Random Permutations and Partition Models

123, 12|3, 13|2, 23|1, 1|2|3. The blocks are unordered, so 2|13 is the same partition as 13|2 and 2|31. A partition B is a sub-partition of B∗ if each block of B is a subset of some block of B∗ or, equivalently, if Bij = 1 implies B∗ ij = 1. This relationship is a partial order denoted by B ≤ B∗, which can be interpreted as B ⊂ B∗ if each partition is regarded as a subset of [n]2. The partition lattice En is the set of partitions of [n] with this partial order. To each pair of partitions B, B′ there corresponds a greatest lower bound B ∧B′, which is the set intersection or Hadamard componentwise matrix product. The least upper bound B ∨B′ is the least element that is greater than both, the transitive completion of B ∪ B′. The least element of En is the partition 0n with n singleton blocks, and the greatest element is the single-block partition denoted by 1n. A permutation σ : [n] → [n] induces an action B 7→ Bσ by composition such that Bσ(i, j) = B(σ(i), σ(j)). In matrix notation, Bσ = σBσ−1, so the action by conjugation permutes both the rows and columns of B in the same way. The block sizes are preserved and are maximally invariant under conjugation. In this way, the 15 partitions of [4] may be grouped into five orbits or equivalence classes as follows: