scholarly journals On Floors and Ceilings of the $k$-Catalan Arrangement

10.37236/4121 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Marko Thiel
Keyword(s):  
Root System ◽  
Catalan Number

The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $\Phi$ is a well-studied object enumerated by the Fuß-Catalan number $Cat^{(k)}(\Phi)$. It is natural to refine this enumeration by considering floors and ceilings of dominant regions. A conjecture of Armstrong states that counting dominant regions by their number of floors of a certain height gives the same distribution as counting dominant regions by their number of ceilings of the same height. We prove this conjecture using a bijection that provides even more refined enumerative information.

scholarly journals $m$-noncrossing partitions and $m$-clusters

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Aslak Bakke Buan ◽  
Idun Reiten ◽  
Hugh Thomas
Keyword(s):  
Positive Integer ◽  
Root System ◽  
Cluster Algebra ◽  
Reflection Group ◽  
Catalan Number ◽  
Explicit Description ◽  
Noncrossing Partitions ◽  
Il Y A ◽  
International Audience ◽  
Exceptional Sequences

International audience Let $W$ be a finite crystallographic reflection group, with root system $\Phi$. Associated to $W$ there is a positive integer, the generalized Catalan number, which counts the clusters in the associated cluster algebra, the noncrossing partitions for $W$, and several other interesting sets. Bijections have been found between the clusters and the noncrossing partitions by Reading and Athanasiadis et al. There is a further generalization of the generalized Catalan number, sometimes called the Fuss-Catalan number for $W$, which we will denote $C_m(W)$. Here $m$ is a positive integer, and $C_1(W)$ is the usual generalized Catalan number. $C_m(W)$ counts the $m$-noncrossing partitions for $W$ and the $m$-clusters for $\Phi$. In this abstract, we will give an explicit description of a bijection between these two sets. The proof depends on a representation-theoretic reinterpretation of the problem, in terms of exceptional sequences of representations of quivers. Soit $W$ un groupe de réflexions fini et cristallographique, avec système de racines $\Phi$. Associé à $W$, il y a un entier positif, le nombre de Catalan généralisé, qui compte les amas dans l'algèbre amassée associée, les partitions non-croisées de $W$, et plusieurs autres ensembles intéressantes. Des bijections entre les amas et les partitions non-croisées ont été données par Reading et Athanasiadis et al. On peut encore généraliser le nombre de Catalan généralisé, obtenant le nombre Fuss-Catalan de $W$, que nous noterons $C_m(W)$. Ici $m$ est un entier positif, et $C_1(W)$ est le nombre Catalan généralisé standard. $C_m(W)$ compte les partitions $m$-non-croisées de $W$ et les $m$-amas de $\Phi$. Dans ce résumé, nous donnerons une bijection explicite entre ces deux ensembles. La démonstration dépend d'une réinterprétation des objets du point de vue des suites exceptionnelles de représentations de carquois.

Distribution of secoiridoid glycosides in the root system of the medicinal plant Gentiana lutea L. subsp. aurantiaca

2017 ◽  
Author(s):  
Ó González-López ◽  
S Mayo ◽  
Á Rodríguez-González ◽  
G Carro-Huerga ◽  
V Suárez Villanueva ◽  
...  
Keyword(s):  
Medicinal Plant ◽  
Root System ◽  
Gentiana Lutea ◽  
Secoiridoid Glycosides

EFFECTS OF DECP PROCESSING ON THE ROOT SYSTEM OF COTTON

2019 ◽  
Vol 2 (1) ◽  
pp. 33-37
Author(s):  
Komiljon Komilov ◽  
◽  
Dilfuzakhon Komilova
Keyword(s):  
Root System

Global Genome Expression Analysis of Transcription Factors under PEG Osmotic Stress in Rice Root System

2009 ◽  
Vol 35 (6) ◽  
pp. 1030-1037 ◽  
Author(s):  
Ting-Chen MA ◽  
Rong-Jun CHEN ◽  
Rong-Rong YU ◽  
Han-Lai ZENG ◽  
Duan-Pin ZHANG
Keyword(s):  
Transcription Factors ◽  
Osmotic Stress ◽  
Root System ◽  
Expression Analysis ◽  
Rice Root ◽  
Genome Expression

Influence of Planting Methods on Root System Morphological and Physiologi-cal Characteristics of Super Rice under High-yielding Cultivation Condition

2012 ◽  
Vol 37 (12) ◽  
pp. 2208-2220
Author(s):  
Jie LI ◽  
Hong-Cheng ZHANG ◽  
Yong CHANG ◽  
Jin-Long GONG ◽  
Ya-Jie HU ◽  
...  
Keyword(s):  
Root System ◽  
Cultivation Condition ◽  
Planting Methods

Variability in Root System Characteristics of Kleingrass Seedlings 1

Crop Science ◽  
1980 ◽  
Vol 20 (3) ◽  
pp. 384-386 ◽  
Author(s):  
C. R. Rischler ◽  
R. L. Monk
Keyword(s):  
Root System ◽  
System Characteristics

Utilization of a hydrate cellulose film for investigation of Arabidopsis thaliana L. root system growth and development

2020 ◽  
Vol 36 (1) ◽  
pp. 36-43
Author(s):  
I.O. Konovalova ◽  
T.N. Kudelina ◽  
S.O. Smolyanina ◽  
A.I. Lilienberg ◽  
T.N. Bibikova
Keyword(s):  
Arabidopsis Thaliana ◽  
Root System ◽  
Life Support ◽  
Higher Plants ◽  
Plant Root ◽  
Lateral Roots ◽  
Basic Research ◽  
Cellulose Film ◽  
A New Technique ◽  
Basic Research Project

A new technique for Arabidopsis thaliana cultivation has been proposed that combines the use of a phytogel-based nutrient medium and a hydrophilic membrane of hydrate cellulose film, separating the root system of the plant from the medium thickness. Growth rates of both main and lateral roots were faster in the plants cultivated on the surface of hydrate cellulose film than in the plants grown in the phytogel volume. The location of the root system on the surface of the transparent hydrate film simplifies its observation and analysis and facilitates plant transplantation with preservation of the root system configuration. The proposed technique allowed us to first assess the effect of exogenous auxin on the growth of lateral roots at the 5-6 developmental stage. methods to study plant root systems, hydrate cellulose film, A. thaliana, lateral roots, differential root growth rate, auxin The work was financially supported by the Russian Foundation for Basic Research (Project Bel_mol_a 19-54-04015) and the basic topic of the Russian Academy of Sciences - IBMP RAS «Regularities of the Influence of Extreme Environmental Factors on the Processes of Cultivation of Higher Plants and the Development of Japanese Quail Tissues at Different Stages of its Ontogenesis under the Conditions of Regenerative Life Support Systems».

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