes its inverse on (x + α , a); • x - α ∈ (-a, x + α ) (is the unique point that) satisfies F α (x - α ) = (1 -α)a. Note that Y(id R ) = {(x, x) : x ∈ R ≥0 }. y = Σ(x) x + α x - α x x+δx x x +δx
f α (x) = f (x), for all x ≤ x - α .
Note that x - α ≥ -a since f (-a) = a and α ≥ 0. Now there is also a real number y M1 such that L M1 α does not intersects the curve of f apart from y M1 , f (y M1 ) . Again by convexity, this y M1 is given by the unique point such that L M1 α is tangent to the curve of f at some point M α = x + α , f (x + α ) . By (4.1) we know that such an M α is unique and is determined by f (x + α ) = α, in other words, by: x + α = f -1 (α) (where f -1 denotes the inverse of f | (-a,a) ). Note that x + α ≥ 0 since α ≥ 0 and f is even, thus f (0) = 0 ≤ α, and f is increasing by strict convexity. Similarly, we have x + α < a since f (a) = 1 > α, moreover the line between x + α , f (x + α ) and (a, a) has slope > α thus

x + α > x - α . Hence, to determine f α , it suffices to determine for each x ∈ (x - α , x + α ) the measure m of (M + L α ) ∩ Y(f ) ∩ (x, a) × R ≥0 , with M = x, f (x)
, and to add it to f (x) in the direction α. More precisely, if:

• x > x is the unique other abscissa where L x α := M + L α crosses the curve of f , • δ x > 0 is the unique positive real number such that x + δ x is the abscissa of the intersection between L x α and the line with equation y = x for x ≥ 0, then the point x, f (x) becomes:

x + δ x , f (x) + x + δ x -f (x ) .
(4.8)

In Figure 8 we give an example of the various quantities that we have introduced so far for f being the map Σ of Example 4.2.

We will now determine x and δ x as functions of x. We first determine x . The line L x α has equation (s -x)α + f (x) for s ∈ R, thus x is given by:

f (x ) = (x -x)α + f (x), thus: f (x ) -αx = f (x) -αx. (4.9)
Now the function F α : s → f (s) -αs has derivative f (s) -α, thus since α ∈ (0, 1) we know that s → f (s) -αs is decreasing for s ≤ x + α and increases (to +∞ by (4.1b)) by for s ≥ x + α . Hence, we deduce that there is indeed a unique x > x + α such that (4.9) is satisfied. Note that F α is invertible on (x + α , +∞) and if F -1 α denotes its inverse on this interval then:

x = F -1 α F α (x)).
Note that (4.9) writes:

F α (x ) = F α (x).
Note also that x < x + α < x so that x = x. Finally, note that x - α satisfies:

F α (x - α ) = F α (a) = f (a) -αa = (1 -α)a,
as announced.

We now determine δ x . First note that the equation of L x α becomes αs + F α (x ). We thus find that δ x satisfies (4.11) in particular (4.10) gives:

x + δ x = α(x + δ x ) + F α (x ), (4.10) = f (x ) + αδ x ,
x + δ x = (1 -α) -1 F α (x ).
(4.12)

Recalling (4.9) and (4.11), the second coordinate in equation (4.8) becomes:

f (x) + x + δ x -f (x ) = f (x) + αδ x ,
as announced.

To prove that the set of points that we obtain is the curve of a function, it suffices to prove that x → x + δ x is increasing on (x - α , x + α ), thus it suffices to prove that x → δ x is increasing. By (4.12) we have: a). But this is clear since the derivative of g is, after multiplication by 1 -α,

δ x = (1 -α) -1 F α (x ) -x . Note that x → x = F -1 α F α (x) is decreasing on (x - α , x + α ). Indeed, we have F α (x) = f (x)-α that is increasing since f is strictly convex, and by definition f (x + α ) = 0 thus F α (x) < 0 for x < x + α thus F α is decreasing on (x - α , x + α ). Now F α is increasing on (x + α , +∞) thus F -1 α is also increasing. By composition, we deduce that x → x = F -1 α F α (x) is decreasing on (x - α , x + α ), with values in (x + α , a). Hence, to prove that x → δ x = (1 -α) -1 F α (x ) -x is increasing on (x - α , x + α ), it suffices to prove that g : t → (1 -α) -1 F α (t) -t is decreasing in t ∈ (x + α ,
F α (t) -(1 -α) = f (t) -α -(1 -α) = f (t) -1,
which is strictly negative since f is increasing (by strict convexity) and f (t) < 1 for all t < a by (4.1b).

In Figure 4 we represent the curve of f α for f = Ω and α = 0.

Proposition 4.13. Let α ∈ [0, 1) and a ∈ R >0 . For any f ∈ Y a we have:

(1 -α)s α = f f -1 (α) -αf -1 (α) ∈ f (0) -αf (x + α ), f (0) .
Proof. Recall that from Theorem 4.7 and its proof that s α = x + δ x for x = x + α . In this setting we have x = x thus (4.12) gives:

(1 -α)s α = F α (x + α ) = f (x + α ) -αx + α = f f -1 (α) -αf -1 (α),
which gives the desired equality. Now by the mean value theorem, there exists c ∈ (0,

x + α ) such that F α (x + α ) = F α (0) + F α (c)x + α .
Hence, since f is strictly convex we know that F α is also strictly convex, thus we have

F α (0) = f (0) -α = -α < F α (c) < F α (x α ) = 0 (where f (0) = 0 since f is even). Recalling that F α (0) = f (0)
gives the announced inequality.

We have the following particular case of Theorem 4.7 and Proposition 4.13 for α = 0 (as pictured in Figure 4). Corollary 4.14. Let a ∈ R >0 . For any f ∈ Y a , the map f 0 : R → R is given by:

f 0 (x) = f (x),
for all x ≤ -a,

f 0 2x + f (x) = f (x),
for all x ∈ (-a, 0),

f 0 (x) = x, for all x ≥ f (0).
We conclude this part by relating the shapes f α and f ± α := (f ± ) α . Proposition 4.15. Let α ∈ [0, 1) and a ∈ R >0 . Let f ∈ Y a and ∈ (0, 1). With the notation of Theorem 4.7 we have:

(f ± ) α = (f α ) ± =: f ± α .
Proof. We denote by δ ± x and F ± α the quantities that appear in Theorem 4.7 for f ± . Note that:

F ± α (x) = f ± (x) -αx = (1 ± 2 )f x 1 ± 2 -(1 ± 2 )α x 1 ± 2 = (1 ± 2 )F α x 1 ± 2 = (F α ) ± (x),
and we deduce that (F ± α ) -1 = (F -1 α ) ± . We deduce that:

δ ± x = (1 ± 2 )δ x 1± /2 . (4.16)
Now note that the equality f (x + δ x ) = f (x) + αδ x holds in fact for any x ∈ R. If g denotes the inverse of x → x + δ x on R (which exists since x → x + δ x is strictly increasing by the proof of Theorem 4.7), if y = x + δ x we thus have x = g(y) and thus δ x = y -g(y). We deduce that:

f α (y) = f g(y) -α y -g(y) = F α g(y) -αy. (4.17)
Now we deduce from (4.16) that the inverse of x → x + δ ± x is g ± (in the sense of Defini-tion 4.4), thus by (4.17) we obtain:

f ± α (y) = F ± α g ± (y) -αy = (1 ± 2 )F α g y 1 ± 2 -αy = (1 ± 2 ) F α g x 1 ± 2 -α y 1 ± 2 = (1 ± 2 )f α y 1 ± 2 = (f α ) ± (y), which is the announced result. Proposition 4.18. Let α ∈ [0, 1) and a ∈ R >0 . Let f ∈ Y a . We have 0 < f α -f ± α ∞,R → 0 as → 0.
Proof. First, note that the infinite norm is positive indeed since f α and f ± α are continuous (even of class C 1 ) by (4.17), by Lemma 4.5 and since the shaking preserve the area (this is a simple consequence of Fubini theorem, see, for instance, [START_REF] Freyer | Interpolating between volume and point enumerator with successive minima[END_REF]Lemma 3.2]). To prove that is goes to 0 as → 0, it suffices to prove that f α is Lipschitz. Indeed, if h : I → R is L-Lipschitz and bounded then:

h ± (x) -h(x) = (1 + 2 )h x 1 + 2 -h(x) = h x 1 + 2 -h(x) + 2 h x 1 + 2 , thus: h + (x) -h(x) ≤ L|x| 1 1 + 2 -1 + 2 h ∞ ≤ (L|x| + h ∞ ) ,
with a similar calculation for h -. Note that it suffices to study f -f α on a segment since this function vanishes for |x| 0. Recalling (4.17), we have:

f α (y) = g (y)F α g(y) -α = g (y) f g(y) -α -α.
We know that f is bounded by (4.1). For g , recall from the proof of Theorem 4.7 that x → δ x is increasing thus the derivative of x → x + δ x is greater than 1, thus g ∞ ≤ 1. This concludes the proof.

Shaking partitions

Let λ be a partition. Proposition 3.1 implies the following result.

Lemma 5.1. We have sh e Y(ω λ ) = sh e Y(ω reg e (λ) ) and sh e Y( ω λ ) = sh e Y( ω reg e (λ) ) .

Hence, to study the shaking operations on partitions it suffices to study the shaking operations on regular partitions. In fact, the shaking of Y( ω reg e (λ) ) for λ an e-regular partition is a bit delicate to determine. Instead, we will bound the latter graph by two close graphs that are stable under the shaking operation.

x y y = ω λ (x) Figure 9: Outer corners (in red) and inner corners (in blue) for λ = (4, 4, 2, 1)

Corners

We say that c ∈ Z is an outer corner (resp. inner corner) of λ if ω λ | (c-1,c) = 1 (resp. -1) and ω λ | (c,c+1) = -1 (resp. 1). We give in Figure 9 an example of outer and inner corners.

The following standard result follows from the definition of inner and outer corners and from the fact that ω λ (x) = |x| for x 0.

Lemma 5.2. Let λ be a partition.

If {o 1 < • • • < o k } (resp. {i 1 < • • • < i })
is the set of outer (resp. inner) corners then = k + 1 and i m < o m < i m+1 for all m ∈ {1, . . . , k}.

In the sequel, fix e ≥ 2 and let L e be the linear subspace of R 2 with slope α e = 1 -2e -1 .

Outer regularisation

(c k-1 , c k ) we have 0 ≤ ρ + k (λ)(s) -ρ + k-1 (λ)(s) ≤ e.
Proof. Since α e < 1, the curve of ω λ can cross L (k) e

only after an inner corner, that is, if c < c k is the inner corner preceding c k (cf. Lemma 5.2) then h < c . Moreover, by definition of an inner corner we have ω λ (c -1) = ω λ (c ) + 1, thus the line joining c k , ω λ (c k ) and c -1, ω λ (c -1) has slope:

α := ω λ (c k ) -ω λ (c ) -1 c k -c + 1 . Since ω λ (c ) = ω λ (c k ) -c k + c , we obtain: α = c k -c -1 c k -c + 1 = 1 - 2 c k -c + 1
.

But now λ is e-regular thus c ≥ c k -e + 1. We thus have c k -c + 1 ≤ e and thus, since

c k -c + 1 > 0 (since c ≤ c k ): α ≤ 1 -2e -1 = α e .
Finally, we have α ≤ α e < 1 thus c -1 ≤ h k < c k , in other words, the line L (k) e meet the curve of ω λ somewhere between the points of abscissa c (excluded) and c -1 (included) thus this concludes the first part of the proof. Now for any s ∈ (c k-1 , h) we have ρ + k (λ)(s) = ρ + k (λ)(s), and by construction for any s ∈ (h, c k ) we have ρ + k (λ)(s) > ρ + k (λ) thus we obtain the first member of the announced inequality. The second one follows from the fact that α e ∈ [0, 1] and from the fact that on (c k-1 , c k ), the function s → ρ + k (λ)(s) -ρ + k (λ)(s) reaches its minimum at s = h k , where:

ω λ (c k ) ≥ ρ + k (λ)(h k ) ≥ ρ + k-1 (λ)(h k ) ≥ ω λ (c ).
Now we already saw that:

ω λ (c ) = ω λ (c k ) -c k + c , c k -c + 1 ≤ e,
thus we obtain:

ω λ (c k ) -ω λ (c ) = c k -c ≤ e -1,
whence the result.

Definition 5.4. Let e ≥ 2 and let λ be an e-regular partition. Let r ≥ 0 be the number of outer corners of λ. We define ρ + (λ) := ρ + r (λ). Note that, by Proposition 5.3, the map ρ + (λ) does not in fact depend on the order that we chose on the outer corners. An example of a map ρ + (λ) is given in Figure 10. (i) For all s ∈ R we have 0 ≤ ρ + (λ) -ω λ (s) ≤ e.

(ii) The graph G + λ := Y ρ + (λ) is stable under the shaking operation with slope α e , that is, we have sh e (G + λ ) = G + λ .

Proof. The first point follows directly from Proposition 5.3 since the intervals (c k-1 , c k ) are pairwise disjoints. For the second point, it follows from the simple observation that the slopes of ρ + (λ) are either -1 or α e (except the part ρ + (λ)(s) = s for s 0). As a consequence, if for x ∈ R the line L := x + L e intersects the curve of ρ + (λ) at a point s ∈ R then L remains below the curve ρ + (λ) on (-∞, s). Observing that L intersects the curve of | • | on (-∞, 0) since α e < 1 gives the result.

Inner regularisation

We now define a similar construction as ρ + (λ) but for inner corners. We give the statements without proofs since they are entirely similar.

i) For all s ∈ R we have -e ≤ ρ -(λ) -ω λ (s) ≤ 0. (ii) The graph G - λ := Y ρ -(λ)
is stable under the shaking operation with slope α e , that is, we have sh e (G - λ ) = G - λ .

Theorem 6.1. Let e ≥ 2 and take α = α e = 1 -2e -1 . Under the Plancherel measure Pl n , the function ω reg e (λ) converges uniformly in probability to Ω α as n → +∞. In other words, for any > 0 we have

Pl n sup R ω reg e (λ) -Ω α > n→+∞ -----→ 0.
Proof. Let > 0. Recalling the notation of Definition 4.4, define:

m ± := inf |Ω ± (s) -Ω(s)| : |s| ≤ 2 ± 2 .
By Lemma 4.5 applied with f = Ω, we have m ± > 0. Now let M n be the set of partitions λ ∈ P n such that:

ω λ -Ω R < min(m + , m -), inf{s ∈ R : ω(s) = |s|} > -2 -2 , sup{s ∈ R : ω(s) = |s|} < 2 + 2 .
Note that by Theorem 2.9 we have Pl n (M n ) → 1 as n → +∞. By Lemma 4.5, for all λ ∈ M n we have:

Ω -(s) ≤ ω λ (s) ≤ Ω + (s),
for all s ∈ R. In other words, for all λ ∈ M n we have:

Y Ω -⊆ Y ω λ ⊆ Y Ω + .
By Proposition 3.4, Theorem 4.7 together with Lemma 5.1, we deduce that:

Y Ω - α ⊆ sh e Y ω reg e (λ) ⊆ Y Ω + α . (6.2)
In particular, note that we can apply Theorem 4.7 for Ω ± indeed by Lemma 4.5, and Ω ± α := (Ω ± ) α . Now by Propositions 5.5 and 5.8, for all s ∈ R we have: Thus, we conclude that for any s ∈ R we have:

Ω - α (s) ≤ ρ + (λ)(s), and ρ -(λ)(s) ≤ Ω + α (s).

We deduce that for any s ∈ R we have:

ρ -(λ)(s) ≤ Ω + α (s) ≤ Ω - α (s) + Ω - α -Ω + α ∞ ≤ ρ + (λ)(s) + Ω - α -Ω + α ∞ ≤ ρ -(λ)(s) + 2e √ n + Ω - α -Ω + α ∞
by Propositions 5.5 and 5.8. Thus, again using Proposition 5.8, we deduce that for any s ∈ R we have:

ω reg e (λ) (s) -Ω α (s) ≤ ω reg e (λ) (s) -ρ -(λ)(s) + ρ -(λ)(s) -Ω + α (s) + Ω + α (s) -Ω α (s)

≤ 2e √ n + 2e √ n + Ω - α -Ω + α ∞ + Ω + α -Ω α ∞ , whence: sup R ω reg e (λ) -Ω α ≤ 4e √ n + 3 Ω + α -Ω α ∞ .
Thus, by Proposition 4.18 deduce that for n 0 we have:

sup R ω reg e (λ) -Ω α ≤ 4 Ω + α -Ω α ∞ .
Recalling that λ ∈ M n with Pl n (M n ) → 1 as n → +∞, we deduce that:

Pl n sup R ω reg e (λ) -Ω α > 4 Ω + α -Ω α ∞ n→+∞ -----→ 0,
which, again by Proposition 4.18, proves the desired convergence.

We illustrate the convergence of Theorem 6.1 for e = 2 in Figure 12 and for e ∈ {3, 4} in Figure 13, with in red the usual limit shape Ω and in blue the map ω reg e (λ) for λ a partition of 3 × 10 4 taken under the Plancherel measure. Proof. We use the same set M n and the same strategy as in the proof of Theorem 6.1.

Finally, we deduce the asymptotic behaviour of the first part and the first column. π sin π e . Proof. The first point is clear by Theorem 6.4. For the second one, by the same theorem it suffices to prove that the announced value is equal to s α . Proposition 4.13 applied for f = Ω we have:

s α = (1 -α) -1 Ω Ω -1 (α) -αΩ -1 (α) .
Recall that α = 1 -2e -1 and Ω (s) = 2 π arcsin( s 2 ). We have Ω (s) = t ⇐⇒ arcsin( s 2 ) = πt 2 ⇐⇒ s = 2 sin πt 2 thus Ω -1 (t) = 2 sin πt 2 . We thus have: We obtain:

Ω -1 (α) =
s α = e 2 Ω Ω -1 (α) -αΩ -1 (α) = e 2 Ω

Figure 1 :Figure 2 :

 12 Figure 1: The limit shape theorem for large partitions under the Plancherel measure 1 All the computations were made using SageMath [SM].

Figure 3 :

 3 Figure 3: Example of 2-regularisation of a large partition, with in green the limit shape Ω 0 of Corollary B and in red the limit shape Ω of Kerov-Vershik-Logan-Shepp Despite the fact that Theorem A is not fully explicit, we are able to deduce the asymptotic behaviour of the length of the first line and of the first column of reg e (λ) (Corollary 6.5). Corollary C. Let e ≥ 2. Under the Plancherel measure Pl n : (i) the rescaled size 1 √ n reg e (λ) 1 of the first row of reg e (λ) converges as n → +∞ in probability to 2; (ii) the rescaled size 1 √ n reg e (λ) 1 of the first column of reg e (λ) converges as n → +∞ in probability to 2e π sin π e .

  Let e ≥ 2 and let λ be a partition. The e-regularisation of λ is the partition reg e (λ) that we obtain after moving each node of Y(λ) as high as possible in its e-ladder.

Figure 5 :

 5 Figure 5: Russian convention for the Young diagram of λ = (4, 4, 2, 1)

√

  n ω λ s √ n . Note that the area between the graphs of ω λ and | • | is 2. Plancherel measure Let λ ∈ P n . A standard tableau of shape λ is a bijection t : Y(λ) → {1, . . . , n} such that t increases along the rows and down the columns, in other words for (a, b) ∈ Y(λ) we have t(a, b)

Figure 7 :

 7 Figure 6: 4-ladders in the Russian convention for λ =(4, 4, 3, 3, 3, 3, 3, 1)

Figure 8 :

 8 Figure 8: Illustration of the proof of Theorem 4.7 with f = Σ and α = 1 4

  If λ has no outer corners then we define ρ + (λ) = ρ + 0 (λ) := ω λ , otherwise let {c 1 < • • • < c r } with r ≥ 1 the set of outer corners. Assuming by induction on k ∈ {1, . . . , r} that we have constructed the piecewise linear function ρ + k-1 (λ) : R → R, then we construct the piecewise linear function ρ + k (λ) : R → R as follows. Let L (k) e be the affine line c k , ω λ (c k ) + L e . (i) For s ≥ c k then ρ + k (λ)(s) := ρ + k-1 (λ)(s); (ii) at s = c k , we follow the line L (k) e (in the negative direction) until we meet the curve of ρ+ k-1 (λ), at point of abscissa h k ; (iii) for s ≤ h k then ρ + k (λ)(s) := ρ + k-1 (λ)(s) again. Proposition 5.3. Assume that the partition λ is e-regular. With the above notation we have h k ∈ [c k-1 , c k ).In particular, the functions ρ + k-1 (λ) and ρ + k (λ) coincide on (at least) R \ (c k-1 , c k ), and for any s ∈

Figure 10 :

 10 Figure 10: The map ρ + (λ) (in thick red) for the 3-regular partition λ = (4, 4, 2, 1). In green are the different lines L (k) e .

Figure 11 :

 11 Figure 11: The map ρ -(λ) (in thick blue) for the 3-regular partition λ = (4, 4, 2, 1). In green are the different lines L (k) e .

(

ρ

  -(λ)(s) ≤ ω λ (s) ≤ ρ + (λ)(s) (where ρ ± (λ)(s) = 1 √ n ρ ± (λ)( √ ns) is rescaled as ω λ , see §2.2), thus: Y ρ -(λ) ⊆ Y( ω reg e (λ) ) ⊆ Y ρ + (λ) .By Propositions 5.5 and 5.8 again and Proposition 3.4, we deduce that:Y ρ -(λ) ⊆ sh e Y( ω reg e (λ) ) ⊆ Y ρ + (λ) . (6.3)By (6.2) and (6.3), we deduce that:Y Ω - α ⊆ Y ρ + (λ) , and Y ρ -(λ) ⊆ Y Ω + .

Figure 12 :

 12 Figure 12: Example of 2-regularisation of a large partition, with in green the limit shape of Corollary B

Corollary 6. 5 .

 5 Let e ≥ 2. Under the Plancherel measure Pl n :

Figure 13 :

 13 Figure 13: Example of 3-and 4-regularisation of a large partition, respectively on the left and right.

Background on partitionsWe recall in §2.1 the combinatorial notion of e-regularisation of a partition. In §2.2 we recall the limit shape result of Kerov-Vershik and Logan-Shepp for large partitions under the Plancherel measure.

Limit shape for regularisation of large partitionsWe can now state our main theorem. Recall from Theorem 4.7 the definition of Ω α for α ≥ 0.

Acknowledgements

The author would like to thank François Bolley and Vincent Beck for useful discussions. The author also thanks the Centre Henri Lebesgue ANR-11-LABX-0020-0. This research was funded, in whole or in part, by the Agence Nationale de la Recherche funding ANR CORTIPOM 21-CE40-001. A CC-BY public copyright license has been applied by the author to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant's open access conditions.