### On Bipartitional Functions

P. V. Sukhatme
1938 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
B ip artitio n al functions are arith m etical functions o f tw o p artitio n s of th e sam e n u m b e r, a n d arise p rim arily in the theory of th e sym m etric function g en eratin g functions. A nalytical m ethods o f ev alu atin g the p artitio n a l functions an d o f studying th em in relatio n to th e th eo ry o f distributions are largely d u e to M acm a h o n (1915). T h e use o f p a rtitio n al n o ta tio n has ren d ered his m ethods distinctly sim pler th a n those o f his
more » ... those o f his predecessors, b u t, sim plified as they are, his m ethods do n o t m ake the p ractical evaluation o f these functions p articu la rly expeditious. I f his m ethods are actu ally p u t in to p ractice, it is found th a t they becom e increasingly laborious an d im p racticab le w ith h igh-ord er sym m etric functions. A n excellent exam ple o f th e difficulties en co untered in th e use o f algebraic m ethods, especially those involving th e action o f differential operators, is to be found in th e en u m eratio n o f th e 5 x 5 an d 6 x 6 L atin Squares (Fisher an d Y ates 1934). In this connexion it is show n by F isher a n d Y ates th a t th e d irect en u m eratio n by trial is a m uch sim pler ap p ro ac h th a n th e developm ent o f th e differential operators o f M acm a h o n 's algebraic solution. B ipartitional functions derive m u ch o f th e ir im p o rtan ce on acco u n t o f th e com b in a to rial problem s of w hich they-supply solutions. T h ey have, consequently, im p o rta n t applications in ap p lied statistics, as, for exam ple, in the d eriv atio n o f,th e m om ents, an d the p ro d u ct m om ents of m o m en t statistics (Fisher 1930), in th e en u m eratio n o f different sam ples o f a given size d raw n from a finite p o p u latio n , an d so on. I t is, therefore, desirable th a t the arith m etic im plied in the algebraic m ethods o f ev alu atin g th e p artitio n al functions should be clearly an d system atically set o u t in stan d ard form for read y evaluation. Professor R . A. F isher suggested to m e to take u p for a system atic study the problem o f ev alu atin g the b ip a rtitio n al functions, an d of fo rm ulating th eir relations to distributions in p i a n o, an d to the com b in ato rial problem s o f w hich th supply solutions. I t is the purpose of this p a p e r to p u t forw ard these relationships in a com parable m an n er, so as to give a com prehensive view of a t least some aspects o f th eir properties. T h e sym m etric functions discussed by M acm ah o n are (1) the m onom ial sym m etric functions, or the ^-functions, in the n o tatio n o f this p ap er 5 (2) the elem entary sym m etric functions called the ^-functions; (3) the hom ogeneous p ro d u ct sums called the A-functions; an d (4) the sums o f pow ers called the ^-functions o f th e given q u a n tities cq, a2, a 3, .... V ol. CCXXXVII. A. 780. (Price 4 48