Lower bounds for periods of Ducci sequences

A Ducci sequence is a sequence of integer $n$-tuples obtained by iterating the map $$D:(a_1,a_2,\dots ,a_n)\mapsto (\mathrm{\mid }{a}_1-a_2\mathrm{\mid },\mathrm{\mid }{a}_2-a_3\mathrm{\mid },\dots ,\mathrm{\mid }{a}_n-a_1\mathrm{\mid })\mathrm{.}$$ 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.