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...
- 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
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. |
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