The Sperner Property in Geometric and Partition Lattices

1990 ◽  
pp. 298-304 ◽  
Author(s):  
Jerrold R. Griggs
Keyword(s):  
Sperner Property ◽  
Partition Lattices

scholarly journals Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Sperner Property ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Noncrossing Partition ◽  
International Audience ◽  
Partition Lattices ◽  
Special Case

International audience We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

scholarly journals The Absolute Orders on the Coxeter Groups $A_n$ and $B_n$ are Sperner

10.37236/8874 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Lawrence H. Harper ◽  
Gene B. Kim ◽  
Neal Livesay
Keyword(s):  
Coxeter Groups ◽  
Symmetric Groups ◽  
Product Theorem ◽  
Reflection Groups ◽  
Dihedral Groups ◽  
Alternate Proof ◽  
Sperner Property ◽  
Noncrossing Partition ◽  
The Absolute ◽  
Partition Lattices

There are several classes of ranked posets related to reflection groups which are known to have the Sperner property, including the Bruhat orders and the generalized noncrossing partition lattices (i.e., the maximal intervals in absolute orders).  In 2019, Harper–Kim proved that the absolute orders on the symmetric groups are (strongly) Sperner.  In this paper, we give an alternate proof that extends to the signed symmetric groups and the dihedral groups.  Our simple proof uses techniques inspired by Ford–Fulkerson's theory of networks and flows, and a product theorem.

Elementary equivalence of partition lattices

1989 ◽  
Vol 29 (3) ◽  
pp. 507-508 ◽  
Author(s):  
A. G. Pinus
Keyword(s):  
Elementary Equivalence ◽  
Partition Lattices

Algebraic Aspects of Partition Lattices

1992 ◽  
pp. 106-122 ◽  
Author(s):  
Ivan Rival ◽  
Miriam Stanford
Keyword(s):  
Partition Lattices

scholarly journals Chains, antichains, and complements in infinite partition lattices

2018 ◽  
Vol 79 (2) ◽  
Author(s):  
James Emil Avery ◽  
Jean-Yves Moyen ◽  
Pavel Růžička ◽  
Jakob Grue Simonsen
Keyword(s):  
Partition Lattices

Noise Barriers with 2-D Partition Lattices

2003 ◽  
Vol 43 (5) ◽  
pp. 722-726
Author(s):  
Sung Soo Jung ◽  
Seung Il Cho ◽  
Yong Bong Lee ◽  
Woo Seop Lee
Keyword(s):  
Noise Barriers ◽  
Partition Lattices

scholarly journals Cumulants and partition lattices II: generalised k-statistics

1986 ◽  
Vol 40 (1) ◽  
pp. 34-53 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Asymptotic Behaviour ◽  
Random Variables ◽  
Möbius Function ◽  
Mobius Function ◽  
Joint Cumulants ◽  
Partition Lattices

AbstractThe role played by the Möbius function of the lattice of all partitions of a set in the theory of k-statistics and their generalisations is pointer out and the main results conscerning these statistics are drived. The definitions and formulae for the expansion of products of generalished k-statistics are presented from this viewpoint and applied to arrays of random variables whos moments satisfy stitable symmentry constraints. Applications of the theory are given including the calculation of (joint) cumulants of k-statistics, the minimum variace estimation of (generalised) moments and the asymptotic behaviour of generalised k-statistics viewed as (reversed) martingales.

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