Partitions, Durfee Symbols, and the Atkin-Garvan Moments of Ranks by

Atkin and Garvan introduced the moments of ranks of partitions in their work connecting ranks and cranks. Here we consider a combina-torial interpretation of these moments. This requires the introduction of a new representation for partitions, the Durfee symbol, and subse-quent refinements. This in turn leads us to a variety of new congru-ences for our ‘marked ’ Durfee symbols much in the spirit of Dyson’s original conjectures on the ranks of partitions.