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 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.

• 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