The Equivalence between the Schur Order and the Stochastic Orders Generated by the Scheme of Allocation of Particles into Cells

In 1982, it was proved that the Schur partial-order relation on the set of distributions on $\{1,2,\dots\}$ is equivalent to the order relation generated the number of nonempty cell distributions by the scheme of independent allocation of particles into cells. Here it is shown that the same partial-order relation is generated by distributions of numbers of cells occupied by at least $r$ particles for any $r\ge 2$.