Matchings in the Partition Lattice

Within the lattice of partitions of a finite set, the kth level consists of partitions having k blocks. A matching between levels $k_1$ and $k_2$ is a one-to-one function assigning to each partition in the smaller level another in the larger level, which is related to the first by refinement. It is shown that matchings between adjacent levels of the partition lattice fail to exist precisely for k in an interval. The endpoints of the matchingless interval are shown to equal asymptotically $n\log 2/\log n$ and $n\log 4/\log n$.