2 2 Fe b 20 00 N – graphs , modular Sidon and sum – free sets , and partition identities ∗

Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum–free Sidon set. Partition identities are also constructed for sets associated with modular sum–free sets. 1 N–graphs for partitions The standard form of a partition n = a1 + a2 + · · ·+ ak is π = (a1, . . . , ak), where the parts a1, . . . , ak are positive integers arranged in descending order. The standard form of a partition is unique. Associated to a partition π = (a1, . . . , ak) of n is an array of dots, called the Ferrers graph of π. This consists of n dots arranged in k rows, with a1 dots on the first row, a2 dots on the second row,. . . , and ak dots on the k–th row. The rows are aligned on the left. The Durfee square D(π) of the graph is the largest square array of dots that appears in the upper left corner of the Ferrers graph. We denote by d(π) the number of dots on a side of the Durfee square, or, equivalently, the number of dots on a diagonal of D(π). 2000 Mathematics Subject Classification. Primary 11P81, 11P83. Secondary 11B05, 11B25, 11B75.