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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 842.11035

**Autor: ** Erdös, Paul; Nicolas, J.L.

**Title: ** On practical partitions. (In English)

**Source: ** Collect. Math. 46, No.1-2, 57-76 (1995).

**Review: ** Let *A* = **{**a_{1} = 1 < a_{2} < ... < a_{k} < ...**}** be an infinite subset of **N**. A partition of n with parts in *A* is a way of writing n = a_{i1}+a_{i2}+...+a_{ij} with 1 \leq i_{1} \leq i_{2} \leq ... \leq i_{j}. An integer a is said to be represented by the above partition, if it can be written a = **sum**^{j}_{r = 1} \epsilon_{r} a_{ir} with \epsilon_{r} = 0 or 1. A partition will be called practical if all a's, 1 \leq a \leq n, can be represented. When *A* = **N**, it has been proved by P. Erdös and M. Szalay that almost all paritions are practical. In this paper, a similar result is proved, first when a_{k} = 2^{k}, secondly when a_{k} \geq ka_{k-1}. Finally an example due to D. Hickerson gives a set *A* and integers n for which a lot of non practical partitions do exist.

**Reviewer: ** J.L.Nicolas (Villeurbanne)

**Classif.: ** * 11P81 Elementary theory of partitions

11B83 Special sequences of integers and polynomials

**Keywords: ** practical partitions; partition

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