An analysis of two related loop-free algorithms for generating integer partitions

Abstract

Two loop-free algorithms, LEX and NEXPAR2, for generating the partitions of a positive integer n, in lexicographic and antilexicographic order, respectively, are investigated, and their structures are shown to be closely related. By utilising a number of combinatorial identities which relate the cardinalities of various classes of partitions, we derive formulae from which operation counts for the two algorithms may be obtained. For large n, corresponding asymptotic formulae are also obtained. A number of modifications and refinements of the two algorithms are discussed, and the relative performance of the algorithms resulting, and also the original algorithms, is analysed for both the non-asymptotic and the asymptotic cases.