Abstract

MUHLE, Henri. Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces. Rev.colomb.mat. [online]. 2018, vol.52, n.1, pp.65-86. ISSN 0034-7426.  https://doi.org/10.15446/recolma.v1n52.74562.

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.

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