Where the Typical Set Partitions Meet and Join

The lattice of the set partitions of [n] ordered by refinement is studied. Suppose r partitions p1, . . . , pr are chosen independently and uniformly at random. The probability that the coarsest refinement of all pi ’s is the finest partition {1}, . . . , {n} is shown to approach 0 for r = 2, and 1 for r ≥ 3. The probability that the finest coarsening of all pi ’s is the one-block partition is shown to approach 1 for every r ≥ 2. Introduction. Let Πn be the set of all set partitions of [n] , ordered by refinement. That is, for two partitions p and p′ , p p′ if each block of p′ is a union of blocks of p . It is well known, Stanley [6], that Πn is a lattice; it means that every pair of partitions p, p′ has the greatest lower bound inf{p, p′} (p inf p′ or p meet p′ ) and the least upper bound sup{p, p′} (p sup p′ or p join p′ ). Namely, inf{p, p′} is the partition whose blocks are the pairwise intersections of blocks of p and p′ , and it is the “coarsest” (simultaneous) refinement of p and p′ . sup{p, p′} is a partition whose every block is both a union of blocks of p and a union of blocks of p′ , with no proper subset of the block having that property; so it is the finest “coarsening” of p and p′ . Assigning to each p the same probability, 1/|Πn| , we transform Πn into the probability space with uniform measure. There is a sizeable literature on the properties of the uniformly distributed partition, see Pittel [5] and the references therein. Closer to the subject of this paper, Canfield and Harper [1] and Canfield [2] used the probabilistic tools to find the surprisingly sharp bounds for the length of the largest antichain in Πn . In [5] we proved that the total number of refinements of the random partition is asymptotically lognormal. In the present paper we study the properties of inf1≤i≤r pi , sup1≤i≤r pi , under the assumption that the uniform partitions p1, . . . , pr are independent. (Formally, we study 1991 Mathematics Subject Classification. 05A18, 05A19, 05C30, 05C80, 06A07, 60C05, 60Fxx.