Parallel and sequential dynamics of two discrete models of signed integer partitions

In this paper we complete and generalize some previous results concerning the computing of the sequential and parallel convergent time for two discrete dynamical system of signed integer partitions. We also refine the concept of parallel convergent time for a finite graded partially ordered set (briefly poset) $X$ which is also a discrete dynamical model. To this aim we define the concept of {it fundamental sequence} of $X$ and we compute this sequence in two particularly important cases . In the first case, when $X$ is the finite lattice $S(n,r)$ of all the signed integer partitions $a_r,dots,a_1,b_1,dots,b_{n-r}$ such that $rge a_rge cdots ge a_1ge 0 ge b_1ge cdots ge b_{n-r}ge -(n-r)$, where $nge rge 0$ and the unique part that can be repeated is $0$. In the second case, when $X$ is the sub-lattice $S(n,d,r)$ of all the signed integer partitions of $S(n,r)$ having exactly $d$ non-zero parts. The relevance of the previous lattices as discrete dynamical models is related to their link with some unsolved extremal combinatorial sum problems.