scholarly journals Symmetric decompositions and the strong Sperner property for noncrossing partition lattices

2016 ◽  
Vol 45 (3) ◽  
pp. 745-775 ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Sperner Property ◽  
Noncrossing Partition ◽  
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.

scholarly journals Shellability of noncrossing partition lattices

2007 ◽  
Vol 135 (04) ◽  
pp. 939-939 ◽  
Author(s):  
Christos A. Athanasiadis ◽  
Thomas Brady ◽  
Colum Watt
Keyword(s):  
Noncrossing Partition ◽  
Partition Lattices

scholarly journals Deformation of Chains via a Local Symmetric Group Action

10.37236/1459 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Patricia Hersh
Keyword(s):  
Symmetric Group ◽  
Group Action ◽  
Maximal Chain ◽  
Type B ◽  
Noncrossing Partition ◽  
Partition Lattices ◽  
Maximal Chains ◽  
Symmetric Chain

A symmetric group action on the maximal chains in a finite, ranked poset is local if the adjacent transpositions act in such a way that $(i,i+1)$ sends each maximal chain either to itself or to one differing only at rank $i$. We prove that when $S_n$ acts locally on a lattice, each orbit considered as a subposet is a product of chains. We also show that all posets with local actions induced by labellings known as $R^* S$-labellings have symmetric chain decompositions and provide $R^* S$-labellings for the type B and D noncrossing partition lattices, answering a question of Stanley.

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 A Decomposition of Parking Functions by Undesired Spaces

10.37236/5940 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Melody Bruce ◽  
Michael Dougherty ◽  
Max Hlavacek ◽  
Ryo Kudo ◽  
Ian Nicolas
Keyword(s):  
Parking Functions ◽  
Partition Lattice ◽  
Self Duality ◽  
Noncrossing Partition ◽  
Symmetric Chain Decomposition ◽  
Order Complexes ◽  
Maximal Chains ◽  
Symmetric Chain ◽  
Noncrossing Partition Lattice

There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests.

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