On the subpartitions of the ordinary partitions

Let a₁≥a₂≥···≥a_ℓ be an ordinary partition. A subpartition with gap d of an ordinary partition is defined as the longest sequence satisfying a₁>a₂>···>a_s and a_s>a_s+1, where a_i-a_j≥d for all i<j≤s. This is a generalization of the Rogers-Ramanujan subpartition which was introduced by L. Kolitsch. In this note, we will study various properties of subpartitions, and as an application we will give a combinatorial proof of two entries which are in Ramanujan's lost notebook.