Ahmed H Alamoud
email: ahmedhalamoud@gmail.com

Emmanuel Baillot

Camila Belabbas

Hadi S Samimi Ardestani

Hamid Bahai

Adrien Baldit

Mahmoud Chizari

Hadi S Samimi

Continuous and Pulsed Experiments with Numerical Simulation to Dissect Pituitary Gland Tumour by Using Liquid Jet

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Abstract-Endoscopic endonasal surgery is a minimal invasive surgery that has been used to dissect pituitary gland tumour via curettes with the help of endoscope. However, this type of surgery has a high risk of failure because curettes may cause damages to blood vessels and optical nerves that lead to more complication for the patient. The aim of this study is to develop a new technique to dissect the tumour by using liquid jet. A series of experimental tests have been performed on animal tissue to study the effect of liquid pressure and nozzle diameter on dissecting and cutting the tissue. Continuous / pulsed liquid jet used with variable nozzle diameters, distances, pressures and angles. The study concluded with promising results on liquid jet to dissect hypophysis tumour and preserving fine blood vessels and optical nerve located near the pituitary gland.

Index Terms-Endoscopic endonasal surgery, tumour removal, liquid jet, continuous and pulsed jet I. INTRODUCTION ne part of the body that is affected by both type of tumour is the brain, which is a crucial region of the body that controls all its functions. According to the latest statistics collected by Cancer Research UK, there were 9365 new cases of brain cancer reported in 2011, where more than 50% of the new cases number dies every year [START_REF]Brain, other CNS and intracranial tumours Key Stats[END_REF]. From these numbers, around 8% of brain tumours are in the pituitary gland, which is a serious medical condition lead to death if not treated [START_REF]Statistics and outlook for brain tumours[END_REF], [START_REF]Types of primary brain tumours[END_REF]. In the United States, the number of patients diagnosed with pituitary gland tumour is around 10,000 patients each year, in which most of these lesions are benign [START_REF]What are the key statistics about pituitary tumors?[END_REF].

The skull base is divided into anterior, middle, and posterior regions that sited behind the eyes and nose, in which the tumours are categorized according to the involved area [START_REF] Hankinson | Skull Base Tumors: Background, History of the Procedure, Problem[END_REF]. Therefore, treating skull base tumours are difficult since they are very near to crucial blood vessels and nerves that can cause health issues [START_REF] Hopkins | What is a Skull Base Tumor?[END_REF]. In addition, when trying to remove it surgically, it may cause, for instance: infection, hematoma, cerebrospinal fluid leak, and the most significant problem is difficulty to reach the lesion in a safe, quick and complete access [START_REF] Hankinson | Skull Base Tumors: Background, History of the Procedure, Problem[END_REF]- [START_REF] Khanna | Skull base tumors: a kaleidoscope of challenge[END_REF]. One of the vital tumour in this region and can cause a severe health problem to the patient when not removed properly is hypophysis tumour, which is located in the skull base and affects the pituitary gland, also called pituitary gland tumour, which is responsible for secretion the main body hormones [START_REF] Hopkins | Pituitary Tumors[END_REF]. Thus, the tumour disorders the normal balance of hormones in the body that cause the patient to become sick [START_REF] Hopkins | Pituitary Tumors[END_REF]. Moreover, malignant type of this tumour affected only few patients [START_REF] Hopkins | Pituitary Tumors[END_REF]. Nevertheless, due to its location, the tumour develops upward, and consequently hypophysis lesion will cause vision problems when it reached and pressed on the optic nerve [START_REF] Hopkins | Pituitary Tumors[END_REF].

Surgery is considered the main treatment for hypophysis tumour and it depends on size, exact location and type of the tumour and whether the tumour is spread to neighbour tissues [START_REF]Surgery for pituitary tumors[END_REF]. Since the pituitary gland is located at the back wall of the sinus, the common technique selected for more than 90% of patients to remove pituitary tumour is transsphenoidal surgery, which is performed from the nose via opening the sphenoid sinus bony walls with small surgical instruments [START_REF]Surgery for pituitary tumors[END_REF]. Moreover, the current minimal invasive technique is done by using endoscope, which is a small camera attached at the tip of lighted thin fibre optic tube that allows the surgeon to do the operation with a small incision to reach the gland and remove the tumour [START_REF]Surgery for pituitary tumors[END_REF].

In recent years a few attempt has been made to remove tumour using liquid jet. Ogawa et al. recently studied on a new technique to remove the hypophysis tumour [START_REF] Ogawa | Pulsed laser-induced liquid jet for skull base tumor removal with vascular preservation through the transsphenoidal approach: a clinical investigation[END_REF]. This technique was applied by pulsed laser-induced liquid jet (LILJ) system that removes the tumour efficiently and safely without any harm to blood vessels and nerves. Moreover, a study has been published in 2015 by Nakagawa et al. regarding the safety of using LILJ system, which concluded that water jet is safe in removing lesions on pituitary gland and surrounding area [START_REF] Nakagawa | Pulsed Laser-induced Liquid Jet System for Treatment of Sellar and Parasellar Tumors: Safety Evaluation[END_REF]

______________________________________________________________________________________

jet stream and its effect on removing pituitary tumour in a minimal invasive procedure.

A. Objective of the study

The study aims to study the use of liquid jet on animal soft tissue. A series of experiments have been performed on continuous and pulsed liquid jet with various nozzle diameters to study its effect on cutting or peeling of the soft tissue. Along with the experimental study a numerical simulation has been done to evaluate the effect of liquid jet with variable nozzle diameters on cutting soft tissue.

II. METHODOLOGY

The study is based on quantitative data taken from two set of experiments and a computer simulation. The first experiment is related to the effect of continuous waterjet with various nozzles diameter on cutting a chicken breast and the second experiment is to study a pulsed water jet on peeling the chicken breast soft tissue. Then, a simple simulation is conducted using Abaqus 6.14, which is finite element analysis (FEA) software, to study the effect of waterjet on penetrating a chicken breast soft tissue.

A. Continuous waterjet with various diameters

To measure the effect of continuous jet with various diameters on soft tissue; the experiment was conducted simply by using a water tab and designing a nozzle that fitted to the outlet of the tab. A manometer has been used to measure the water pressure that had a value of 3.2 bar ± 0.2.

Thus, 20 pieces have been created and a nozzle hole has been drilled for each piece via a variable speed rotary tool with drill bits that has a diameter range from 0.3 mm to 1.6 mm, which resulted in 14 circles.

After that, the soft tissue has been placed on a plexiglass board and fixed with rubber bands to avoid any movement caused by the flow of water. Furthermore, the distant between the nozzle and the soft tissue is 66 mm, which is constant for all samples (Fig. 1).

B. Pulsed liquid jet experiment

The second experiment is conducted by a pulsed liquid jet to peel a sample of soft tissue from a chicken breast that has been done with 4 variables, which are angle and distant between the nozzle and the sample, liquid jet pressure, and nozzle diameter, to study the effect of these parameter on peeling soft tissue.

To do this test, the following parts were needed: First, a water pump with a reservoir to create a variable nozzle pressure. Second, a flexible hose that used to freely adjust the angle and a changeable nozzle to have more than one diameter. Therefore, to avoid building a new device from scratch and save time, a search has been done to find a suitable device that meet the requirements with less modification, which resulted in buying a Waterpik® Ultra Water Flosser that has 10 adjustable pressure settings with 1200 pulses per minute and a changeable nozzle [START_REF]Waterpik Ultra Water Flosser WP-120 -Electric Flosser[END_REF]. The nozzle has been created by using a 3D fused deposition modelling (FDM) printer, Dimension Elite 3D from Stratasys Ltd., with production-grade thermoplastic ABSplus-P430 that provides a precise measurement with a layer thickness of 0.178 mm to make multiple nozzles with various diameters in a short time [START_REF]ABSplus Spec Sheet[END_REF].

Then, the soft tissue has been placed on a plexiglass board that can be adjusted forward and backward according to the distance setting. The soft tissue then fixed with rubber bands on the plexiglass board. In addition, the distance between the nozzle and the sample was adjusted by the nozzle holder. The experimental setup is shown in Fig 2.

C. Computer simulations

To study the effect of waterjet on penetrating chicken breast soft tissue, a 2D model was created using finite element analysis software, Abaqus/CAE 6.14, which is capable of analysing the model and evaluate results [START_REF]Abaqus/CAE User's Guide[END_REF].

The model carefully created in the software to get accurate results. To create the model firstly parts created. Then defining a property for each part, assemble parts, create step and a field output, define the interaction between parts, choose the load and boundary condition, mesh each part, create a job and submit it, monitor the job for any error, have respectively done to get the results. The model, as shown in Fig. 3, has been created for each waterjet diameter; 0.3 mm, 0.4 mm, 0.5 mm, 0.6 mm and 0.7 mm, which each has been simulated separately.

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In two-dimensional FEA modelling, a 2D deformable shell part representing the water jet has been sketched. The nozzle diameter was related to the waterjet width and the height of waterjet column was set to 30 mm. Also, another 2D deformable shell part representing the chicken soft tissue with geometry of 1mm height and 20 mm width created.

Different material properties were defined for chicken breast soft tissue (TABLE I) [START_REF]Bulk Density Averages[END_REF], [START_REF] Misra | Needle-tissue interaction forces for bevel-tip steerable needles[END_REF], and for water jet. Equation of State (EoS) type U s -U p (shock velocity, U s , and particle velocity, U p ) has been used for water jet in the finite element model (TABLE II) [START_REF] Chizari | An explicit numerical modelling of the water jet tube forming[END_REF], [START_REF] Chizari | Experimental and numerical study of water jet spot welding[END_REF]. In next step, the two parts were assembled by creating instances from parts; Chicken and Water as dependent, and then, the water has been positioned in the centre of the sample by using translate instance function.

Moreover, a step has been created for the model as Dynamic, Explicit with a time period 0.1 and time scaling factor 1. After that, field output was created for the step selecting the following variables: S for stress, PE, PEEQ and LE for strain, U for displacement, V for velocity, A for acceleration, RF for reaction forces and moments, and CSTRESS for contact stress.

A surface to surface contact (Explicit) has been created as an interaction type between the water jet and the sample with a kinematic contact method as a mechanical constrain formulation.

For the load, a boundary condition was created as pinned that hold the two lower edges of the sample. Next, a predefined field was created to apply a velocity for the water jet toward the sample, which differs for each nozzle diameter. The velocity values were taken from experiment1. After that, the two parts have been meshed independently with approximate global size of 0.082 and the element shapes for both parts were Quad-dominated with a structured technique. Next, the element type for both pieces was "CPE4R", which is a 4-node bilinear plane strain quadrilateral with reduced integration and viscous hourglass control that has a linear geometric order with enabled element deletion.

III. RESULT AND DISCUSSION

After analysing the results, comparison charts have been generated from experiment 1, experiment 2 and simulation to study the effect of various nozzle diameters on dissecting soft tissue.

A. Experimental results

Experiment 1

In the first experiment, velocity values were decreasing when the nozzle diameter increased. Thus, resulted in a deeper cut with the same cutting time, 20 seconds, that reaches to total cut (10) from 0.8 mm nozzle diameter and onward (Fig. 4). However, the cutting quality was best for the smallest nozzle diameter. The cutting quality, as shown in Fig. 4, started to reduce from 0.7 mm, which at this point both cutting quality and cutting depth were crossing that concludes the 0.7 mm is a critical point for the chicken breast soft tissue target. Additionally, 0.6 mm nozzle had better quality and cutting depth compared to the remaining diameters, while the velocity for 0.6 mm and 0.7 mm were almost the same (Fig. 4), which may be caused by turbulent flow of water jet.

Moreover, smaller nozzle diameters; from 0.3 mm to 0.6 mm, had more precise cutting quality compared to bigger nozzle diameters. The bigger nozzles increase the width of the cut which is significantly bigger than the actual diameter of the nozzle due to the scattered water generated from hitting the sample. As shown in Fig. 4, the velocity was going down, oppositely, the flow rate is rising, which caused an unpredicted dissecting width to the target sample.

In conclusion, continuous water jet is a good solution to cut soft tissues. However, the quantity of liquid consumed and the time needed to cut the soft tissue might considered as a disadvantage to dissect pituitary gland tumour because of the tumour critical location near the optical nerve and between blood vessels. Therefore, experiment 2 has been performed to study the effect of variable pressure with pulsed liquid jet.

Experiment 2

This experiment has been done by using pulsed liquid jet with various: distances, angles and pressures. Hence, to study their effect on peeling soft tissue chicken breast soft tissue.

After preparing the experiment and measuring the peeling depth that has been created from applying the liquid jet. Therefore, results obtained for the samples dissecting depths have been plotted in two graphs providing a clear comparison between each parameter. The results have been summarized for 10º and 30º angles (Fig. 5 and Fig. 6). Each point in Fig. 5 and Fig. 6 graphs represents a distance in millimetres between the nozzle tip and the sample. The pressure set 1.0 bar for the first try (1P) and 2.0 bar for the second try (2P). As a result, each nozzle diameter has eight markers with twenty four points in total that represents the total number of test done in this experiment.

Engineering Letters, 25:3, EL_25_3_18 (Advance online publication: 23 August 2017) ______________________________________________________________________________________ From Fig. 5 and Fig. 6, 0.9 mm nozzle values had almost the highest dissecting depth compared to 0.5 mm and 0.7 mm, which were due to the thickness of the liquid jet that provides more flow and bigger splash compared to 0.7 mm and 0.5 mm nozzles. In addition, the 2 mm distance with 30º angle had the highest value for 0.9 mm nozzle that resulted in a thick cut. Furthermore, the sample for 0.7 mm nozzle with 4mm, 2.0 Pa, 10ᵒ angle setting creates a thin peeled layer of the tissue that reduced the dissecting depth, which effect the penetration to match closely with the depth of 2mm, 2.0 Pa, 10ᵒ angle setting.

Moreover, it has been observed that when the pressure increased the peeling depth get bigger. In addition, 0.7 mm nozzle had a steady increment compared to the other two nozzles. Moreover, the penetration distance was greater for 30ᵒ angle because the liquid jet can go deeper into the soft tissue. Therefore, the dissecting width was greater because of the scattered liquid.

Finally, it is evident from experiment 2 that the soft tissue property can affect the peeling depth. With this in mind, the pituitary gland tumour has varying size and hardness that required variable pressure setting, which can be adjusted during the surgery. Even more, the simplicity and the size of the used device in experiment 2 provides an advantage compare to Ogawa et al. [START_REF] Ogawa | Pulsed laser-induced liquid jet for skull base tumor removal with vascular preservation through the transsphenoidal approach: a clinical investigation[END_REF] device, which is bigger in size and uses an expensive laser beam device to generate the pulsed liquid jet.

B. Numerical results

After creating 2D models and 3D models, the analyses for these models were carried out, and then the results were detailed afterward. Thus, a comparison charts were made from these results to discuss the effect of various nozzle diameter on cutting a soft tissue of chicken breast. The 2D and 3D models focus on the time required for each nozzle to penetrate the sample, the performance of water velocity when cutting the sample, and the dissecting width created from the water jet. Additionally, a plot has been generated for 2D models to show the distance that water splash reaches after hitting the sample.

2D models

To study the influence of nozzle diameters on cutting penetration, a plot as shown in Fig. 7, has been created. The plots illustrate the simulation time to cut through a 1.0 mm chicken breast soft tissue. It shows that 0.1 mm of thickness was dissected at the same time for all nozzles. After that, the lines were diverse except for 0.3 mm and 0.4 mm nozzles which dissected the same depth until near 0.8 mm of thickness, where they separated. Similarly, 0.6 mm and 0.7 mm nozzles cut the sample at the same time till 0.4 mm of displacement, where they separated and continue to penetrate the sample in parallel line with the shortest time, unexpectedly, for 0.7 mm nozzle. On the other hand, 0.5 mm nozzle had its own cutting time that started from the sample thickness of 0.1 mm, and then increased the dissecting depth gradually until fully penetrated the sample. However, the water jet velocity for each nozzle (Fig. 8) had dropped to more than half its value at the same time, 0.01s, which is the time to dissect 0.1 mm. Moreover, the three smallest nozzle diameters had rose their velocity 2 m/s after dissecting 0.1 mm, which means that 0.5 mm nozzle is considered as a critical diameter for the water jet. Furthermore, the two 0.6 and 0.7 mm nozzle diameters continued to drops more of their velocity, then remained approximately at the same points until penetrating the sample. To make a clear measurement, a new plot has been generated (Fig. 10) that shows the X-axis from 8 mm to 12 mm long, which can be extracted that the maximum width of 2.5 mm was created by 0.7 mm nozzle followed by 0.6 mm nozzle that penetrated 2.2 mm of width. Meanwhile, the three smallest nozzle diameters had almost the same cutting width of 1.9 mm; therefore, 0.5 mm nozzle can be consider a better choice since it is easier to create and has the same effect on penetration width. Moreover, a water splash chart has been generated that shows the maximum distance the water particle had travelled after hitting the sample for each nozzle diameter (Fig. 11). As a result, it has been observed that the distance of the water splash is not linked to the nozzle diameter, except for the biggest diameter that had the extreme water splash. Hence, 0.7 mm nozzle had the greatest total splash distance that equal to nearly double its diameter from the right side (1.36 mm) and the same for the left side (1.32 mm). Surprisingly, 0.4 mm nozzle diameter had the next wider splash that spreads 1.02 mm from the right side and 1.33 mm from the left side. Afterward, the splash reduced in size for 0.6 mm nozzle, followed by 0.5 mm and 0.3 mm nozzle diameters that had approximately the same splash distance. Furthermore, only 0.5 mm nozzle and 0.7 mm nozzle had an even splash distance that were equally travelled in both directions.

C. Summary

The dissecting width from both experiments and simulation are affected by the nozzle diameter, as larger nozzle will get a wider cut but takes more time to reach the desired point. In addition, the liquid consumed volume will be higher for bigger nozzle diameter that must be taken into account when working in a small and narrowed area; for instance, pituitary gland area, where it needs smaller liquid jet diameter for more precise tumour dissecting in a quick time and less liquid volume.

After analysing the results from both experiments and simulation, 0.5 mm nozzle is the best diameter to dissect pituitary gland tumour because it has steady depth with an even water splash that creates constant cutting width for less water volume.

Moreover, the size of the device and easily controlling of the liquid pressure are the advantages of using water pump compared to using laser to create the liquid jet that had been used in Ogawa et al. [START_REF]Brain, other CNS and intracranial tumours Key Stats[END_REF] study, which is bigger in size and needs to control the laser power to vary the liquid pressure. Furthermore, the device in experiment 2 can be modified to produce lower pressure with different pulsed ratio, which can be done for future work.

In addition, there are several benefits of using pulsed liquid jet compared to current surgical methods of dissecting pituitary gland tumour. Accordingly, preserving fine blood vessels and optical nerve located near the pituitary gland, less harm to surrounding anatomical structures, and provides a safe and easy use to the surgeon that eliminate damages to pituitary gland and neighbouring tissues.

IV. CONCLUSION

Since the current methods to dissect pituitary gland tumour have many disadvantages, designing a new medical device to dissect the tumour in less harm to the patient with minimal invasive procedure that provides a high success rate is essential to reduce the complexity of cutting the lesion that lead to physician faults, which can cause more medical complications to the patient. Therefore, liquid jet has the ability to remove the tumour with no side effects when designed with a safeguard configuration that avoid any damage to blood vessels and optical nerve by limiting the jet pressure. In addition, it provides a precise cutting depth with various nozzle angles that chosen according to the surgeon requests.

Moreover, results from the continuous water jet experiment has demonstrates that the cutting quality and Engineering Letters, 25:3, EL_25_3_18 (Advance online publication: 23 August 2017) ______________________________________________________________________________________ depth were influenced by the nozzle diameter when fixing the water inlet pressure, in which, the cutting quality were decreased when the jet diameter get bigger. In contrast, the cutting depth increased proportional to the nozzle diameter until 0.8 mm nozzle, where the water jet penetrated the sample. Next, pulsed liquid jet was conducted with more parameters that improve the dissecting depth with less liquid volume compared to continuous water jet experiment. Furthermore, the simulation shows clear outcomes to the effect of various nozzle diameters related to the dissecting width and the penetration time that is close to the experiment results, which gives estimated values when selecting the jet diameter.

Due to the project's time limitation, the number of nozzle diameter variables for experiment 2 has been reduced to three diameters. Also, additional work is required for pulsed liquid jet experiment to control the jet pulse rate, provide more variables for the jet pressure, and test nozzle diameter with the same length required during the surgery. In addition, simulate a finite element model with Eulerian type for water jet to provide outcomes that are more precise.

Finally, pulsed liquid jet technology will replace current surgical methods in the future, not only for hypophysis tumour, but also for other body parts because it consume less water, which improves protecting small body structures with better accessibility that affords stable well cutting ability in narrowed area [START_REF] Alamoud | Dissect Pituitary Gland Tumour Using Liquid Jet: Experimental and Numerical Simulation[END_REF].

Fig. 1

 1 Fig. 1 Setup of the first experiment

Fig 2

 2 Fig 2 Setup of the second experiment

Fig. 3 A

 3 Fig. 3 A two-dimensional FEA model of the liquid jet penetrating into the soft target

Fig. 4

 4 Fig. 4 Comparison chart for velocity, cutting depth and quality for all nozzle diameters

Fig. 5 2 Fig. 6

 526 Fig. 5 Dissecting depth values for 10º angle parameter in experiment 2

Fig. 7

 7 Fig. 7 Penetration depths related to each nozzle diameter

Fig. 8 Fig. 9

 89 Fig.8Velocity effect on each nozzle diameter when dissecting the sample

Fig. 10 A

 10 Fig. 10 A close look chart for penetration width created by each nozzle diameter

Fig. 11

 11 Fig. 11 Water splash traveling distance for each nozzle diameter

Engineering Letters, 25:3, EL_25_3_18 (Advance online publication: 23 August 2017)

  . As a result, more studies are needed to be conducted regarding water jet technology with different nozzle diameters and methods to create the water

	Continuous and Pulsed Experiments with
	Numerical Simulation to Dissect Pituitary
	Gland Tumour by Using Liquid Jet
	Ahmed H. Alamoud, Emmanuel Baillot, Camila Belabbas, Hadi S. Samimi Ardestani, Hamid Bahai,
	Adrien Baldit, Mahmoud Chizari
	O

TABLE II WATER

 II PROPERTIES[START_REF] Chizari | An explicit numerical modelling of the water jet tube forming[END_REF],[START_REF] Chizari | Experimental and numerical study of water jet spot welding[END_REF]

	Density (Kg/m 3 )	Co (m/s)	s (Pa.s)	Gamma o
	983	1450	1.5 x 10-8	0


Temporomandibular joint plays a major role in everyday life allowing speaking and eating. This work aims to describe a method that combine several mechanical tests highlighting soft tissue properties, both local and global, through local compression and traction. It provides "first order" evaluation of mechanical properties and allow setting optimisation loop with more complex models.

Mechanical characterisation of the temporomandibular joint disc through local compression and traction

L.K. Tappert 1a , A. Baldit 1b , R. Do Nascimento 1,2 , P. Lipinski 1 and R. Rahouadj

Introduction

Temporomandibular joint plays a major role in everyday life allowing speaking and eating. Many events can affect its behaviour while lifespan therefore a better understanding of the mechanical behaviour of its components, as well as their interactions, is required [START_REF] Creuillot | Mechatronic Design Towards Investigation of the Temporo-Mandibular Joint Behaviour in Mechatronics: Ideas, Challenges, Solutions and Applications[END_REF][START_REF] Tappert | Local elastic properties characterization of the temporo-mandibular joint disc through macro-indentation[END_REF]. This work is focused on the soft element of this joint: the disc. It is a biphasic material with up to 66 -80% of water [START_REF] Detamore | Motivation, Characterization, and Strategy for Tissue Engineering the Temporomandibular Joint Disc[END_REF] and mainly made of elastin and collagens type I and II. This composite like structure confers to the tissue an anisotropic, non-linear and viscous behaviour that has to be mechanically characterised to allow simulating the whole articulation comportment. Although, its complex behaviour leads to a large range of mechanical properties [START_REF] Tappert | Local elastic properties characterization of the temporo-mandibular joint disc through macro-indentation[END_REF][START_REF] Detamore | Motivation, Characterization, and Strategy for Tissue Engineering the Temporomandibular Joint Disc[END_REF][START_REF] Detamore | Tensile Properties of the Porcine Temporomandibular Joint Disc[END_REF][START_REF] Kim | Biomechanical Tissue Characterization of the Superior Joint Space of the Porcine Temporomandibular Joint[END_REF][START_REF] Tanaka | Stress analysis in the TMJ during jaw opening by use of a three-dimensional finite element model based on magnetic resonance images[END_REF] that mismatch with the aim of predicting its evolution through planned events like surgery. This work aims to describe a method that combine several mechanical tests highlighting soft tissue properties, both local and global, through local compression and traction.

Material and methods

Porcine temporomandibular discs (Figure 1a) were extracted from slaughterhouse remaining parts and directly frozen within physiological solution until experimentation. The tests performed within physiological solution were sequenced as: i) first local spherical compressions [START_REF] Tappert | Local elastic properties characterization of the temporo-mandibular joint disc through macro-indentation[END_REF], ii) tensile sample extraction and iii) tensile test through fibres directions. Each test aims to characterise a specific property of the tissue. Actually, local compressions in direction perpendicular to fibres reflect the composite like matrix behaviour. Then, parallelepiped samples were extracted from the central disc structure along fibres directions with 9.9x2.1x4.1 mm 3 final dimensions measured thanks to a calliper. Eventually, cyclic tensile tests were performed thanks to a Zwick Zwicki 0.5 tensile machine equipped with a 100 N load cell and driven by displacement during 3 cycles reaching an imposed strain near 10% at a strain rate of 𝜀~8×10 '( 𝑠 '+ similar to [START_REF] Tappert | Local elastic properties characterization of the temporo-mandibular joint disc through macro-indentation[END_REF]. This final test aims to characterise the fibre effect on tissue behaviour. The Young's modulus has been computed through Python curve fitting on the last measurement points' quarter of each loading phases to observe viscosity effect on this elastic property.

Results

The tensile sample's extraction highlighted the pre-constrained condition of the disc structure as shown in Figure 1a with an opening angle q increasing progressively while tissue relaxed and reaching 35 o . Figure 1 Both tensile and spherical compression curves had the same characteristics: nonlinear with an accommodation and hysteresis along the cycles as presented in Figures 1b and1c. Being located at the disc centre, those results are related to the same location with a spherical radius of 3 mm for the compression presented in Figure 1b. In Figure 2a, the spherical compression test presents a Young's modulus of 24.31 kPa and a Poisson's ratio of 0.5. On the other hand, the tensile test allows characterizing the Young's modulus of 9.09 MPa at final loading phase. Figure 2: a) spherical compression and b) tensile characterisations and c) strain energy density as function of time during cyclic tensile test. The tensile test strain energy density presents a non-linear evolution with an accommodation following the stress-strain curve shape and decreasing values for each loading cycle applied.

Discussion

Biological tissue growing is done under mechanical constrains leading to an optimised musculoskeletal structure. Tissue residual stresses have been released and observed (Figure 1a) during sample extraction but are not taken into account in the mechanical analysis. In Figures 1b and1c, the complex mechanical behaviour of the disc tissue is presented under compression and traction loads. The mechanical properties obtained from local compressions are consistent with the literature [START_REF] Tappert | Local elastic properties characterization of the temporo-mandibular joint disc through macro-indentation[END_REF][START_REF] Kim | Biomechanical Tissue Characterization of the Superior Joint Space of the Porcine Temporomandibular Joint[END_REF]. It characterises the tissue through a quasi-incompressible behaviour reflected by a Poisson's ratio close to 0.5 and a low Young's modulus value related to the matrix made of collagen type II and elastin. The Young's modulus obtained through tensile cyclic test is as well comparable to the literature [START_REF] Detamore | Tensile Properties of the Porcine Temporomandibular Joint Disc[END_REF][START_REF] Tanaka | Stress analysis in the TMJ during jaw opening by use of a three-dimensional finite element model based on magnetic resonance images[END_REF] with two orders of magnitude higher value than those obtained from compression tests. This strong variation of elastic properties highlights the anisotropy of the material due to unidirectional orientation of collagen type I fibres in central area of the disc. The strain energy density evolution points out the dissipative behaviour of the tissue.

Conclusion

Eventually, the complex behaviour of disc structure requires combined methods to be characterised. Simple methods like Hertz contact theory [START_REF] Tappert | Local elastic properties characterization of the temporo-mandibular joint disc through macro-indentation[END_REF] and linear characterisation provide "first order" evaluation of mechanical properties and allow setting optimisation loop with more complex models [START_REF] Creuillot | Mechatronic Design Towards Investigation of the Temporo-Mandibular Joint Behaviour in Mechatronics: Ideas, Challenges, Solutions and Applications[END_REF]. This procedure can be improved using more experimental data through digital image correlation during tensile test [START_REF] Baldit | Experimental analysis of the transverse mechanical behaviour of annulus fibrosus tissue[END_REF] and hyper elastic model to characterise more accurate tissue's mechanical properties. The evaluation of the residual stresses through image analyses (as described in [START_REF] Floch | Residual strain and stress in pig intervertebral disc[END_REF]) is currently under investigation.

  Figure 1: a) Porcine temporomandibular disc with an antero-posterior cut related to the tensile test preparation, b) spherical compression and c) tensile cyclic force-displacement curves.


Cyril Banderier

Jean-Luc Baril

Céline Moreira

Dos Santos

Right-jumps and pattern avoiding permutations

Keywords: Permutation pattern, left-to-right maximum, insertion sort, generating function, analytic combinatorics, D-finite function, supercongruence

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

Introduction

In computer science, many algorithms related to sorting a permutation have been analysed and shown to have behaviours linked to nice combinatorial properties (see e.g. [START_REF] Knuth | The art of computer programming[END_REF]). Their complexity can be analysed in terms of memory needed, or number of key operations (like comparisons or pointer swaps). An important family of algorithms, like the so-called insertion algorithms, or in situ permutations, are quite efficient in terms of the number of pointer swaps (but are not the fastest ones in terms of comparisons). Due to this higher cost, they have been much less studied than the faster stack sorting algorithms. Like for the stack algorithms, instead of seeing them as an input/output pair, we can see them like a process: input and set of intermediate steps. This opens a full realm of questions on such processes, and they often lead to nice links with other parts of mathematics (like the link between trees, birth and death processes, random walks in probability theory, or permutations and Young tableaux in algebraic combinatorics). Our article will investigate a link between a sorting algorithm, patterns in permutations, and their asymptotics counterparts.

Another motivation to analyse such processes comes from the field of bioinformatics. Indeed, in genomics, a crucial part of study is to estimate the similarity of two genomes. This consists in finding the length of a shortest path of evolutionary mutations that transforms one genome into another. Usually, the main operations used in the rearrangement of a genome are of three different types: substitutions (one gene is replaced with another), insertions (a gene is added) and deletions (a gene is removed). For instance, we refer to [START_REF] Neil | An Introduction to Bioinformatics Algorithms[END_REF]; [START_REF] Meidanis | Introduction to Computational Molecular Biology[END_REF] for an explanation of these operations, and the notions of transposons or jumping genes.

As the problem is too hard in full generality, many simpler mathematical models of the genome are used (see [START_REF] Ewens | The chromosome inversion problem[END_REF]): one of them is using permutations of {1, 2, . . . , n} where each gene is assigned a number. Following the idea of transposition mutations (see [START_REF] Neil | An Introduction to Bioinformatics Algorithms[END_REF]), our motivation is to find some combinatorial properties in terms of pattern avoiding permutations whenever one element is deleted and inserted in a position to its right. This operation will be called a right-jump. 1 2 3 4 5 6 7

1 2 4 5 6 3 7

Figure 1: A right-jump in the permutation σ = 1234567. In this article, we investigate the structure of permutations obtained after several iterations of such right-jumps.

This operation is a variant of genome duplication, which consists of copying a part of the original genome inserted into itself, followed by the loss of one copy of each of the duplicated genes. In particular, it is comparable to the whole duplication-random loss model studied in [START_REF] Chaudhuri | On the tandem duplication-random loss model of genome rearrangement[END_REF]. Although there are many connections between these models, it is surprising that the behaviour of their combinatorial properties depends on different parameters: the right-jump model reveals some links with left-to-right maximum statistics (see [START_REF] Baril | Gray code for permutations with a fixed number of left-to-right minima[END_REF]; [START_REF] Bóna | Combinatorics of permutations[END_REF]), while the whole duplication-random loss model reveals links with descent statistics (see [START_REF] Baril | Whole mirror duplication-random loss model and pattern avoiding permutations[END_REF]; [START_REF] Bouvel | On the enumeration of d-minimal permutations[END_REF]; [START_REF] Bouvel | Posets and permutations in the duplication-loss model: minimal permutations with d descents[END_REF]; [START_REF] Bouvel | A variant of the tandem duplication-random loss model of genome rearrangement[END_REF]; [START_REF] Chaudhuri | On the tandem duplication-random loss model of genome rearrangement[END_REF]; [START_REF] Mansour | Minimal permutations with d descents[END_REF]).

In the literature, such right-jumps in permutations are also found in the domain of sorting theory. Indeed, it corresponds (modulo a mirror symmetry) to the insertion-sorting algorithms on permutations (see [START_REF] Knuth | The art of computer programming[END_REF]). Since the seminal work of Knuth on this subject, many articles related to sorting with a stack exhibit links with pattern avoiding permutations. In contrast, for insertion sorting algorithms (also the subject of a vivid literature), only one study exhibits links with pattern avoiding permutations. Indeed, in his thesis, [START_REF] Magnússon | Sorting Operators and Their Preimages[END_REF] proves that the set of permutations that can be sorted with one step of the insertion-sorting operator is the class of permutations avoiding the three patterns 321, 312 and 2143.

Plan of the article. In Section 2, we recall some basic facts on permutations patterns. In Section 3, we generalize the result of Magnússon by studying the iteration of right-jumps in terms of pattern avoiding permutations: we prove that the set of permutations obtained from the identity after a given number of right-jumps is the class of permutations avoiding some patterns, which we characterize. In Section 4, we enumerate these forbidden patterns by giving their bivariate exponential generating function (involving an additional parameter: the number of left-to-right maxima), and we give the corresponding asymptotics and limit law. We also give some modular congruences for our main enumeration sequence. In Section 5, we conclude with several possible extensions of this work.

Patterns in permutations

In this section, we give some classical definitions and properties on patterns in permutations. For any permutation σ ∈ S n (the set of permutations of length n), the graphical representation of σ = σ 1 σ 2 . . . σ n is the set of points in the plane at coordinates (i, σ i ) for i ∈ n (i) . For instance, the permutation 53621487 has the graphical representation illustrated in Figure 2. A left-to-right maximum of σ ∈ S n is a value σ i , 1 ≤ i ≤ n, such that σ j ≤ σ i for j ≤ i. A value σ i of σ, 1 ≤ i ≤ n which is not a left-to-right maximum will be called a non-left-to-right-maximum of σ. For instance, if σ = 53621487 then the left-to-right maxima are 5, 6, 8 and the non-left-to-right-maxima are 1, 2, 3, 4, 7. A permutation π of length k, is a pattern of a permutation σ ∈ S n if there is a subsequence of σ which is order-isomorphic to π, i.e., if there is a subsequence σ i1 . . . σ i k of σ with 1 ≤ i 1 < • • • < i k ≤ n and such that σ ij < σ i whenever π j < π . We write π ≺ σ to denote that π is a pattern of σ. A permutation σ that does not contain π as a pattern is said to avoid π. For example, σ = 2413 contains the patterns 231, 132, 213 and 312, but σ avoids the patterns 123 and 321. The set of all permutations avoiding the patterns π 1 , . . . , π m is denoted by Avoid(π 1 , . . . , π m ). We say that Avoid(π 1 , . . . , π m ) is a class of pattern avoiding permutations with basis {π 1 , . . . , π m }. For instance, we refer to the book of [START_REF] Kitaev | Patterns in permutations and words[END_REF] and [START_REF] Bóna | Combinatorics of permutations[END_REF] to deepen these notions. A set C of permutations is stable for the involvement relation ≺ if, for any σ ∈ C, for any π ≺ σ, then we also have π ∈ C. Now, we formulate a definition that is crucial for the present study. (i) In this article, we write n for {1, 2, . . . , n}.

Definition 1 (Permutation basis and basis permutations). If a set C of permutations is stable for the involvement relation ≺, then C is a class of pattern avoiding permutations: C = Avoid(B). The basis B of forbidden patterns is then given by B = {σ / ∈ C, ∀π ≺ σ with π = σ, π ∈ C} . In other words, the basis B is the set of minimal permutations σ that do not belong to C, where minimal is intended in the sense of the pattern-involvement relation ≺ on permutations, that is: if π ≺ σ and π = σ then π ∈ C. Notice that B might be infinite. We call basis permutations the permutations belonging to B.

Equipped with these definitions, our mission consists now in giving a description of the permutations belonging to the basis B p (the permutations which are the minimal forbidden patterns) for the set C p of permutations at distance at most p from the identity, i.e., permutations obtained from the identity after at most p right-jumps.

3 Iteration of right-jumps in permutations: a structural description of the forbidden patterns

In this section we study the iteration of right-jumps in terms of pattern avoiding permutations. We establish that the set C p of permutations obtained from the identity after at most p right-jumps is a class of permutations avoiding some patterns that we characterize.

Lemma 1 (Characterization of the distance). A permutation obtained from the identity after p right-jumps contains at most p non-left-to-right-maxima.

Proof: The result holds for p = 1; indeed a right-jump transformation of the identity permutation creates the permutation 1 2 . . . (i -1) (i + 1) . . . (j -1) i j . . . n for 1 ≤ i < j, where i is the only one non-leftto-right-maximum. Now, let us assume that each permutation π obtained from the identity after (p -1) right-jumps contains at most p -1 non-left-to-right-maxima. Let σ be a permutation obtained from the identity after p right-jumps. Using the recurrence hypothesis, σ is obtained from a permutation π with at most p -1 non-left-to-right-maxima by moving an element π i , 1 ≤ i < n, in a position to its right. We distinguish two cases: (1) π i is a non-left-to-right-maximum, and (2) π i is a left-to-right maximum.

Case (1): Since π i is a non-left-to-right-maximum, there exists j < i such that π j is a left-to-right maximum satisfying π j > π i . Since we move π i to its right, π j remains on the left of π i in σ which implies that π i is a non-left-to-right-maximum in σ. Using the same argument, any non-left-to-right-maximum π k in π remains a non-left-to-right-maximum in σ. Moreover, let π k be a left-to-right maximum in π, i.e., π j < π k for all j < k. Since the right-jump transformation moves to the right of a non-left-to-rightmaximum, all values on the left of π k in σ are lower than π k , which proves that π k remains a left-to-right maximum in σ. Therefore, σ contains at most p -1 non-left-to-right-maxima (as π does).

Case (2): π i is a left-to-right maximum, i.e., π j < π i for all j < i. Since π i is moved to its right, any left-to-right maximum located on the left of π i in π remains a left-to-right maximum in σ. On the other hand, any left-to-right maximum located on the right of π i in π is greater than π i and thus, it remains a left-to-right maximum in σ. Therefore, the number of left-to-right maxima in σ is at least the number of left-to-right maxima in π minus one (we do not consider π i ). This means that the number of non-left-toright-maxima in σ is at most p.

Considering the two previous cases allows to complete the proof, by induction.

We now derive our first enumeration result for the set D p of permutations at distance p from the identity, i.e., the set of permutations reachable from the identity with p right-jumps, but that one cannot reach with less than p right-jumps: C p = ∪ k∈ p D k , and this union is disjoint.

Theorem 1 (Permutations after p right-jumps). The set D p of permutations at distance p from the identity is the set of permutations with exactly p non-left-to-right-maxima. Accordingly, the number d n,p of permutations of length n in D p is counted by the Stirling numbers s(n, n -p):

d n,p = s(n, n -p) = 0≤j≤h≤p (-1) j h j n -1 + h p + h n + p p -h (j -h) p+h h! .
Proof: After considering Lemma 1, it suffices to prove that any permutation σ with at most p non-leftto-right-maxima can be obtained from the identity after p right-jumps. Let σ be a permutation with p ≥ 1 non-left-to-right-maxima. Let us assume that the leftmost non-left-to-right-maximum is σ i and let j < i be the position of the smallest left-to-right maximum σ j such that σ j > σ i . Then we set σ = σ 1 . . . σ j-1 σ i σ j . . . σ i-1 σ i+1 . . . σ n . Since we have σ j > σ i and also σ i > σ j-1 (if σ j-1 exists), σ i becomes a left-to-right maximum in σ . Thus, σ contains exactly p -1 non-left-to-right-maxima and by construction, σ can be obtained from σ by a right-jump. This proves that permutations at distance p from the identity are exactly the permutations with n -p left-to-right-maxima, which are known to be counted by s(n, n -p), the signless Stirling number of the first kind (see [START_REF] Flajolet | Analytic combinatorics[END_REF] for the closed-form formula due to Schlömilch, and sequence OEIS A094638 in Sloane and collaborators (2016) for many occurrences of the corresponding triangular array).

For instance, the values of d n,p for n = 7 and 0 ≤ p < 7 are 1, 21, 175, 735, 1624, 1764, 720.

The following corollary says a little more on the lattice structure associated to our process "a particle jumps to the right".

Corollary 1 (Changing the starting point and sorting algorithms). For any permutation p, one denotes by t p its number of non-left-to-right-maxima. Let σ and π be two permutations, then t σ -1 •π right-jumps are necessary to obtain π from σ. In particular, t σ -1 right-jumps are necessary and sufficient to sort by insertion the permutation σ into the identity.

Proof: Firstly, t σ right-jumps are necessary and sufficient to obtain σ from the identity. Therefore, t σ -1 •π transformations are sufficient and necessary to obtain σ -1 • π from the identity. We set t = t σ -1 •π and let Id = χ 0 , χ 1 , . . . , χ t-1 , χ t = σ -1 • π be a shortest path between the identity and σ -1 • π. Now, let us prove that if a permutation β is obtained from α by one right-jump, then for any permutation γ, γ • β is also obtained from γ • α by one right-jump. Indeed, if we have α = α 1 α 2 . . . α n then β can be written as

β = α 1 . . . α i-1 α i+1 . . . α j-1 α i α j . . . α n . Composing by a permutation γ, we obtain γ • α = γ(α 1 )γ(α 2 ) . . . γ(α n ) and γ • β = γ(α 1 ) . . . γ(α i-1 )γ(α i+1 ) . . . γ(α j-1 )γ(α i )γ(α j ) . . . γ(α n )
which proves that γ • β is also obtained from γ • α by one right-jump. So if we compose by σ at each step of the above shortest path, then we obtain a shortest path of t σ -1 •π right-jumps from σ to π, which completes the proof.

Since the set C p of permutations obtained after p right-jumps is stable for the relation ≺, C p is also a class Avoid(B p ) of pattern avoiding permutations where B p is the basis consisting of minimal permutations σ that are not in C p (see Definition 1). Theorem 2 gives the explicit description of these basis permutations.

Theorem 2 (Structural description of the basis permutations). A permutation σ ∈ S n belongs to the basis B p of forbidden patterns, if and only if the following conditions hold:

(i) σ contains exactly p + 1 non-left-to-right-maxima. (ii) n -1 is a non-left-to-right-maximum. (iii) σ 2 is a non-left-to-right-maximum.
(iv) For any three left-to-right maxima, σ i , σ j and σ k (with i < j < k) such that there is no left-to-right maximum between them, there exists a non-left-to-right-maximum σ t (with j < t < k) satisfying σ t > σ i .

Proof: N.B.: Figure 3 on next page illustrates the different claims and notations of this theorem. Let σ ∈ S n be a permutation belonging to the basis B p , i.e., σ / ∈ C p and π ≺ σ implies π ∈ C p . Throughout this proof, we refer to Figure 3 for an illustration of the three conditions (ii), (iii) and (iv).

-First, the deletion of a non-left-to-right-maximum in σ decreases the number of non-left-to-rightmaxima by one exactly. Therefore, the minimality of σ implies that σ necessarily contains exactly p + 1 non-left-to-right-maxima, which proves (i).

-For a contradiction, assume that (ii) is not satisfied, i.e., n -1 is a left-to-right maximum. Since n is always a left-to-right maximum, n is on the right of n -1 in σ. Thus, the permutation π obtained by deleting n from σ also contains p + 1 non-left-to-right-maxima (a non-left-to-right-maximum on the right of n in σ remains a non-left-to-right-maximum on the right of n -1 in π). Therefore, π does not belong to C p ; this gives a contradiction with the minimality of σ.

-For a contradiction, assume that (iii) is not satisfied, i.e., σ 2 is a left-to-right maximum and thus, σ 1 is smaller than σ 2 . Thus, the permutation π obtained by deleting σ 1 from σ also contains p + 1 non-left-to-right-maxima. Indeed, a non-left-to-right-maximum σ i in σ such that σ i < σ 1 becomes a non-left-to-right-maximum σ i < σ 2 -1 in π. Moreover, a non-left-to-right-maximum σ i in σ such that σ i > σ 1 (there is , 2 ≤ < i, with σ i < σ ) becomes a non-left-to-right-maximum σ i -1 in π with σ i -1 < σ -1. Therefore, π does not belong to C p ; this gives a contradiction with the minimality of σ.

-For a contradiction, assume that (iv) is not satisfied; i.e., there are (i, j, k), 1 ≤ i < j < k ≤ n, such that σ i , σ j and σ k are three consecutive left-to-right maxima of σ (consecutive means that there is no other left-to-right maximum between σ i and σ j and between σ j and σ k ), and such that there is no non-left-to-right-maximum σ , j < < k, satisfying σ i < σ . Let π be the permutation obtained from σ by deleting σ j . It is clear that any non-left-to-right-maximum on the left of σ j in σ remains a non-left-toright-maximum in π. Let σ , > k, be a non-left-to-right-maximum on the right of σ k in σ. If σ < σ j then σ < σ j ≤ σ k -1 and σ remains a non-left-to-right-maximum in π. If σ > σ j then there is σ t , t ≥ k, such that σ ≤ σ t ≥ σ k , and thus, there is σ t -1 on the left of σ k -1 in π with σ t -1 > σ k -1 which means that σ -1 is a non-left-to-right-maximum in π. Let σ , j < < k, be a non-left-to-rightmaximum between σ j and σ k in σ. Assuming that (iv) is not satisfied, we deduce that σ < σ i , and σ remains a non-left-to-right-maximum in π. Finally, π also contains p + 1 non-left-to-right-maxima; this gives a contradiction with the minimality of σ.

Conversely, let σ be a permutation satisfying (i), (ii), (iii) and (iv) and π be a permutation obtained by deleting σ i , 1 ≤ i ≤ n, from σ. Let us prove that π belongs to C p , that is, π contains at most p non-left-to-right-maxima.

-If σ i is a non-left-to-right-maximum of σ, then π has p non-left-to-right-maxima and thus, π ∈ C p .

-Now, let us assume that σ i is a left-to-right maximum of σ. If σ i = n, then (ii) implies that n -1 is a non-left-to-right-maximum of σ and a left-to-right maximum in π; this implies that π contains p non-leftto-right-maxima and thus, π ∈ C p . If σ i = σ 1 , then (iii) implies that σ 2 is a non-left-to-right-maximum of σ and σ 2 -1 is a left-to-right maximum in π; this implies that π contains p non-left-to-right-maxima and thus, π ∈ C p . If there exists (j, k), 1 ≤ j < i < k ≤ n, such that σ j and σ k are left-to-right maxima (we choose j the greatest possible and k the lowest possible with this property). Then, (iv) implies that there is σ , i < < k, such that σ j < σ < σ i (we choose the lowest possible > i). Thus, σ is a non-left-to-right-maximum in σ and becomes a left-to-right maximum in π, which implies that π contains exactly p non-left-to-right-maxima, and thus π ∈ C p .

Finally, the permutation π necessarily belongs to C p , which completes the proof.

(iii)

σ 2 σ i σ j σ t σ k (ii) n -1 (iv)
Figure 3: An illustration of Theorem 2 that characterizes the basis permutations of Bp. Condition (i) states such a basis permutation has p + 1 non-left-to-right maxima (drawn with a small black point, while left-to-right maxima are drawn with a big green point), condition (ii) states that n -1 is not a left-to-right maximum, condition (iii) states that σ2 is not a left-to-right maximum, and condition (iv) states that there is a "higher" non-left-to-right maximum between 3 left-to-right maxima.

Corollary 2 (Length of the forbidden patterns). Permutations in B p have length ≤ 2(p + 1) and ≥ p + 2. As a consequence, B p is a finite set.

Proof: Theorem 2 implies that the number of left-to-right maxima in a basis permutation is at most the number of non-left-to-right-maxima. Since a basis permutation of B p has p + 1 non-left-to-right-maxima, its length is at most 2(p + 1).

For instance, the basis for p = 0, 1, 2 are respectively B 0 = {21}, B 1 = {312, 321, 2143} (recovering the result of Magnússon), and B 2 = {4123, 4132, 4213, 4231, 4312, 4321, 21534, 21543, 31254, 32154, 31524, 31542, 32514, 32541, 214365}.

Enumerative results for basis permutations

In order to obtain a recursive formula for the number b n,p of permutations of length n in the basis B p , we present the following preliminary lemma.

Lemma 2 (A recursive description). Let σ ∈ S n be a basis permutation having p ≥ 1 non-left-to-rightmaxima and such that σ k+1 = n, k ≥ 0. Let α be the subsequence σ 1 σ 2 . . . σ k and π be the permutation in S k isomorphic to α. Then, π is a basis permutation with p -n + k + 1 non-left-to-right-maxima.

Proof: Any permutation σ can be uniquely written as σ = αnβ where α and β are two subsequences of n -1 . Let k be the length of α and let π = π 1 π 2 . . . π k be the permutation of k that is isomorphic to the subsequence α. Let us prove that π is minimal.

Since σ is minimal, it satisfies the three conditions (ii), (iii) and (iv) of Theorem 2. Since all elements in β are non-left-to-right-maxima in σ, π contains exactly p -(n -1 -k) non-left-to-right-maxima and thus, n -p -1 left-to-right maxima.

-The condition (iii) of Theorem 2 on σ does not involve the part nβ. Therefore, π satisfies (iii).

-The deletion of nβ from σ preserves the condition (iv) on π. Thus, π satisfies (iv).

-Let σ i , σ j and σ k+1 = n, 1 ≤ i < j ≤ k, be the last three left-to-right maxima of σ. After the deletion of nβ, the two left-to-right maxima of σ, σ i and σ j , are respectively transported in π into π i and π j = k. Condition (iv) on σ ensures that there is σ , between σ j and n such that σ > σ i . The greatest value σ satisfying this property is then transported in π into k -1, which proves that k -1 is on the right of k in π. Thus, π satisfies (ii).

Using Theorem 2, the permutation π is a basis permutation with p + -n + 1 non-left-to-rightmaxima.

Theorem 3 (An infinite recursion). The number b n,p of basis permutations of length n in B p (or equivalently having exactly p + 1 non-left-to-right-maxima) is given by the following recurrence relation (for

p < n -2): b n,p = p-1 =0 ( + 1)! • n -2 • b n--2,p--1
anchored with b n,p = 0 if p < (n -2)/2 or p > n -2, and b n,n-2 = (n -1)! for n > 1.

Proof: Any permutation σ of length n ≥ 1 contains at least one left-to-right maximum and thus, at most n -1 non-left-to-right-maxima which implies that b n,p = 0 for n ≤ p + 1. Using the proof of Corollary 2, we also have b n,p = 0 for n > 2(p + 1). Moreover, the basis permutations of length n with n -1 non-leftto-right-maxima are the permutations of the form nα where α ∈ S n-1 . So, we have b n,n-2 = (n -1)! for n > 1. Now, let us prove the recursive relation. Let σ ∈ S n be a basis permutation with p + 1 non-left-toright-maxima. We consider its unique decomposition σ = αnβ where α and β are some subsequences of n -1 . Let + 2, ≥ 0, be the length of nβ and let π be the permutation in S n--2 isomorphic to α. Using Lemma 2 with k = n --2, π is minimal with p --1 non-left-to-right-maxima. So, we can associate to σ = αnβ the pair (π, γ) where π ∈ S n--2 is minimal with p--1 non-left-to-right-maxima and γ ∈ S +1 is isomorphic to β.

Conversely, let π be a basis permutation of length n --2 with p --1 non-left-to-right-maxima and γ ∈ S +1 . We construct a basis permutation σ of length n with p + 1 non-left-to-right-maxima as follows. From γ ∈ S +1 , we construct a subsequence β of n -1 of length + 1 such that β contains the value n -1 and such that β is isomorphic to γ. Since n -1 belongs to β, its position in β also is the position of the greatest value of γ. So, β is characterized by the choice of values among n -2 . Now, we define the unique subsequence α of n -2 \X isomorphic to π where X is the set of values used in β. This construction ensures that σ = αnβ is a basis permutation of length n with p + 1 non-leftto-right-maxima, and so σ ∈ B p . So, there are n-2 possibilities to choose the values of β and ( + 1)! possibilities to choose γ and b n--2,p--1 possibilities to choose a basis permutation π ∈ S n--2 with p --1 non-left-to-right-maxima. Varying from 0 to p -1, we obtain the recursive formula.

Theorem 3 allows us to find the bivariate exponential generating function for the number of basis permutations according to the number of non-left-to-right-maxima.

Theorem 4 (Closed-form for the bivariate generating function). Consider the bivariate exponential generating function B(x, y) = n≥0,p≥0 b n,p

x n y p n! where the coefficient of x n y p n! is the number b n,p of basis permutations of length n in B p . Then, we have

B(x, y) = 1 2y 1 - 1 V (1 -xy) 1 2 (1+V ) + 1 2y 1 + 1 V (1 -xy) 1 2 (1-V ) - 1 y ,
where V := 1 + 4/y .

Proof:

Setting F p (x) := n≥0 b n,n-p x n
n! and F (x, y) := p≥0 F p (x)y p , we have B(x, y) = F (xy, 1/y). (We work with the generating function F p (x) of the (b n,n-p )'s rather than the generating function of the (b n,p )'s because then the derivation of the proof is simpler to write).

Taking the second derivative of F (x, y) with respect to x gives

∂ 2 x F (x, y) = ∂ 2 x   p≥0 F p (x)y p   = ∂ 2 x F 0 (x) + ∂ 2 x F 1 (x)y + ∂ 2 x F 2 (x)y 2 + p≥3 ∂ 2 x F p (x)y p . (1)
Now, the recursive relation of Theorem 3 for b n+2,n-p+2 implies for p ≥ 3:

∂ 2 x F p (x) = n≥0 b n+2,n-p+2 x n n! = n≥0 x n n! n-p+1 =0 ( + 1)! • n • b n-,(n-)-(p-1) = n≥0 (n + 1)! x n n! • n≥0 b n,n-p+1 x n n! = 1 (1 -x) 2 F p-1 (x).
Plugging this recurrence into the equation (1) (and using F 0 (x) = F 1 (x) = 0) gives:

∂ 2 x F (x, y) = ∂ 2 x F 2 (x)y 2 + p≥2 y (1 -x) 2 F p (x)y p .
It remains to simplify F 2 (x); the initial conditions of Theorem 3 (b n,n-2 = (n -1)! and b n,p = 0 for n < 2) imply that

F 2 (x) = n≥2 b n,n-2 x n n! = n≥2 (n -1)! n! x n = -ln(1 -x) -x .
This leads to the main differential equation:

∂ 2 x F (x, y) = ∂ 2 x F 2 (x)y 2 + y (1 -x) 2 F (x, y) = y (1 -x) 2 (y + F (x, y)) . (2)
First, by plug & prove, the solutions of ∂ 2 x F (x) = yF (x)/(1-x) 2 are a linear combination of (1-x) α , with α = (1 + √ 1 + 4y)/2, or α = (1 -√ 1 + 4y)/2. Now, F (x, y) = -y is a trivial particular solution of the non-homogeneous differential equation (2), so the general solution of this differential equation is of the form:

F (x, y) = K(y) • (1 -x) 1 2 (1+ √ 1+4y) + L(y) • (1 -x) 1 2 (1- √ 1+4y) -y.
Since F (0, y) = 0 and ∂F (x,y) ∂x x=0 = 0, we respectively deduce the two equations

K(y) + L(y) -y = 0 and -K(y) • (1 + 1 + 4y) -L(y) • (1 -1 + 4y) = 0 ,
thus we obtain

K(y) = y 2 √ 1 + 4y -1 √ 1 + 4y and L(y) = y 2 √ 1 + 4y + 1 √ 1 + 4y .
This gives

F (x, y) = y 2 √ 1 + 4y -1 √ 1 + 4y • (1 -x) 1 2 (1+ √ 1+4y) + y 2 √ 1 + 4y + 1 √ 1 + 4y • (1 -x) 1 2 (1- √ 1+4y) -y ,
and therefore the theorem, as B(x, y) = F (xy, 1/y).

Theorem 5 (Asymptotics). The exponential generating function for the number of basis permutations with respect to their length is given by

B(x) = B(x, 1) = n≥0 b n x n n! = √ 5 -1 2 √ 5 • (1 -x) 1+ √ 5 2 + √ 5 + 1 2 √ 5 • (1 -x) 1- √ 5 2 -1 = x 2 2! + 2 x 3 3! + 7 x 4 4! + 32 x 5 5! + 179 x 6 6! + 1182 x 7 7! + 8993 x 8 8! + O(x 9 ).
It is a D-finite transcendental function satisfying the following differential equation

B(x) -(1 -x) 2 ∂ 2 x B(x) + 1 = 0, B(0) = B (0) = 0 . (3)
Equivalently, its coefficients b n satisfy the recurrence

b n+2 = 2nb n+1 + (1 + n -n 2 )b n , b 0 = b 1 = 0, b 2 = 1 , (4)
and the asymptotics are given by

b n n! ∼ φ √ 5 Γ(φ -1) 1 n 2-φ (1 + o(1)) ,
where φ is the golden ratio φ = (1 + √ 5)/2, and Γ(z) := +∞ 0 t z-1 exp(-t)dt is the Euler gamma function.

Accordingly, a permutation of length n has a probability asymptotically 0 to be an element of the basis of forbidden patterns, however, this probability is not "very small" as it decays only polynomially:

Prob(s ∈ S n belongs to ∪ p∈N B p ) =
b n n! ≈ 0.499/n 0.381 (1 + o(1)) .

Proof: Setting y = 1 in the bivariate exponential generating function given in Theorem 4 gives B(x).

If a function B(z) = b n z n is D-finite (it is satisfying a linear differential equation, with polynomial coefficients in z), then its coefficients b n are polynomially recursive (in short, P-recursive): they satisfy a linear recurrence, with polynomial coefficients in n. See e.g. [START_REF] Flajolet | Analytic combinatorics[END_REF]; [START_REF] Stanley | Enumerative combinatorics[END_REF] for more on these two equivalent notions. Starting from the building blocks (1 -x) a , which are D-finite, and then using the closure properties of D-finite functions (by sum and product) gives the differential equation (3) (this is e.g. implemented in the Gfun Maple package, see [START_REF] Salvy | Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable[END_REF]). The recurrence is obtained by extracting the coefficient of x n on both sides of the differential equation. The asymptotics follows from a singularity analysis (see [START_REF] Flajolet | Analytic combinatorics[END_REF]) on each term of the shape (1 -x) a , indeed, for any a ∈ R which is not an integer, one has:

[x n ](1 -x) a = 1 Γ(-a)n 1+a 1 + 1 2 a(a + 1) 1 n + O( 1 n 2 ) .
Note that B(x, y) is D-finite in the variable x:

1 + yB(x, y) -(1 -xy) 2 ∂ 2 x B(x, y) = 0 with B(0, y) = (∂ x B)(0, y) = 0,
but it is not D-finite in the variable y. This follows from a saddle point analysis on B(1, y) = n β n y n , indeed the asymptotics of β n involve arbitrarily large (ln n) d , while the asymptotics of a D-finite function can only have a finite sum of such powers of log, see [START_REF] Flajolet | Analytic combinatorics[END_REF]. This argument is thus similar to a proof that (1 -y) 1-y is not D-finite.

Remark [Irrational critical exponent]: It is very seldom that a combinatorial problem leads to some asympotics involving an irrational number as exponent. In fact, in combinatorics and in statistical physics, most of the asymptotics of integer sequences are of the shape b n ∼ Cn α A n , and the exponent α which appears there is a key quantity: its value is often the signature of some universal phenomena (in physics, it is called a critical exponent). For D-finite sequences, the theory implies that it is an algebraic number, however, this exponent is very often -3/2, or a dyadic number (for the reasons explained in [START_REF] Banderier | Formulae and asymptotics for coefficients of algebraic functions[END_REF]), or a rational number (due to a result on G-functions). Indeed, a theorem (resulting from the works of Katz, André, Chudnovsky & Chudnovsky, see [START_REF] Chambert-Loir | Théorèmes d'algébricité en géométrie diophantienne[END_REF]) states that G-functions (D-finite functions with integer coefficients and non-zero radius of convergence) have a rational critical exponent. Now, instead of considering the exponential generating function B(x) = b n x n /n!, we may consider its inverse Borel transform, i.e., the ordinary generating function b n x n . It is also a D-finite function, because D-finite functions are closed by Hadamard product, and therefore the Borel transform (and the inverse Borel transform) of a D-finite function is D-finite (i.e., if the sequence b n is P-recursive, so are n!b n and b n /n!). We have thus a new D-finite function with integer coefficients and irrational critical exponent (involving the golden ratio φ), but this is not contradicting the G-function theorem, because, due to the multiplication by n!, we now have a 0 radius of convergence. In conclusion, we have the pleasure to have here one of the few examples in combinatorics of a problem leading to an irrational critical exponent. Other examples are given via the KPZ formula in physics, or via quantities related to quadtrees, see [START_REF] Flajolet | Analytic combinatorics[END_REF].

Pushing further the asymptotics from Theorem 5, we get the following limit law: Theorem 6 (Limit Law). In the model where all permutations of length n are equidistributed, a random permutation of length n in ∪ p∈N B p is typically a member of B p , for p ∼ n -(ln n)/ √ 5, with Gaussian fluctuations. Equivalently, the average number of left-to-right-maxima in a random basis permutation is p ∼ (ln n)/ √ 5 with Gaussian fluctuations.

Proof: This follows from the closed-expression for B(x, y), or from a singularity analysis of the differential equation. Indeed, the average and standard deviation follow from the computation of ∂ y B(x, y) and ∂ 2 y B(x, y) at y = 1. The Gaussian limit law follows from the quasi-power theorem applied to a variable exponent perturbation or to our non-confluent differential equation (see Theorem IX.11 and Theorem IX.18 from [START_REF] Flajolet | Analytic combinatorics[END_REF]).

As a random permutation of S n has ln n left-to-right maxima on average, the above theorem quantifies to what extent the right-jump process kills the left-to-right maxima when one starts from the identity permutation. For the combinatorial structure B p , it could be possible that its complementary set has a nicer structure. Those permutations not in the basis are for sure counted by u n = n! -b n ; this sequence satisfies u n+3 = (n + 1)(n 2 -n -1)u n -(3n 2 + 3n -1)u n+1 + 3(n + 1)u n+2 , which is still a nice recurrence but of order one more than the recurrence for b n , so it is a heuristic confirmation than b n is a more fundamental sequence than u n .

The first values of b n (the number of basis permutations of length n) are 1, 2, 7, 32, 179, 1182, 8993, 77440 for 2 ≤ n ≤ 9. We added this sequence to the On-line Encyclopedia of Integer Sequences (hereafter abbreviated OEIS), see Sloane and collaborators ( 2016 There is a vast literature in number theory analysing the modular congruences of famous sequences

Conclusion

In this article, we analysed the iteration of the process "a particle jumps to the right" in a permutation, and we gave the typical properties of the patterns which are not reached after p moves. We expect our approach (introducing a catalytic variable and getting a D-finite function) to work in many other cases. However, we already know a nice permutation class for which the basis is not D-finite. Indeed, as an extension of this work, an interesting question is to consider a model in which both right-jumps and left-jumps are allowed: this is a very natural process, also related to sorting algorithms and bioinformatics processes.

In a forthcoming work, we show that for this new model, the basis of forbidden minimal patterns for permutations obtained by p iterations of the process is related to Young Tableaux with 2 equally long first rows (but it is no longer D-finite, unlike the pure right-jump iteration process that we considered in this article). Another natural question is: is it the case that using e.g. the correspondence between records and cycles in permutations, there is an elegant process corresponding to a "particle jumps to the right", with permutations at distance p from identity being counted in terms of cycles in the permutation? To get direct "bijective" proofs of our formulae is also an interesting question: as a credo, it cannot be the case that such nice formulae/recurrences are only reached by solving differential equations (like we did in this article). It may be the case that a generating tree approach leads to the simple recurrence (4) we get for b n (see e.g. [START_REF] Banderier | Generating functions for generating trees[END_REF] or [START_REF] Barcucci | ECO: a methodology for the enumeration of combinatorial objects[END_REF]). With respect to the asymptotics, it is noteworthy that the process analysed in the present article involves the golden ratio (this is very unusual in combinatorics to have this constant as critical exponent): is it the trace of some universality class? (Like it is sometimes the case for problems coming from statistical mechanics, see our remark after Theorem 5.)

Last but not least, we already mentioned a vast literature of publications in number theory analysing the modular congruences of famous sequences (Pascal triangle, Fibonacci, Catalan, Motzkin, Apéry numbers, . . . , [START_REF] Deutsch | Congruences for Catalan and Motzkin numbers and related sequences[END_REF]; [START_REF] Xin | A short approach to Catalan numbers modulo 2 r[END_REF]). It seems to us that our approach to tackle them at the level of D-finite functions is new (see also [START_REF] Kauers | A method for determining the mod-2 k behaviour of recursive sequences, with applications to subgroup counting[END_REF]; [START_REF] Rowland | A case study in meta-automation: automatic generation of congruence automata for combinatorial sequences[END_REF]), and it would be worth analysing these properties in full generality. In this article, we proved by a mixture of Ansatz and brute force proof that b n mod m (where m can be any integer) is a periodic function (of period bounded by a polynomial in m). In fact, we prove in [START_REF] Banderier | On the period mod m of polynomially-recursive sequences: a case study[END_REF] that this period and the values of b n mod m for any given m can be made explicit.

Figure 2 :

 2 Figure2: The graphical representation of σ = 53621487. We show an occurrence of a pattern 213 with a dashed line; big green points are the left-to-right maxima and small black points are non-left-to-right-maxima.

Figure 4 :

 4 Figure 4: This histogram illustrates Theorem 6: the average number of left-to-right-maxima in a random basis permutation is p ∼ (ln n)/ √ 5 with Gaussian fluctuations. However the speed of convergence and the small amplitude of the variance makes that large values of n have to be considered to recognize clearly the bell curve of the Gaussian (above, this is a histogram for n = 4000).

Table 1 :

 1 ): Number bn,p of basis permutations of length n (the "minimal forbidden patterns" of Bp, or equivalently, with p + 1 non-left-to-right-maxima) where 2 ≤ n ≤ 11 and 0 ≤ p ≤ 6 (OEIS A265163). The last column contains βp := n bn,p (OEIS A265164); the last line contains bn := p bn,p (OEIS A265165).

	p\n 2 3 4 5	6	7	8	9	10	11	#B p
	0	1							1
	1	2 1							3
	2	6 8	1						15
	3	24 58	18	1				101
	4		120 444	244	32	1		841
	5			720 3708 3104	700	50	8232
	6				5040 33984 39708	13400	78732
	Σ	1 2 7 32 179 1182 8993 77440 744425 7901410

Acknowledgments: We thank our laboratories (LE2I and LIPN) for funding this collaboration, and presentation of preliminary results at the conferences AofA'15 and Permutation Patterns'15. We also thank the two referees for their careful reading.

-Moreira-Dos-Santos-

(Pascal triangle, Fibonacci, Catalan, Motzkin, Apéry numbers, see [START_REF] Deutsch | Congruences for Catalan and Motzkin numbers and related sequences[END_REF]; [START_REF] Xin | A short approach to Catalan numbers modulo 2 r[END_REF]; [START_REF] Rowland | A case study in meta-automation: automatic generation of congruence automata for combinatorial sequences[END_REF]; [START_REF] Kauers | A method for determining the mod-2 k behaviour of recursive sequences, with applications to subgroup counting[END_REF]). The properties of b n mod m are sometimes called "supercongruences" when m is the power of a prime number: many articles consider m = 2 r , or m = 3 r . We now give a result which holds for any m (not necessarily the power of a prime number).

Theorem 7 (Supercongruences for D-finite functions). Consider any P-recurrence of order r:

If the polynomial P 0 (n) is ultimately invertible mod m (i.e., gcd(P 0 (n), m) = 1, for all n large enough), then the sequence (u n ) is ultimately periodic (ii) mod m, and there is an algorithm to get this period.

In particular, recurrences such that P 0 (n) = 1 are periodic mod m. Accordingly, our sequence b n mod m (defined by recurrence (4)) is periodic for any m.

Proof: Indeed, as the leading term P 0 is invertible, we can write:

in which each term has just a finite set of possible values. What is more, for any polynomial P (n) with integer coefficients, P (n) mod m is of period p, for some p|m. (This follows from the fact that the sum and the product is preserving periodicity mod m, as we did not require in the definition of "period" that m is the smallest m such that the sequence is m periodic). Therefore, one can then construct a Markov chain (an automaton): the states are all the possible 2r + 1-tuples of values mod m for (P 0 (n), . . . , P r (n), u n-1 , . . . , u n-r ), and the recurrence dictates the transitions in this Markov chain.

The pigeonhole principle implies that there is a loop in this finite graph, and our period is thus one of the divisors of the length of this loop. Besides, the smallest period p and the first integer n 1 satisfying u n = u n+p mod m for n ≥ n 1 are such that p + n 1 is smaller than the number of states in the automaton, i.e. smaller than m 2r+1 , or more precisely smaller than φ(m)m 2r where φ(m) is the Euler totient function (the number of invertible elements modulo m, as P 0 (n) is ultimately invertible). This theorem explains the periodic behaviour of b n mod m. By brute-force computation, we can get b n mod m, for any given m. For example b n mod 15 is periodic of period 12: for n ≥ 9, one has b n mod 15 = (10, 5, 10, 10, 0, 10, 5, 10, 5, 5, 0, 5) ∞ .

The period can be quite large, for example b n mod 3617 has period 26158144. We say more on this phenomenon for P-recursive sequences in [START_REF] Banderier | On the period mod m of polynomially-recursive sequences: a case study[END_REF]. (ii) In the sequel, we will omit the word "ultimately": a periodic sequence of period p is thus a sequence for which u n+p = un for all large enough n. Some authors use the terminology "eventually periodic" instead.