On bipartitional functions

Abstract
Bipartitional functions are arithmetical functions of two partitions of the same number, and arise primarily in the theory of the symmetric function generating functions. Analytical methods of evaluating the partitional functions and of studying them in relation to the theory of distributions are largely due to Macmahon (1915). The use of partitional notation has rendered his methods distinctly simpler than those of his predecessors, but, simplified as they are, his methods do not make the practical evaluation of these functions particularly expeditious. If his methods are actually put into practice, it is found that they become increasingly laborious and impracticable with high-order symmetric functions. An excellent example of the difficulties encountered in the use of algebraic methods, especially those involving the action of differential operators, is to be found in the enumeration of the 5 x 5 and 6 x 6 Latin Squares (Fisher and Yates 1934). In this connexion it is shown by Fisher and Yates that the direct enumeration by trial is a much simpler approach than the development of the differential operators of Macmahon’s algebraic solution.

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