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Supersolvable LL-lattices of binary trees

Riccardo Biagioli, Frédéric Chapoton
2005 Discrete Mathematics  
Some posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given.  ...  A finite graded lattice of rank n is supersolvable if and only if it is S n EL-shellable. Poset of forests A tree is a leaf-labeled rooted binary tree and a forest is a set of such trees.  ...  Introduction The aim of this article is to study some posets on forests of binary leaf-labeled trees.  ... 
doi:10.1016/j.disc.2005.03.005 fatcat:tpcwygoibrg5nbpzy5rezv4jba

Supersolvable LL-lattices of binary trees [article]

Riccardo Biagioli, Frederic Chapoton
2003 arXiv   pre-print
Some posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given.  ...  Poset of forests A tree is a leaf-labeled rooted binary tree and a forest is a set of such trees. Vertices are either inner vertices (valence 3) or leaves and roots (valence 1).  ...  The Möbius function of P , µ : Int(P ) → Z, is defined recursively by supersolvable lattices include modular lattices, the partition lattice Π n and the lattice of subgroups of a finite supersolvable group  ... 
arXiv:math/0304132v1 fatcat:oqrl7jdr6fbv5gs42bmbspbdka

Author index to volume 296

2005 Discrete Mathematics  
Chapoton, Supersolvable LL-lattices of binary trees (1) 1-13 Bonoli, G. and O. Polverino, The twisted cubic in PGð3; qÞ and translation spreads in HðqÞ , T. and C.G.  ...  Zhang, Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees (2-3) 187-197 Hung, L.X., see N.D. Tan (1) 59-72 Hurlbert, G.H., see B.  ... 
doi:10.1016/s0012-365x(05)00315-8 fatcat:m2gwquhb4zcw5h3uc4nmsxjc2u

Factorization of the Characteristic Polynomial

Joshua Hallam, Bruce Sagan
2014 Discrete Mathematics & Theoretical Computer Science  
We will see that Stanley's Supersolvability Theorem is a corollary of this result.  ...  International audience We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers.  ...  Additionally, he showed these roots where counted by the sizes of blocks in a partition of the atom set of the lattice. Blass and Sagan [BS97] extended this result to LL lattices.  ... 
doi:10.46298/dmtcs.2386 fatcat:6ywjqlek6jawvizpm4w5kphspa

Whitney numbers of some geometric lattices

E Damiani, O D'Antona, F Regonati
1994 Journal of combinatorial theory. Series A  
It is worth noting that, from the work of Aigner [Aig2] , this problem has an affirmative answer when restricted to the case of binary lattices.  ...  ,L ..... be a chain of supersolvable geometric lattices of rank O, 1, 2, ..., n .... in ql.  ... 
doi:10.1016/0097-3165(94)90034-5 fatcat:aoqcqqs2cnfs7p2g7z3tfts3wm

Factoring the characteristic polynomial of a lattice

Joshua Hallam, Bruce Sagan
2015 Journal of combinatorial theory. Series A  
We introduce a new method for showing that the roots of the characteristic polynomial of certain finite lattices are all nonnegative integers.  ...  We will see that Stanley's Supersolvability Theorem is a corollary of this result. Additionally, we will prove a theorem which gives three conditions equivalent to factorization.  ...  Acknowledgments The authors would like to thank Torsten Hoge for pointing out the connection of Theorem 11 with Terao's theorem about nice partitions in central hyperplane arrangements.  ... 
doi:10.1016/j.jcta.2015.06.006 fatcat:3776hbua5jbvhkskfsccqhoa4m

A LL-lattice reformulation of arithmetree over planar rooted trees. Part II [article]

Leroux Philippe
2004 arXiv   pre-print
We propose a 'deformation' of a vectorial coding used in Part I, giving a LL-lattice on rooted planar trees according to the terminology of A. Blass and B. E. Sagan.  ...  Our parenthesis framework allows a more tractable reformulation to explore the properties of the underlying lattice describing operations and simplify a proof of a fundamental theorem related to arithmetics  ...  Blass [8, 1] , we compute the Möbius function of these lattices and proved they are of type LL (like for the Tamari lattices Y n orN n associated with planar binary trees).  ... 
arXiv:math/0408349v1 fatcat:e2xqeoq3zbe6fkupxwvo6sxbru

Factoring the characteristic polynomial of a lattice [article]

Joshua Hallam and Bruce E. Sagan (Department of Mathematics, Michigan State University)
2015 arXiv   pre-print
We introduce a new method for showing that the roots of the characteristic polynomial of certain finite lattices are all nonnegative integers.  ...  We will see that Stanley's Supersolvability Theorem is a corollary of this result. Additionally, we will prove a theorem which gives three conditions equivalent to factorization.  ...  Acknowledgments The authors would like to thank Torsten Hoge for pointing out the connection of Theorem 11 with Terao's theorem about nice partitions in central hyperplane arrangements.  ... 
arXiv:1403.0666v2 fatcat:hpbjhhtzmfdp7lkabwzk6cffly

Applications of Quotient Posets [article]

Joshua Hallam
2014 arXiv   pre-print
Blass and Sagan's result about LL lattices will also be shown to be a consequence of our factorization theorems.  ...  Our factorization theorems will then be used to show that any interval of the Tamari lattice has a characteristic polynomial which factors in this way.  ...  We will use this fact to prove Blass and Sagan's result about LL lattices [2] which is a generalization of the supersolvability result.  ... 
arXiv:1411.3022v1 fatcat:rkcla744m5atzg7tsv552bjgpy

Factorization of the Characteristic Polynomial of a Lattice using Quotient Posets

Fpsac, Usa Chicago, Dmtcs
2014 unpublished
We will see that Stanley's Supersolvability Theorem is a corollary of this result.  ...  We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers.  ...  Additionally, he showed these roots where counted by the sizes of blocks in a partition of the atom set of the lattice. Blass and Sagan [BS97] extended this result to LL lattices.  ... 
fatcat:iiwyyk6fczdqbcfzxjzercnapq