  We set up a combinatorial framework for inclusion-exclusion on the partitions
into distinct parts to obtain an alternative generating function of partitions
into distinct and non-consecutive parts. In connection with Rogers-Ramanujan
identities, the generating function yields two formulas in Slater's list. The
same formulas were constructed by Hirschhorn. Similar formulas were obtained by
Bringmann, Mahlburg and Nataraj. We also use staircases to give alternative
triple series for partitions into $d-$distinct parts for any $d \geq 2$.
