Pub Date : 2014-05-20DOI: 10.4310/JOC.2017.V8.N4.A2
Suho Oh, David E. Speyer
International audience Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other.
{"title":"Links in the complex of weakly separated collections","authors":"Suho Oh, David E. Speyer","doi":"10.4310/JOC.2017.V8.N4.A2","DOIUrl":"https://doi.org/10.4310/JOC.2017.V8.N4.A2","url":null,"abstract":"International audience\u0000 \u0000 Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other.\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2014-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70474378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-04DOI: 10.1007/978-3-319-00200-2_13
Vincent Pilaud, Christian Stump
{"title":"EL-labelings and canonical spanning trees for subword complexes","authors":"Vincent Pilaud, Christian Stump","doi":"10.1007/978-3-319-00200-2_13","DOIUrl":"https://doi.org/10.1007/978-3-319-00200-2_13","url":null,"abstract":"","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2012-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51000093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description and a relevant Minkowski sum decomposition of generalized associahedra.
{"title":"Generalized associahedra via brick polytopes","authors":"Vincent Pilaud, Christian Stump","doi":"10.15488/1671","DOIUrl":"https://doi.org/10.15488/1671","url":null,"abstract":"We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description and a relevant Minkowski sum decomposition of generalized associahedra.","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2012-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67149041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cesar Ceballos, Jean-Philippe Labbé, Christian Stump
We present a family of simplicial complexes called emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex.
{"title":"Multi-cluster complexes","authors":"Cesar Ceballos, Jean-Philippe Labbé, Christian Stump","doi":"10.15488/1672","DOIUrl":"https://doi.org/10.15488/1672","url":null,"abstract":"We present a family of simplicial complexes called emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex.","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67149102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Idan Eisner, Alek Vainshtein. Domination analysis for scheduling on non preemptive uniformly related machines. Broutin, Nicolas and Devroye, Luc. 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’12), 2012, Montreal, Canada. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA’12), pp.81-82, 2012, DMTCS Proceedings.
{"title":"Domination analysis for scheduling on non preemptive uniformly related machines","authors":"Idan Eisner, A. Vainshtein","doi":"10.46298/DMTCS.2985","DOIUrl":"https://doi.org/10.46298/DMTCS.2985","url":null,"abstract":"Idan Eisner, Alek Vainshtein. Domination analysis for scheduling on non preemptive uniformly related machines. Broutin, Nicolas and Devroye, Luc. 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’12), 2012, Montreal, Canada. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA’12), pp.81-82, 2012, DMTCS Proceedings.","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70481004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated.
{"title":"Enumerating projective reflection groups","authors":"Riccardo Biagioli, Fabrizio Caselli","doi":"10.46298/DMTCS.2898","DOIUrl":"https://doi.org/10.46298/DMTCS.2898","url":null,"abstract":"Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated.","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2011-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70480901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present $textit{type preserving}$ bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner. The bijections for the abstract Coxeter types $B$, $C$ and $D$ are new in the literature. To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all cases. The statistics can be then seen to be counted by the generalized Catalan numbers Cat$(W)$ when $W$ is a classical reflection group. In particular, the statistics of type $A$ appear as a new explicit example of objects that are counted by the classical Catalan numbers.
{"title":"Bijections between noncrossing and nonnesting partitions for classical reflection groups","authors":"Alex Fink, Benjamin Iriarte Giraldo","doi":"10.4171/PM/1869","DOIUrl":"https://doi.org/10.4171/PM/1869","url":null,"abstract":"We present $textit{type preserving}$ bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner. The bijections for the abstract Coxeter types $B$, $C$ and $D$ are new in the literature. To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all cases. The statistics can be then seen to be counted by the generalized Catalan numbers Cat$(W)$ when $W$ is a classical reflection group. In particular, the statistics of type $A$ appear as a new explicit example of objects that are counted by the classical Catalan numbers.","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2010-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/1869","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70891795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions.
{"title":"Hyperplane Arrangements and Diagonal Harmonics","authors":"D. Armstrong","doi":"10.46298/DMTCS.2889","DOIUrl":"https://doi.org/10.46298/DMTCS.2889","url":null,"abstract":"In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to \"extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions.","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2010-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70480847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-02-19DOI: 10.1090/S0002-9947-2011-05494-2
Sheng Chen, Nan Li, Steven V. Sam
Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of $P(n)$ to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in $P(n)$ is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related.
{"title":"Generalized Ehrhart polynomials","authors":"Sheng Chen, Nan Li, Steven V. Sam","doi":"10.1090/S0002-9947-2011-05494-2","DOIUrl":"https://doi.org/10.1090/S0002-9947-2011-05494-2","url":null,"abstract":"Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of $P(n)$ to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in $P(n)$ is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related.","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2010-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0002-9947-2011-05494-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60960271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Anagnostopoulos, C. Dombry, N. Guillotin-Plantard, I. Kontoyiannis, E. Upfal
We consider a stochastic version of the $k$-server problem in which $k$ servers move on a circle to satisfy stochastically generated requests. The requests are independent and identically distributed according to an arbitrary distribution on a circle, which is either discrete or continuous. The cost of serving a request is the distance that a server needs to move to reach the request. The goal is to minimize the steady-state expected cost induced by the requests. We study the performance of a greedy strategy, focusing, in particular, on its convergence properties and the interplay between the discrete and continuous versions of the process.
{"title":"Stochastic Analysis of the $k$-Server Problem on the Circle","authors":"A. Anagnostopoulos, C. Dombry, N. Guillotin-Plantard, I. Kontoyiannis, E. Upfal","doi":"10.46298/DMTCS.2791","DOIUrl":"https://doi.org/10.46298/DMTCS.2791","url":null,"abstract":"We consider a stochastic version of the $k$-server problem in which $k$ servers move on a circle to satisfy stochastically generated requests. The requests are independent and identically distributed according to an arbitrary distribution on a circle, which is either discrete or continuous. The cost of serving a request is the distance that a server needs to move to reach the request. The goal is to minimize the steady-state expected cost induced by the requests. We study the performance of a greedy strategy, focusing, in particular, on its convergence properties and the interplay between the discrete and continuous versions of the process.","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70481083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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