Resumen de On practical partitions

P. Erdös, J.L. Nicolas

• Let $\mathcal{A} = \{a_1 = 1 < a_2 ?\dots ? a_k ?\dots\}$ be an infinite subset of $\mathbb{N}$. A partition of $n$ with parts in $\mathcal{A}$ is a way of writing $n = a_{i_1} + a_{i_2} +\dots + a_{i_j}$ with $1\leq i_1\leq i_2\leq \dots\leq i_j$ . An integer a is said to be represented by the above partition, if it can be written $a =\sum^j_{ r=1}\varepsilon_ra_{i_r}$ with $\varepsilon_r = 0$ or 1. A partition will be called practical if all $a's, 1 \leq a \leq n$ can be represented. When $\mathcal{A} = \mathbb{N}$, it has been proved by P. Erdös and M. Szalay that almost all partitions 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 $\mathcal{A}$ and integers $n$ for which a lot of non practical partitions do exist.