Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces

Consider the noncrossing set partitions of an n-element set which, either do not use the block {n - 1, n} or which do not use both the singleton block {n} and a block containing 1 and n - 1. In this article we study the subposet of the noncrossing partition lattice induced by these elements, and sho...

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Autores:
Mühle, Henri
Tipo de recurso:
Article of journal
Fecha de publicación:
2018
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
spa
OAI Identifier:
oai:repositorio.unal.edu.co:unal/66425
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/66425
http://bdigital.unal.edu.co/67453/
Palabra clave:
51 Matemáticas / Mathematics
noncrossing partition
supersolvable lattice
left-modular lattice
parking function
lexicographic shellability
NBB base
Möbius function
Particiones sin cruces
retículo supersoluble
retículo modular izquierdo
funciones de parqueo
descascarabilidad lexicográfica
bases NBB
función Möbius
Rights
openAccess
License
Atribución-NoComercial 4.0 Internacional
Description
Summary:Consider the noncrossing set partitions of an n-element set which, either do not use the block {n - 1, n} or which do not use both the singleton block {n} and a block containing 1 and n - 1. In this article we study the subposet of the noncrossing partition lattice induced by these elements, and show that it is a supersolvable lattice, and therefore lexicographically shellable. We give a combinatorial model for the NBB bases of this lattice and derive an explicit formula for the value of its Möbius function between least and greatest element.This work is motivated by a recent article by M. Bruce, M. Dougherty, M. Hlavacek, R. Kudo, and I. Nicolas, in which they introduce a subposet of the noncrossing partition lattice that is determined by parking functions with certain forbidden entries. In particular, they conjecture that the resulting poset always has a contractible order complex. We prove this conjecture by embedding their poset into ours, and showing that it inherits the lexicographic shellability.