scholarly journals The representation of the symmetric group on $m$-Tamari intervals (conference version)

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Guillaume Chapuy ◽  
Louis-François Préville-Ratelle
Keyword(s):  
Symmetric Group ◽  
Lattice Structure ◽  
Parking Functions ◽  
Tamari Lattice ◽  
The North ◽  
Starting Point ◽  
International Audience ◽  
Associated Representation ◽  
Frobenius Series ◽  
Refined Version

International audience 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 $\mathcal{T}{_n}^{(m)}$, which generalizes the usual Tamari lattice $\mathcal{T}n$ obtained when $m=1$. This lattice was introduced by F. Bergeron in connection with the study of diagonally coinvariant spaces in three sets of $n$ variables. The representation of the symmetric group $\mathfrak{S}_n$ on these spaces is conjectured to be closely related to the natural representation of $\mathfrak{S}_n$ on (labelled) intervals of the $m$-Tamari lattice studied in this paper. An interval $[P,Q$] of $\mathcal{T}{_n}^{(m)}$ 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 $\mathfrak{S}_n$ acts on labelled intervals of $\mathcal{T}{_n}^{(m)}$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 $\mathfrak{S}_n$. 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. The form of this equation is highly non-standard: it 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.

scholarly journals Tamari Lattices for Parabolic Quotients of the Symmetric Group

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Henri Mühle ◽  
Nathan Williams
Keyword(s):  
Symmetric Group ◽  
Coxeter Groups ◽  
Weak Order ◽  
Set Partitions ◽  
Tamari Lattice ◽  
Nous Montrons ◽  
International Audience ◽  
Tamari Lattices

International audience We present a generalization of the Tamari lattice to parabolic quotients of the symmetric group. More precisely, we generalize the notions of 231-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients, and show bijectively that these sets are equinumerous. Furthermore, the restriction of weak order on the parabolic quotient to the parabolic 231-avoiding permutations is a lattice quotient. Lastly, we suggest how to extend these constructions to all Coxeter groups. Nous présentons une généralisation du treillis de Tamari aux quotients paraboliques du groupe symétrique. Plus précisément, nous généralisons les notions de permutations qui évitent le motif 231, les partitions non-croisées, et les partitions non-emboîtées aux quotients paraboliques, et nous montrons de façon bijective que ces ensembles sont équipotents. En restreignant l’ordre faible du quotient parabolique aux permutations paraboliques qui évitent le motif 231, on obtient un quotient de treillis d’ordre faible. Enfin, nous suggérons comment étendre ces constructions à tous les groupes de Coxeter.

scholarly journals On a Subposet of the Tamari Lattice

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sebastian A. Csar ◽  
Rik Sengupta ◽  
Warut Suksompong
Keyword(s):  
Symmetric Group ◽  
Partial Order ◽  
Tamari Lattice ◽  
Nous Montrons ◽  
International Audience ◽  
The Common ◽  
Binary Functions

International audience We discuss some properties of a subposet of the Tamari lattice introduced by Pallo (1986), which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We relate this poset to a partial order on the symmetric group studied by Edelman (1989). Nous discutons d'un subposet du treillis de Tamari introduit par Pallo. Nous appellons ce poset le comb poset. Nous montrons que trois fonctions binaires qui ne se comptent pas bien dans le trellis de Tamari se comptent bien dans un intervalle du comb poset : distance dans le trellis de Tamari, le supremum et l'infimum et les parsewords communs. De plus, nous discutons un rapport entre ce poset et un ordre partiel dans le groupe symétrique étudié par Edelman.

scholarly journals Extending the parking space

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Andrew Berget ◽  
Brendon Rhoades
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Algebraic Combinatorics ◽  
Parking Space ◽  
Parking Functions ◽  
Dyck Paths ◽  
International Audience ◽  
Special Case

International audience The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $\mathrm{Park}_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization. L’action du groupe symétrique $S_n$ sur l’ensemble $\mathrm{Park}_n$ des fonctions de stationnement de longueur $n$ a reçu beaucoup d’attention dans la combinatoire algébrique. Nous démontrons que l’action de $S_n$ sur $\mathrm{Park}_n$ s’étend à une action de $S_{n+1}$. Plus précisément, nous construisons un gradué $S_{n+1}$-module $V_n$ telles que la restriction de $S_n$ est isomorphe à $\mathrm{Park}_n$. Nous décrivons la $S_n$-Frobenius caractères des modules $V_n$ à tous les degrés et décrivent le $S_{n+1}$-Frobenius caractères de $V_n$ en degrés extrêmes. Nous donnons une généralisation bivariée $V_n^{(\ell, m)}$ de notre module $V_n$ dont la représentation théorie est régie par une généralisation bivariée des chemins de Dyck. Une généralisation Fuss de nos résultats est un cas particulier de cette généralisation bivariée.

scholarly journals An extension of Tamari lattices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Louis-François Préville-Ratelle ◽  
Xavier Viennot
Keyword(s):  
Symmetric Group ◽  
Binary Tree ◽  
Prime Numbers ◽  
Binary Trees ◽  
Square Lattice ◽  
Reverse Order ◽  
Tamari Lattice ◽  
Nous Montrons ◽  
International Audience ◽  
Tamari Lattices

International audience For any finite path $v$ on the square lattice consisting of north and east unit steps, we construct a poset Tam$(v)$ that consists of all the paths lying weakly above $v$ with the same endpoints as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. In particular this solves the problem of extending the $m$-Tamari lattice to any pair $(a; b)$ of relatively prime numbers in the context of the so-called rational Catalan combinatorics.For that purpose we introduce the notion of canopy of a binary tree and explicit a bijection between pairs $(u; v)$ of paths in Tam$(v)$ and binary trees with canopy $v$. Let $(\overleftarrow{v})$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order and exchanging east and north steps. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$ and that Tam$(v)$ is isomorphic to the set of binary trees having the canopy $v$, which is an interval of the ordinary Tamari lattice. Thus the usual Tamari lattice is partitioned into (smaller) lattices Tam$(v)$, where the $v$’s are all the paths of length $n-1$ on the square lattice.We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group. Pour tout chemin $v$ sur le réseau carré formé de pas Nord et Est, nous construisons un ensemble partiellement ordonné Tam $(v)$ dont les éléments sont les chemins au dessus de $v$ et ayant les mêmes extrémités. Pour certains choix de $v$ nous retrouvons le classique treillis de Tamari ainsi que son extension $m$-Tamari. En particulier nous résolvons le problème d’étendre le treillis $m$-Tamari à toute paire $(a; b)$ d’entiers premiers entre eux dans le contexte de la combinatoire rationnelle de Catalan.Pour ceci nous introduisons la notion de canopée d’un arbre binaire et explicitons une bijection entre les paires $(u; v)$ de chemins dans Tam$(v)$ et les arbres binaires ayant la canopée $v$. Soit $(\overleftarrow{v})$ le chemin obtenu en lisant les pas en ordre inverse et en échangeant les pas Est et Nord. Nous montrons que Tam$(v)$ est isomorphe au dual de Tam$(\overleftarrow{v})$ et que Tam$(v)$ est isomorphe à l’ensemble des arbres binaires ayant la canopée $v$, qui est un intervalle du treillis de Tamari ordinaire. Ainsi le traditionnel treillis de Tamari admet une partition en plus petits treillis Tam$(v)$, où les $v$ sont tous les chemins de longueur $n-1$ sur le réseau carré. Enfin nous explicitons les liens possibles entre l’ensemble ordonné Tam$(v)$ et (la combinatoire des) espaces diagonaux coinvariants généralisés du groupe symétrique.

Vegetation structure of tundra and taiga on the map of the natural vegetation of Europe

2007 ◽  
pp. 13-22 ◽  
Author(s):  
T. K. Yurkovskaya
Keyword(s):  
Vegetation Cover ◽  
Abiotic Factors ◽  
Self Regulation ◽  
Space Structure ◽  
Middle Taiga ◽  
Russian Plain ◽  
Forest Communities ◽  
The North ◽  
Tundra Vegetation ◽  
Starting Point

I have focused only on some features of structure in the taiga vegetation cover. In conclusion I would like to tell some words about the causes of complicated space structure of the taiga and tundra vegetation cover. The causes of latitudinal differentiation are climatic undoubtedly, but heterogeneity of vegetation cover within the limits of tundra and taiga subzones is accounted for different factors. In tundra abiogenic factors prevail, first of all the permafrost processes. That is the reason why tundra vegetation cover is so sensible to any disturbances and so hard regenerates after various transformations. In taiga the space structure is mostly the result of self-regulation and self- restoration of biota. The abiotic factors, certainly, play significant role, but they recede to the second plan. So we showed that in the north and middle taiga the structure of vegetation cover, during the Holocene up to present time, is determined in many respects by the increasing role of mires. Suffice it to look at the map of distribution of mires in order to estimate their role in vegetation cover of the easteuropean taiga (Yurkovskaya, 1980). So, the increase of mire area on the Russian Plain in m2/year per 1000 ha varies between 200 and 700, the average increas is ca 300—400 m2/year (Elina et all., 2000). The mires favour peniplenization and unite the separate areas of forest communities into the whole by means of forming the buffer paludificated territories (various hydrophilous variants of forest communities). But if mires, at all their stability, after destroying practically don't restore, the forests even after continuous cuttings restore their structure and composition through the series of successional stages unless an ecotope is damaged completely. Hence the space structure of taiga is the result, first of all, self development and self regulation of its vegetation cover. But, as it is known, at present time the process of destruction of natural biota has gone too far that the question arises not only about supporting its state and structure but also about the survival of the mankind itself. In this regard the vegetation map of Europe is the invaluable basis, which gives the starting point for all conservational, ecological and economical measures. But it is important to learn reading and using the map. And this is one of our actual goals.

Nachhaltigkeit im Wald – vor und hinter der Welle der Globalisierung | Sustainable Forest Management – In Front and Behind the Wave of Globalization

2000 ◽  
Vol 151 (12) ◽  
pp. 502-507
Author(s):  
Christian Küchli
Keyword(s):  
Forest Management ◽  
Cultural Landscape ◽  
Sustainable Forest Management ◽  
Essential Element ◽  
Specific Situation ◽  
The South ◽  
The North ◽  
Time Space ◽  
Starting Point ◽  
The Impact

Are there any common patterns in the transition processes from traditional and more or less sustainable forest management to exploitative use, which can regularly be observed both in central Europe and in the countries of the South (e.g. India or Indonesia)? Attempts were made with a time-space-model to typify those force fields, in which traditional sustainable forest management is undermined and is then transformed into a modern type of sustainable forest management. Although it is unlikely that the history of the North will become the future of the South, the glimpse into the northern past offers a useful starting point for the understanding of the current situation in the South, which in turn could stimulate the debate on development. For instance, the patterns which stand behind the conflicts on forest use in the Himalayas are very similar to the conflicts in the Alps. In the same way, the impact of socio-economic changes on the environment – key word ‹globalisation› – is often much the same. To recognize comparable patterns can be very valuable because it can act as a stimulant for the search of political, legal and technical solutions adapted to a specific situation. For the global community the realization of the way political-economic alliances work at the head of the ‹globalisationwave›can only signify to carry on trying to find a common language and understanding at the negotiation tables. On the lee side of the destructive breaker it is necessary to conserve and care for what survived. As it was the case in Switzerland these forest islands could once become the germination points for the genesis of a cultural landscape, where close-to-nature managed forests will constitute an essential element.

From single-species advice to mixed-species management: taking the next step

2004 ◽  
Vol 61 (8) ◽  
pp. 1398-1409 ◽  
Author(s):  
Morten Vinther ◽  
Stuart A. Reeves ◽  
Kenneth R. Patterson
Keyword(s):  
North Sea ◽  
Fishery Management ◽  
Single Species ◽  
Mixed Species ◽  
The North ◽  
Recent Approach ◽  
Starting Point ◽  
The North Sea ◽  
Management Advice ◽  
Management Priorities

Abstract Fishery management advice has traditionally been given on a stock-by-stock basis. Recent problems in implementing this advice, particularly for the demersal fisheries of the North Sea, have highlighted the limitations of the approach. In the long term, it would be desirable to give advice that accounts for mixed-fishery effects, but in the short term there is a need for approaches to resolve the conflicting management advice for different species within the same fishery, and to generate catch or effort advice that accounts for the mixed-species nature of the fishery. This paper documents a recent approach used to address these problems. The approach takes the single-species advice for each species in the fishery as a starting point, then attempts to resolve it into consistent catch or effort advice using fleet-disaggregated catch forecasts in combination with explicitly stated management priorities for each stock. Results are presented for the groundfish fisheries of the North Sea, and these show that the development of such approaches will also require development of the ways in which catch data are collected and compiled.

scholarly journals Southeast Asia as One of World’s Primary Sources of Biotic Recolonization Following Anthropocene Extinctions

2021 ◽  
Vol 9 ◽  
Author(s):  
Şerban Procheş ◽  
Syd Ramdhani ◽  
Alice C. Hughes ◽  
Lian Pin Koh
Keyword(s):  
Distribution Patterns ◽  
Climatic Conditions ◽  
Habitat Destruction ◽  
Conservation Priorities ◽  
High Survival ◽  
The North ◽  
Starting Point ◽  
Ancient Lineage ◽  
The Face ◽  
Lineage Diversity

The plight of Southeast Asia’s animals, plants and ecosystems in the face of unsustainable exploitation and habitat destruction has been illustrated in several recent studies, despite often falling outside the global discourse on global conservation priorities. Here, we collate biogeographic and phylogenetic information to argue that this beleaguered region is one of world’s primary macrorefugia, and possibly its best chance of regaining its natural biodiversity distribution patterns after the current Anthropocene upheaval. The region uniquely combines top diversity values in (a) ancient lineage diversity and (b) cosmopolitan lineage diversity, suggesting that it has acted in the past as a biodiversity museum and source of global colonization. This is at least partly due to the interplay between latitudinal diversity gradients and continental connectivity patterns. However, the peak values in South China/North Indochina for cosmopolitan tetrapods and their sister lineages suggest that a key feature is also the availability of diverse climatic conditions. In particular, the north-south orientation of the mountain ranges here has allowed for rapid recolonization within the region following past climatic changes, resulting in high survival values and overall exceptional relict lineage diversity. From this starting point, global colonization occurred on multiple occasions. It is hoped that, with urgent action, the region can once again fulfill this function.

scholarly journals Partitions of an Integer into Powers

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Matthieu Latapy
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Tree Structure ◽  
Dynamical Model ◽  
International Audience ◽  
Partitions Of Integers ◽  
Natural Way

International audience In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.

scholarly journals ANALYSIS ON 3D RECONSTRUCTION OF THE MONUMENT TO HEROES AS A TOOL FOR A CONCEPTUAL AND METHODOLOGICAL APPROACH IN THE PATRIMONIZATION AND EVALUATION OF CULTURAL INTEREST GOODS

2019 ◽  
Vol XLII-2/W17 ◽  
pp. 279-286
Author(s):  
M. I. Rodriguez-Laiton ◽  
H. A. León-Vega ◽  
E. Upegui
Keyword(s):  
Public Space ◽  
Low Cost ◽  
Methodological Approach ◽  
The Public ◽  
Future Developments ◽  
The North ◽  
North Side ◽  
Starting Point ◽  
Simon Bolivar ◽  
The City

Abstract. The following article describes the implementation of a methodology for the structural reconstruction of the Heroes monument and the statue on the north side of Simon Bolivar Ecuestre located between the intersection of the north highway and 80th Street in Bogota (Colombia) from the acquisition of SFM photogrammetry methods and images, using low-cost sensors for this process and making use of drones from the obtaining of frames of a video to for areas with lower altimetric reach, and thereby creating an analysis in their accuracy, sizing and quality within the framework of appropriation and documentation of the cultural assets in the public space of the city Bogotá taking this data as a starting point for future developments in the process of 3D reconstructions Colombia.

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