Department of Mathematics, Howard University, Washington DC, USA

The Riordan group is an easy yet powerful tool for looking at a large number of results in combinatorial enumeration. At the first level it provides quick proofs for many binomial identities as well as a systematic way to invert them. We will see how they arise naturally when looking at the uplift principle as applied to classes of ordered trees. We will also discuss some recent results including the Double Riordan group, summer – winter trees, spoiled child trees, and will mention a few open problems as well. The main tools involved are generating functions, matrix multiplication, and elementary group theory.

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