The Rand and Block Distances of Pairs of Set Partitions

Abstract

AbstractThe Rand distance of two set partitions is the number of pairs {x,y} such that there is a block in one partition containing both x and y, but x and y are in different blocks in the other partition. Let R(n,k) denote the number of distinct (unordered) pairs of partitions of n that have Rand distance k. For fixed k we prove that R(n,k) can be expressed as \(\sum_j c_{k,j} {n \choose j} B_{n-j}\) where c k,j is a non-negative integer and B n is a Bell number. For fixed k we prove that there is a constant K n such that \(R(n,{n \choose 2}-k)\) can be expressed as a polynomial of degree 2k in n for all n ≥ K n . This polynomial is explicitly determined for 0 ≤ k ≤ 3.The block distance of two set partitions is the number of elements that are not in common blocks. We give formulae and asymptotics based on N(n), the number of pairs of partitions with no blocks in common. We develop an O(n) algorithm for computing the block distance.