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 ar, …, a1, b1, …, bn−r such that r ≥ ar ≥ ⋯≥a1 ≥ 0 ≥ b1 ≥ ⋯≥bn−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.