Where the Typical Set Partitions Meet and Join

Abstract

The lattice of the set partitions of $[n]$ ordered by refinement is studied. Suppose $r$ partitions $p_1,\dots,p_r$ are chosen independently and uniformly at random. The probability that the coarsest refinement of all $p_i$'s is the finest partition $\bigl\{\{1\},\dots,\{n\}\bigr\}$ is shown to approach $0$ for $r=2$, and $1$ for $r\ge 3$. The probability that the finest coarsening of all $p_i$'s is the one-block partition is shown to approach $1$ for every $r\ge 2$.