Let a1≥a2≥⋅⋅⋅≥aℓ be an ordinary partition. A subpartition with gap d of an ordinary partition is defined as the longest sequence satisfying a1>a2>⋅⋅⋅>as and as>as+1, where ai−aj≥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.
Partition Subpartition Partial theta function
Mathematics Subject Classification (2000)
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