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Linear Inequalities for Flags in Graded Partially Ordered Sets

Louis J. Billera, Gábor Hetyei
2000 Journal of combinatorial theory. Series A  
We describe the strongest possible inequalities for graded partially ordered sets of rank at most 5. Academic Press  ...  The closure of the convex cone generated by all flag f-vectors of graded partially ordered sets is shown to be polyhedral.  ...  ACKNOWLEDGMENT We are grateful to Margaret Bayer, Vesselin Gasharov and Swapneel Mahajan for pointing out misprints in previous versions of this paper.  ... 
doi:10.1006/jcta.1999.3008 fatcat:h3vntjbogrezjne2cmcjgr2z44

Flag Vectors of Eulerian Partially Ordered Sets

Margaret M. Bayer, Gábor Hetyei
2001 European journal of combinatorics (Print)  
The closed cone of flag vectors of Eulerian partially ordered sets is studied. A new family of linear inequalities valid for Eulerian flag vectors is given. Half-Eulerian posets are defined.  ...  The cone of Eulerian flag vectors is completely determined up through rank seven.  ...  G Hetyei was partially supported by Hungarian National Foundation for Scientific Research grant no. F 032325.  ... 
doi:10.1006/eujc.2000.0414 fatcat:pao7v4tqqrax3fwzzsyb7u2554

Flag vectors of Eulerian partially ordered sets [article]

Margaret M. Bayer
1999 arXiv   pre-print
A new family of linear inequalities valid for flag vectors of Eulerian posets is given.  ...  The closed cone of flag vectors of Eulerian partially ordered sets is studied. It is completely determined up through rank seven. Half-Eulerian posets are defined.  ...  graded partially ordered sets.  ... 
arXiv:math/9907144v1 fatcat:eokcix5euvdktpbsqup6m2m7se

Generalizations of Eulerian partially ordered sets, flag numbers, and the Mobius function [article]

Margaret M. Bayer, Gabor Hetyei
2001 arXiv   pre-print
A partially ordered set is r-thick if every nonempty open interval contains at least r elements.  ...  The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets.  ...  Introduction In this paper we study certain classes of graded partially ordered sets (posets), defined by conditions on the sizes of rank sets in intervals.  ... 
arXiv:math/0101075v1 fatcat:tfoxlbvievfl5fa7qkmunpyzau

Page 2 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
The paper studies linear inequalities that are valid for the flag f- vector of all Eulerian partially ordered sets of fixed rank.  ...  For Eulerian posets the authors obtain a set of linear inequalities that have to be satisfied by the flag f-vector.  ... 

Linear inequalities for enumerating chains in partially ordered sets

Samuel K. Hsiao, Lauren Rose, Rachel Stahl, Ezra M. Winston
2011 Transactions of the American Mathematical Society  
We characterize the linear inequalities satisfied by flag f -vectors of all finite bounded posets. We do the same for semipure posets.  ...  In particular, the closed convex cone generated by flag f -vectors of bounded posets of fixed rank is shown to be simplicial, and the closed cone generated by flag f -vectors of semipure posets of fixed  ...  Acknowledgements We thank Gábor Hetyei for inspiring discussions and the anonymous referee for several valuable comments and suggestions, including a simplification of the proof of Theorem 2.1.  ... 
doi:10.1090/s0002-9947-2010-05253-5 fatcat:a3oh6qyaufcrdhrp4esf3l5raa

Page 9060 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews  
(H-AOS; Budapest) Linear inequalities for flags in graded partially ordered sets.  ...  Summary: “The closure of the convex cone generated by all flag / - vectors of graded partially ordered sets is shown to be polyhedral.  ... 

Weighted PBW degenerations and tropical flag varieties [article]

Xin Fang, Evgeny Feigin, Ghislain Fourier, Igor Makhlin
2020 arXiv   pre-print
Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.  ...  We study algebraic, combinatorial and geometric aspects of weighted PBW-type degenerations of (partial) flag varieties in type A.  ...  The work was partially supported by the grant RSF-DFG 16-41-01013.  ... 
arXiv:1711.00751v2 fatcat:p2t6dgf6r5aavdoy3oxv6tixve

Weighted PBW degenerations and tropical flag varieties

X. Fang, E. Feigin, G. Fourier, I. Makhlin
2018 Communications in Contemporary Mathematics  
Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.  ...  We study algebraic, combinatorial and geometric aspects of weighted PBWtype degenerations of (partial) flag varieties in type A.  ...  Acknowledgments The work was partially supported by the grant RSF-DFG 16-41-01013.  ... 
doi:10.1142/s0219199718500165 fatcat:3lfs74cfifb6njaloyf4bzbgf4

Flag Enumeration in Polytopes, Eulerian Partially Ordered Sets and Coxeter Groups

Louis J. Billera
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
We discuss the enumeration theory for flags in Eulerian partially ordered sets, emphasizing the two main geometric and algebraic examples, face posets of convex polytopes and regular CW -spheres, and Bruhat  ...  One result is a direct expression for the Kazhdan-Lusztig polynomial of a Bruhat interval in terms of a new invariant, the complete cd-index.  ...  The Bruhat order on (W, S) is a partial order on the set W , defined as follows.  ... 
doi:10.1142/9789814324359_0151 fatcat:rztckstykvhixhrvlmihbxnsky

Linear Inequalities for Rank 3 Geometric Lattices

Kathryn L. Nyman
2004 Discrete & Computational Geometry  
We give a collection of inequalities which imply all the linear inequalities satisfied by the flag Whitney numbers of rank 3 geometric lattices.  ...  We further describe the smallest closed convex set containing the flag Whitney numbers of rank 3 geometric lattices as well as the smallest closed convex set containing the flag Whitney numbers of those  ...  I thank him for his guidance and suggestions.  ... 
doi:10.1007/s00454-003-0807-6 fatcat:5patoufjobdevgie73lnbg3e3e

The cd-Index: A Survey [article]

Margaret M. Bayer
2020 arXiv   pre-print
This is a survey of the cd-index of Eulerian partially ordered sets. The cd-index is an encoding of the numbers of chains, specified by ranks, in the poset.  ...  Eulerian posets include the face posets of regular CW spheres (in particular, of convex polytopes), intervals in the Bruhat order on Coxeter groups, and the lattices of regions of oriented matroids.  ...  The authors also described one-sided ideals of the graded algebra A representing flag vector relations on simplicial polytopes and on cubical polytopes, and gave dimension arguments from the resulting  ... 
arXiv:1901.04939v2 fatcat:ny7333qmyrf3dl64hjt3y4moju

Generalizations of Eulerian partially ordered sets, flag numbers, and the Möbius function

Margaret M. Bayer, Gábor Hetyei
2002 Discrete Mathematics  
A partially ordered set is r-thick if every nonempty open interval contains at least r elements.  ...  The generalized Dehn-Sommerville equations are proved for ag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets.  ...  Introduction In this paper we study certain classes of graded partially ordered sets (posets), deÿned by conditions on the sizes of rank sets in intervals.  ... 
doi:10.1016/s0012-365x(02)00336-9 fatcat:7s3kqgxum5bvbloipxnzntw7nq

Inequalities for cd -Indices of Joins and Products of Polytopes

Richard Ehrenborg, Harold Fox
2003 Combinatorica  
Posets and the cd-index For terminology on partially ordered sets (posets), we refer the reader to [22] .  ...  In order to obtain more succinct expressions for the recursions, we were forced to include a new element of degree minus one.  ...  Acknowledgments We thank Louis Billera, Margaret Readdy and the two referees for their generous comments.  ... 
doi:10.1007/s00493-003-0026-z fatcat:e4iqzdg3pnblzb7ioigyhcjkji

Page 659 of Mathematical Reviews Vol. , Issue 96b [page]

1996 Mathematical Reviews  
The author conjectures that this result holds for arbitrary Cohen-Macaulay Eulerian posets and proves that (if true) this conjecture gives all linear inequalities satisfied by the flag f-vector of a Cohen-Macaulay  ...  Modularity for L can be char- acterized using prime annihilator conditions. In this paper, the author studies annihilators and related topics in a partially or- dered set.  ... 
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