Abstract
A permutation is partitioned into its cycles. If a permutation is random and if all the permutations are equally probable, a random partition is defined. The probability distribution of the partition is obtained and its properties are examined. Six geneses of the random partition show that it is very fundamental and appears in various situations of statistics, computer science and physics. The distance between two independent random partitions is studied for the possible applications to cluster analysis.
This paper deals mainly with the case where the elements of a set are distinguishable. If they are undistinguishable, the size index of the random partition has nice combinatorial properties. Here, some new results, which can be shown in more general setup, are shortly reported.
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Sibuya, M. Random partition of a finite set by cycles of permutation. Japan J. Indust. Appl. Math. 10, 69–84 (1993). https://doi.org/10.1007/BF03167203
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DOI: https://doi.org/10.1007/BF03167203