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Revista Colombiana de Matemáticas
versão impressa ISSN 0034-7426
Rev.colomb.mat. vol.52 no.1 Bogotá jan./jun. 2018
https://doi.org/10.15446/recolma.v1n52.74562
Original articles
Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces
Dos posets de particiones sin cruces provenientes de espacios de parqueo prohibidos
1 Technische Universität Dresden, Dresden - Germany
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.
Keywords: noncrossing partition; supersolvable lattice; left-modular lattice; parking function; lexicographic shellability; NBB base; Möbius function
Considere las particiones sin cruces de un conjunto de n elementos que no usan el bloque {n-1, n}, ni usan a la vez el bloque {n} y un bloque que contenga a 1 y n - 1. En este artículo estudiamos el subposet del retículo de particiones sin cruces inducido por estos elementos. Probamos que este retículo es supersoluble, y por lo tanto es lexicográficamente descascarable. También damos un modelo combinatorio de las bases NBB de este retículo y derivamos una fórmula explicita para el valor de su función de Möbius entre el elemento mínimo y el máximo.
Este trabajo es motivado por un artículo reciente de M. Bruce, M. Dougherty, M. Hlavacek, R. Kudo, e I. Nicolas en el cual introducen un subposet del retículo de particiones sin cruces que es determinado por funciones de parqueo con ciertas entradas prohibidas. En particular, ellos conjeturan que el poset resultante siempre tiene un complejo de orden contráctil. En este artículo probamos esta conjetura, sumergiendo su poset en el nuestro y mostrando que esta inmersión hereda la descascarabilidad lexicográfica.
Palabras clave: Particiones sin cruces; retículo supersoluble; retículo modular izquierdo; funciones de parqueo; descascarabilidad lexicográfica; bases NBB; función Möbius
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Received: March 15, 2017; Accepted: January 31, 2018
This is an open-access article distributed under the terms of the Creative Commons Attribution License
