The enumeration of generalized Tamari intervals

Let $v$ be a grid path made of north and east steps. The lattice $\rm{T{\scriptsize AM}}(v)$, based on all grid paths weakly above $v$ and sharing the same endpoints as $v$, was introduced by Pr\'eville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case $v=(NE)^n$. Our main contribution is that the enumeration of intervals in $\rm{T{\scriptsize AM}}(v)$, over all $v$ of length $n$, is given by $\frac{2 (3n+3)!}{(n+2)! (2n+3)!}$. This formula was first obtained by Tutte(1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection from these intervals in $\rm{T{\scriptsize AM}}(v)$ to non-separable planar maps.Comment: 19 pages, 11 figures. Title changed, originally titled "From generalized Tamari intervals to non-separable planar maps (extended abstract)", submitte

An extension of Tamari lattices

For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same number of north and east steps as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. Let $\overleftarrow{v}$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$. We do so by showing bijectively that the poset Tam$(v)$ is isomorphic to the poset based on rotation of full binary trees with the fixed canopy $v$, from which the duality follows easily. This also shows that Tam$(v)$ is a lattice for any path $v$. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height $n$, is a partition of the (smaller) lattices Tam$(v)$, where the $v$ are all the paths on the square grid that consist of $n-1$ unit steps. We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.Comment: 18 page

The representation of the symmetric group on m-Tamari intervals

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables.Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398, which will not be submitted to any journa

From generalized Tamari intervals to non-separable planar maps

International audienceLet v be a grid path made of north and east steps. The lattice TAM(v), based on all grid paths weakly above the grid path v sharing the same endpoints as v, was introduced by Pre ́ville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case v = (NE)n. They showed that TAM(v) is isomorphic to the dual of TAM(←−v ), where ←−v is the reverse of v with N and E exchanged. Our main contribution is a bijection from intervals in TAM(v) to non-separable planar maps. It follows that the number of intervals in TAM(v) over all v of length n is 2(3n+3)! (n+2)!(2n+3)! . This formula was first obtained by Tutte(1963) for non-separable planar maps

Tamari lattices and parking functions: proof of a conjecture of F. Bergeron

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^(m), which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of coinvariant spaces. He conjectured several intriguing formulas dealing with the enumeration of intervals in this lattice. One of them states that the number of intervals in T_n^(m) is $$\frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}.$$ This conjecture was proved recently, but in a non-bijective way, while its form strongly suggests a connection with plane trees. Here, we prove another conjecture of Bergeron, which deals with the number of labelled, intervals. An interval [P,Q] of T_n^(m) is labelled, if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. We prove that the number of labelled intervals in T_n^(m) is $${(m+1)^n(mn+1)^{n-2}}.$$ The form of these numbers suggests a connection with parking functions, but our proof is non-bijective. It is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. This equation involves a derivative and a divided difference, taken with respect to two additional variables. Solving this equation is the hardest part of the paper. Finding a bijective proof remains an open problem.Comment: 21 pp. This paper is now subsumed by arXiv:1202.5925, and will not be submitted to any journa

Électroarthrographie : mesure et analyse des potentiels électriques à la surface du genou pour l’évaluation de l’arthrose

RÉSUMÉ L’électroarthrographie (EAG) est une technique qui fut récemment découverte par un groupe de chercheurs de l’École Polytechnique de Montréal pour enregistrer et analyser les signaux électriques à la surface du genou qui sont produits par la compression du cartilage articulaire. Auparavant, il fut démontré que des potentiels électriques dits ‘d’écoulement’ peuvent être mesurés directement sur le cartilage articulaire lors de mesures invasives, et que ces potentiels diminuent lorsqu’il y a une dégradation du cartilage produite par une réduction de la concentration de glycosaminoglycanes ou lorsque le collagène perd de son intégrité. L’électroarthrographie constitue alors une méthode non-invasive de mesure des potentiels d’écoulement qui reflètent l’état de santé du cartilage articulaire. Le premier objectif de ce mémoire était de concevoir un protocole de prise de mesures EAG répétable et adapté à des populations gériatriques. Pour ce faire, nous avons fait des études de cas selon plusieurs protocoles différents. Pour le protocole qui a été retenu, la mise en charge du cartilage articulaire se fait par un simple transfert de poids d’une jambe à l’autre, alors que le sujet se tient debout. Ce transfert de poids est répété durant deux minutes pour permettre de réduire le niveau de bruit en calculant des valeurs EAG moyennes. Les potentiels électriques sont mesurés à l’aide d’électrodes collées sur la surface du genou en utilisant un système d’enregistrement sans fil comprenant huit canaux en courant continu. Pour traiter les signaux mesurés, un programme avec une interface graphique fut mis au point pour filtrer les signaux, corriger la dérive de la ligne de base et moyenner les multiples épisodes de mise en charge. La répétabilité des potentiels moyennés pour chacune des électrodes fut évaluée par une procédure de test-retest pour 14 sujets. Il s’est avéré que quatre sites de mesure, situés surtout au niveau de la ligne d’articulation, ont montré des coefficients de corrélation intra-classe statistiquement significatifs. Le deuxième objectif était de déterminer la provenance des signaux et d’évaluer la sensibilité de l’EAG à la dégradation du cartilage. Pour ce faire, nous avons comparé deux populations; un groupe Contrôle, avec des genoux sains, et un groupe dont un des genoux était diagnostiqué arthrosique, tandis que le genou controlatéral portait une prothèse totale du genou. Chaque groupe comprenait vingt sujets. Dans le groupe Contrôle, l’amplitude moyenne des signaux EAG était statistiquement plus élevée du côté médial que du côté latéral, ce qui reflète des forces au niveau de l’articulation qui sont reconnues être plus élevées du côté médial que du----------ABSTRACT Electroarthrography (EAG) is a technique that was recently discovered by a group of researchers from the Ecole Polytechnique de Montreal, to record and analyze the electrical signals on the surface of the knee that are produced by the compression of articular cartilage. Previously, it has been shown that electrical potentials called 'streaming potentials' can be measured directly on the surface of the articular cartilage and that cartilage degradation, produced by a reduction in the concentration of glycosaminoglycans or collagen content, can decrease the amplitude of the streaming potentials. Eectroarthrography is therefore a non-invasive method for measuring electrical potential that reflect streaming potentials and the status of articular cartilage. The first objective of this project was to design a protocol that can provide repeatable EAG measurements and that is suitable for geriatric populations. To do this, we performed case studies using several different protocols. For our final protocol, the loading of articular cartilage is performed with a simple transfer of weight from one leg to another while the subject is standing. This weight transfer is repeated for two minutes to reduce the noise level by calculating average EAG values. Electrical potentials are measured with electrodes placed on the surface of the knee using a wireless recording system with eight channels. To process the measured signals, a program with a graphic user interface was developed to filter the signals, correct the baseline drift and average the multiple loading episodes. The repeatability of the averaged EAG for each of the electrodes was evaluated by a test-retest procedure for 14 subjects. We found that four measurement sites, mostly located over the joint line, showed statistically significant intraclass correlation coefficients. The second objective was to determine the origin of the EAG signals and to evaluate the sensitivity of the EAG for the detection of cartilage degradation. We thus compared two populations: a control group with healthy knees and a group of patients with osteoarthritis diagnosed for one knee, while the contralateral knee had a total knee replacement. Each group consisted of twenty subjects. In the control group, the mean amplitude of the EAG signal was statistically higher for the medial side than for the lateral side, reflecting joint forces that are known to be higher in the medial side than on the lateral side. For EAG signals recorded over the total knee replacement, their mean amplitude was statistically zero, reflecting the absence of cartilage. For signals collected over the arthritic knees, their mean amplitude was statisticall

The number of intervals in the m-Tamari lattices

An m-ballot path of size n is a path on the square grid consisting of north and east steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice, which generalizes the usual Tamari lattice obtained when m=1. We prove that the number of intervals in this lattice is $$\frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}.$$ This formula was recently conjectured by Bergeron in connection with the study of coinvariant spaces. The case m=1 was proved a few years ago by Chapoton. Our proof is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. The solution of this equation is an algebraic series, obtained by a guess-and-check approach. Finding a bijective proof remains an open problem.Comment: 19 page

Synthèse de ligands disaccharidiques de la lectine PA-IIL de Pseudomonas aeruginosa impliquée dans la fibrose kystique

La fibrose kystique (FK) est la maladie génétique mortelle la plus répandue chez les jeunes \ud Canadiens. La colonisation des poumons par une bactérie opportuniste, Pseudomonas aeruginosa, est la principale cause de morbidité et de mortalité chez les patients FK. La maladie est causée par des mutations du gène codant pour la protéine CFTR (Cystic Fibrosis Transmembrane conductance Regulator) qui agit comme canal à ions chlorures. Ces modifications entraînent notamment une surexpression d'oligosaccharides fucosylés à la surface de l'épithélium pulmonaire. Le processus d'adhésion de la bactérie à la surface des cellules de l'épithélium pulmonaire est causé par la présence de deux lectines à la surface de la bactérie. Nous nous intéressons principalement à l'une d'entre elles, une lectine calcium dépendante qui reconnaît particulièrement le L-fucose (PA-IlL). Des études cristallographiques menées sur PA-IlL, en complexe avec divers ligands naturels tel que le Lewis a, ont permis d'identifier plusieurs éléments essentiels à l'obtention d'une forte interaction ligand-lectine. Basé sur ces études, le projet de recherche a consisté en la synthèse de différents analogues d'un disaccharide composé d'une unité fucose et glucosamine du type L-Fuc-a(1→4)-D-GIcNAcβ intimement impliqué dans le site de liaison de la protéine. Différents glycoclusters du disaccharide ont été synthétisés en utilisant la 'Click Chemistry'. De plus, quelques disaccharides modifiés en position C-2 et C-6 ont aussi été synthétisés. Les disaccharides ainsi que les glycoclusters ont été testés sur la PA-IlL en utilisant un test d'inhibition compétitive (ELLA). Les dérivés disaccharidiques ont montrés une constante de dissociation (Kd = 310 nM) dans le même ordre de grandeur que celle du meilleur ligand naturel Lewis a (Kd = 210 nM) connu jusqu'à maintenant pour la PA-IlL. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Pseudomonas aeruginosa, Fibrose kystique, Lectine PA-IlL, Lewis a, Glycosylation, «click chemistry», Glycoclusters

DNA methylation and single nucleotide variants in the brain-derived neurotrophic factor (BDNF) and oxytocin receptor (OXTR) genes are associated with anxiety/depression in older women

Background: Environmental effects and personal experiences could be expressed in individuals through epigenetic non-structural changes such as DNA methylation. This methylation could up- regulate or down-regulate corresponding gene expressions and modify related phenotypes. DNA methylation increases with aging and could be related to the late expression of some forms of mental disease. The objective of this study was to evaluate the association between anxiety disorders and/or depression in older women and DNA methylation for four genes related to anxiety or depression. Methods: Women aged 65 and older with (n = 19) or without (n = 24) anxiety disorders and/or major depressive episode (DSM-IV), were recruited. DNA methylation and single nucleotide variant (SNV) were evaluated from saliva, respectively by pyrosequencing and by PCR, for the following genes: brain-derived neurotrophic factor (BDNF; rs6265), oxytocin receptor (OXTR; rs53576), serotonin transporter (SLC6A4; rs25531), and apolipoprotein E (APOE; rs429358 and rs7412). Results: A greater BDNF DNA methylation was observed in subjects with anxiety/depression compared to control group subjects (Mean: 2.92 SD ± 0.74 vs. 2.34 ± 0.42; p= 0.0026). This difference was more pronounced in subjects carrying the BDNF rs6265 CT genotype (2.99 ± 0.41 vs. 2.27 ± 0.26; p= 0.0006) than those carrying the CC genotype (p= 0.0332); no subjects with the TT genotype were observed. For OXTR, a greater DNA methylation was observed in subjects with anxiety/depression, but only for those carrying the AA genotype of the OXTR rs53576 SNV, more particularly at one out of the seven CpGs studied (7.01 ± 0.94 vs. 4.44 ± 1.11; p= 0.0063). No significant differences were observed for APOE and SLC6A4. Conclusion: These results suggest that DNA methylation in interaction with SNV variations in BDNF and OXTR, are associated with the occurrence of anxiety/depression in older women