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 ≥ 2.

中文翻译：

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Slater 列表中两个公式的组合构造

我们建立了一个组合框架，用于将分区划分为不同的部分，以获得划分为不同和非连续部分的替代生成函数。结合 Rogers-Ramanujan 恒等式，生成函数在 Slater 的列表中产生两个公式。Hirschhorn 构建了相同的公式。Bringmann、Mahlburg 和 Nataraj 获得了类似的公式。我们还使用楼梯为分区提供替代的三重系列d- 任何不同的部分d ≥ 2.