The Rand and Block Distances of Pairs of Set Partitions

  • Frank Ruskey
  • Jennifer Woodcock
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)


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 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.


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  1. 1.
    Canfield, E.R.: Engel’s inequality for the Bell numbers. Journal of Combinatorial Theory, Series A 72, 184–187 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Filkov, V., Skiena, S.: Integrating Microarray Data by Consensus Clustering. International Journal on Artificial Intelligence Tools 13(4), 863–880 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Addison-Wesley (1989)Google Scholar
  4. 4.
    Hubert, L.: Comparing Partitions. Journal of Classification 2, 193–218 (1985)CrossRefGoogle Scholar
  5. 5.
    Knuth, D.E.: The Art of Computer Programming, vol 4: Combinatorial Algorithms, Part 1. Addison-Wesley, Reading (2011)zbMATHGoogle Scholar
  6. 6.
    Moser, L., Wyman, A.: An asymptotic formula for the Bell numbers. Transactions of the Royal Society of Canada III 49, 49–54 (1955)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Munagi, A.O.: Set Partitions with Successions and Separations. Int. J. Math and Math. Sc. 2005(3), 451–463 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Rand, W.: Objective criteria for the evaluation of clustering methods. J. American Statistical Assoc. 66(336), 846–850 (1971)CrossRefGoogle Scholar
  9. 9.
    Ruskey, F.: Simple Combinatorial Gray Codes Constructed by Reversing Sublists. In: Ng, K.W., Balasubramanian, N.V., Raghavan, P., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 201–208. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  10. 10.
    Stanley, R.R.: Enumerative Combinatorics, vol. 1. Wadsworth (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Frank Ruskey
    • 1
  • Jennifer Woodcock
    • 1
  1. 1.Dept. of Computer ScienceUniversity of VictoriaCanada

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