Two Posets of Noncrossing Set Partitions

Abstract

In this manuscript we consider a subposet of the lattice of noncrossing set partitions of an n-element set under refinement order. This subposet is induced by those noncrossing set partitions, which 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. We prove that the resulting poset is in fact a supersolvable lattice, and we give a combinatorial proof for the value of its Möbius function between least and greatest element by using Blass and Sagan's theory of NBB bases.As a corollary we prove a conjecture by Bruce, Dougherty, Hlavacek, Kudo and Nicolas about the homotopy type of a certain subposet of our poset, which comes from parking functions with undesired positions.