Dynamics of the Picking transformation on integer partitions
نویسندگان
چکیده
The aim of this work is to study the iteration of a transformation defined for families of sets. Let P be a family of disjoint and nonempty sets S1, ..., Sk, then pick from each set an element and remove it, suppress the sets that become empty and add one new set composed of all the picked elements. It leads to a new family of sets, and our concern is the way the cardinalities of the sets evolve for such a transformation. Figure 1 shows an example of this transformation iterated several times. At each step, the new set is represented with a dotted border. The right part of the figure shows the evolution of the cardinalities of the sets. First note that this transformation preserves the total number of elements in the family, which will be denoted by n. It is also clear that if we focus on the cardinalities of the sets in the family, the choice of the picked elements does not matter. Thus each family of sets P may be seen as a partition p1, . . . , pk of the integer n, where pi is the cardinality of Si. A partition of n is a collection of integers p1, . . . , pk satisfying ∀i, 1 ≤ i ≤ k, pi ≥ 1 and ∑i=1 pi = n. Then the image of P by the transformation is a partition of n, uniquely defined, which will be denoted by f (P).
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