Two posets of noncrossing partitions coming from undesired parking spaces

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Consider the noncrossing set partitions of an n-element set which, either do not use the block fn-1; ng or which do not use both the singleton block fng 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.


Original languageEnglish
Pages (from-to)65-86
Number of pages22
JournalRevista Colombiana de Matematicas
Issue number1
Publication statusPublished - 2018


ASJC Scopus subject areas


  • Left-modular lattice, Lexicographic shellability, Möbius function, NBB base, Noncrossing partition, Parking function, Supersolvable lattice

Library keywords