The Lattice of integer partitions and its infinite extension

Details The Lattice of integer partitions and its infinite extension The Lattice of integer partitions and its infinite extension

2000-08-02

arxiv.org/abs/math/0008020v1

Mathematics

Combinatorics

To appear in LNCS special issue, proceedings of ORDAL'99. See http://www.liafa.jussieu.fr/~latapy

Scientific paper

In this paper, we use a simple discrete dynamical system to study the integers partitions and their lattice. The set of the reachable configurations equiped with the order induced by the transitions of the system is exactly the lattice of integer partitions equiped with the dominance ordering. We first explain how this lattice can be constructed, by showing its strong self-similarity property. Then, we define a natural extension of the system to infinity. Using a self-similar tree, we obtain an efficient coding of the obtained lattice. This approach gives an interesting recursive formula for the number of partitions of an integer, where no closed formula have ever been found. It also gives informations on special sets of partitions, such as length bounded partitions.

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