Title: Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces,
Abstract: Consider the noncrossing set partitions of an $n$-element set which either do
not contain the block $\{n-1,n\}$, or which do not contain the singleton block
$\{n\}$ whenever $1$ and $n-1$ are in the same block. 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.