The Rand and Block Distances of Pairs of Set Partitions

@inproceedings{Ruskey2011TheRA,
  title={The Rand and Block Distances of Pairs of Set Partitions},
  author={Frank Ruskey and Jennifer Woodcock},
  booktitle={IWOCA},
  year={2011}
}
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 $\sum_j c_{k,j} {n \choose j} B_{n-j}$ where ck,j is a non-negative integer and Bn is a Bell number. For fixed k we prove that there is a… CONTINUE READING

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