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We develop a recursive method for computing the $r$-removed $P$-orderings and $P$-orderings of order $h,$ the characteristic sequences associated to these and limits associated to these sequences for subsets $S$ of a Dedekind domain $D.$ This method is applied to compute these objects for $S=\mathbb{Z}$ and $S=\mathbb{Z}\setminus p\mathbb{Z}$.