Counting and computing the Rand and block distances of pairs of set partitions

Authors:
Frank Ruskey;Jennifer Woodcock;Yuji Yamauchi
Affiliations:
Dept. of Computer Science, University of Victoria, Victoria, BC, V8W 3P6, Canada;Dept. of Computer Science, University of Victoria, Victoria, BC, V8W 3P6, Canada;Dept. of Computer Science, University of Victoria, Victoria, BC, V8W 3P6, Canada
Venue:
Journal of Discrete Algorithms
Year:
2012

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Abstract

The 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 @?"jC(j,k)(nj)B"n"-"j where C(j,k) is a non-negative integer, B"n is the nth Bell number, and the summation range is of size less than 2k. If n=2k+2 then R(n,(n2)-k) can be expressed as a polynomial of degree 2k in n. This polynomial is explicitly determined for 0=