A combinatorial construction for two formulas in Slater’s list

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. We also use staircases to give alternative triple series for partitions into $d-$distinct parts for any $d \geq 2$.