One of the fundamental invariants under conjugation of Ferrers graphs of partitions is the number of different parts. However, this invariant has not been expoited in the literature. Using simple combinatorial arguments we will show how this invariant can be used to obtain a variety of new results including a new proof and variant of Cauchy's q-binomial theorem, a six parameter extension of Heine's fundamental transformation, a variant of the Rogers-Fine identity, and an extension of a fifth order mock-theta function identity of Ramanujan. The talk will be accessible to non-experts and graduate students.

 
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