scholarly journals Tableaux combinatorics for two-species PASEP probabilities

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Olya Mandelshtam
Keyword(s):  
Steady State ◽  
Stationary Distribution ◽  
Heavy Particle ◽  
Finite Lattice ◽  
Nous Proposons ◽  
Combinatorial Interpretation ◽  
One Dimensional ◽  
International Audience ◽  
The Right ◽  
Steady State Probabilities

International audience The goal of this paper is to provide a combinatorial expression for the steady state probabilities of the twospecies PASEP. In this model, there are two species of particles, one “heavy” and one “light”, on a one-dimensional finite lattice with open boundaries. Both particles can hop into adjacent holes to the right and left at rates 1 and $q$. Moreover, when the heavy and light particles are adjacent to each other, they can switch places as if the light particle were a hole. Additionally, the heavy particle can hop in and out at the boundary of the lattice. Our first result is a combinatorial interpretation for the stationary distribution at $q=0$ in terms of certain multi-Catalan tableaux. We provide an explicit determinantal formula for the steady state probabilities, as well as some general enumerative results for this case. We also describe a Markov process on these tableaux that projects to the two-species PASEP, and hence directly explains the connection between the two. Finally, we extend our formula for the stationary distribution to the $q=1$ case, using certain two-species alternative tableaux. Le but de ce document est de fournir une expression combinatoire décrivant les probabilités de l’état d’équilibre de PASEP à deux espèces. Dans ce modèle, il existe deux espèces de particules, une “lourde” et une “légère”, disposées sur un réseau fini unidimensionnel. Les deux particules peuvent sauter dans les trous adjacents à droite et à gauche, avec des probabilités proportionnelles à 1 et $q$. Par ailleurs, lorsque les particules lourdes et légères sont à côté l’une de l’autre, elles peuvent changer de place, comme si la particule légère était un trou. En outre, la particule lourde peut sauter dans et hors de la frontière du réseau. Notre premier résultat est une interprétation combinatoire de la distribution stationnaire dans le cas $q=0$, en termes de certains tableaux “multi-Catalan”. Nous proposons une formule explicite déterminantale pour les probabilités stationnaires, ainsi que plusieurs résultats énumératifs généraux pour ce cas. Nous décrivons aussi un processus de Markov sur ces tableaux, qui se projette sur le PASEP à deux espèces, et qui fournit donc directement une connexion entre les deux. Enfin, nous exprimons notre formule pour la distribution stationnaire dans le cas $q=1$, en utilisant certains tableaux alternatifs de deux espèces.

scholarly journals Rhombic alternative tableaux, assemblees of permutations, and the ASEP

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Olya Mandelshtam ◽  
Xavier Viennot
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Generating Functions ◽  
Finite Lattice ◽  
One Dimensional ◽  
Open Boundaries ◽  
Bijective Proof ◽  
Insertion Algorithm ◽  
International Audience ◽  
Steady State Probabilities

International audience In this paper, we introduce therhombic alternative tableaux, whose weight generating functions providecombinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there aretwo species of particles, oneheavyand onelight, on a one-dimensional finite lattice with open boundaries, and theparametersα,β, andqdescribe the hopping probabilities. The rhombic alternative tableaux are enumerated by theLah numbers, which also enumerate certainassembl ́ees of permutations. We describe a bijection between the rhombicalternative tableaux and these assembl ́ees. We also provide an insertion algorithm that gives a weight generatingfunction for the assemb ́ees. Combined, these results give a bijective proof for the weight generating function for therhombic alternative tableaux.

scholarly journals Enumeration of Cylindric Plane Partitions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Robin Langer
Keyword(s):  
Natural Generalization ◽  
Path Model ◽  
Plane Partition ◽  
Combinatorial Interpretation ◽  
Plane Partitions ◽  
Generating Series ◽  
International Audience ◽  
The Individual ◽  
The Right ◽  
Reverse Plane

International audience Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. As in the reverse plane partition case, the right hand side of this identity admits a simple factorization form in terms of the "hook lengths'' of the individual boxes in the underlying shape. The first result of this paper is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result of this paper is an explicit combinatorial interpretation of the Macdonald weight occurring in the $(q,t)$-analog in terms of the non-intersecting lattice path model for cylindric plane partitions. Les partitions planes cylindriques sont une généralisation naturelle des partitions planes renversées. Une série génératrice pour énumération des partitions planes cylindriques a été donnée récemment par Borodin. Comme dans le cas des partitions planes renversées, la partie droite de cette identité peut être factoriser en terme de "longueur d’équerres'' des carrés dans la forme sous-jacente. Le premier résultat de cet article est une nouvelle preuve bijective de l'identité de Borodin qui utilise le cadre de "diagramme de croissance'' de Fomin pour la correspondance de RSK généralisée. Le deuxième résultat de cette article est une $(q,t)$-déformation d'identité de Borodin qui généralise un résultat de Okada dans le cas des partitions planes renversées. Le troisième résultat de cet article est une formule combinatoire explicite pour le poids de Macdonald qui utilise le modèle des chemins non-intersectant pour les partitions planes cylindriques.

scholarly journals A product formula for the TASEP on a ring

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Erik Aas ◽  
Jonas Sjöstrand
Keyword(s):  
Stationary Distribution ◽  
Random Permutation ◽  
Product Formula ◽  
Combinatorial Interpretation ◽  
International Audience

International audience For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the $\textit{order}$ of the first entries is independent of the $\textit{order}$ of the last entries. The proof uses multi-line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated. Pour une permutation randomisée tirée de la mesure stationnaire du TASEP, nous démontrons, conditionnée à l’évènement que les premières lettres sont plus grandes que les dernières lettres, que l’ordre des petites lettres est indépendant de l’ordre des grandes lettres. La preuve utilise les files d’attente multilignes de Ferrari et Martin, et le théorème a une interprétation combinatoire énumérative dans ce contexte. Finalement, nous présentons une conjecture pour le cas où les petits et les grandes lettres ne sont pas séparées.

scholarly journals Statistics on staircase tableaux, eulerian and mahonian statistics

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Sylvie Corteel ◽  
Sandrine Dasse-Hartaut
Keyword(s):  
Nous Proposons ◽  
Combinatorial Interpretation ◽  
Eulerian Polynomials ◽  
Nous Permet ◽  
International Audience ◽  
Mahonian Statistics

International audience We give a simple bijection between some staircase tableaux and tables of inversion. Some nice properties of the bijection allows us to define some q-Eulerian polynomials related to the staircase tableaux. We also give a combinatorial interpretation of these q-Eulerian polynomials in terms of permutations. Nous proposons une bijection simple entre certains tableaux escalier et les tables d'inversion. Cette bijection nous permet de montrer que les statistiques Euleriennes et Mahoniennes sont naturelles sur les tableaux escalier. Nous définissons des polynômes q-Eulériens et en donnons une interprétation combinatoire.

Selection of one-dimensional sedimentation: models for on-line use

1995 ◽  
Vol 31 (2) ◽  
pp. 193-204 ◽  
Author(s):  
Koen Grijspeerdt ◽  
Peter Vanrolleghem ◽  
Willy Verstraete
Keyword(s):  
Steady State ◽  
Selection Criteria ◽  
Data Sets ◽  
Concentration Profiles ◽  
A Posteriori ◽  
One Dimensional ◽  
On Line ◽  
Dynamic Concentration ◽  
Selection Of ◽  
Modelling Task

A comparative study of several recently proposed one-dimensional sedimentation models has been made. This has been achieved by fitting these models to steady-state and dynamic concentration profiles obtained in a down-scaled secondary decanter. The models were evaluated with several a posteriori model selection criteria. Since the purpose of the modelling task is to do on-line simulations, the calculation time was used as one of the selection criteria. Finally, the practical identifiability of the models for the available data sets was also investigated. It could be concluded that the model of Takács et al. (1991) gave the most reliable results.

scholarly journals Reduced Visual Magnocellular Event-Related Potentials in Developmental Dyslexia

2021 ◽  
Vol 11 (1) ◽  
pp. 48
Author(s):  
John Stein
Keyword(s):  
Steady State ◽  
Developmental Dyslexia ◽  
Reading Ability ◽  
Event Related Potentials ◽  
Black And White ◽  
Reverse Pattern ◽  
Related Potentials ◽  
The Right ◽  
Visual Hemifields ◽  
Electroencephalogram Eeg

(1) Background—the magnocellular hypothesis proposes that impaired development of the visual timing systems in the brain that are mediated by magnocellular (M-) neurons is a major cause of dyslexia. Their function can now be assessed quite easily by analysing averaged visually evoked event-related potentials (VERPs) in the electroencephalogram (EEG). Such analysis might provide a useful, objective biomarker for diagnosing developmental dyslexia. (2) Methods—in adult dyslexics and normally reading controls, we recorded steady state VERPs, and their frequency content was computed using the fast Fourier transform. The visual stimulus was a black and white checker board whose checks reversed contrast every 100 ms. M- cells respond to this stimulus mainly at 10 Hz, whereas parvocells (P-) do so at 5 Hz. Left and right visual hemifields were stimulated separately in some subjects to see if there were latency differences between the M- inputs to the right vs. left hemispheres, and these were compared with the subjects’ handedness. (3) Results—Controls demonstrated a larger 10 Hz than 5 Hz fundamental peak in the spectra, whereas the dyslexics showed the reverse pattern. The ratio of subjects’ 10/5 Hz amplitudes predicted their reading ability. The latency of the 10 Hz peak was shorter during left than during right hemifield stimulation, and shorter in controls than in dyslexics. The latter correlated weakly with their handedness. (4) Conclusion—Steady state visual ERPs may conveniently be used to identify developmental dyslexia. However, due to the limited numbers of subjects in each sub-study, these results need confirmation.

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