A combinatorial construction for two formulas in Slater's List

TL;DR

This paper develops a combinatorial framework for partitions, deriving new generating functions related to Rogers-Ramanujan identities, and presents alternative series representations for partitions with distinct parts.

Contribution

It introduces a novel combinatorial approach to derive formulas in Slater's list and provides alternative series for partitions with distinct parts.

Findings

Derived new generating functions for partitions into distinct and non-consecutive parts.

Connected the formulas to Rogers-Ramanujan identities and Slater's list.

Provided alternative triple series for partitions into d-distinct parts.

Abstract

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 $d \geq 2$.