Lower bounds for periods of Ducci sequences

A Ducci sequence is a sequence of integer $n$-tuples obtained by iterating the map $D : (a_1, a_2, \ldots, a_n) \mapsto \big(|a_1-a_2|,|a_2-a_3|,\ldots,|a_n-a_1|\big).$ Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given $n$. We prove lower bounds for $P(n)$ by counting certain partitions.

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