Parallel rank of two sandpile models of signed integer partitions

We introduce the concept of fundamental sequence for a finite graded poset x which is also a discrete dynamical model. the concept of fundamental sequence is a refinement of the concept of parallel convergence time for these models. we compute the parallel convergence time and the fundamental sequence when x is the finite lattice p (n,r) of all the signed integer partitions a r,.,a 1,b 1,.,b n - r such that r ≥ a r ≥ ≥ a 1 ≥ 0 ≥ b 1 ≥ ≥ b n - r ≥ - (n - r),where n ≥ r ≥ 0,and when x is the sublattice p (n,d,r) of all the signed integer partitions of p (n,r) having exactly d nonzero parts. © 2013 g. chiaselotti et al.