100 MAJOR P. A. MAcMAHON: MEMOIR ON THE 



The Fundamental Relation. 

 Art. 23. We must now examine the fundamental relation 



GF(/, m,n) = 



for brevity, where 

 and 



/. ="(1-8+1) (t-s+2) ... (l-s+mn) 

 p. = (m l)(n 1). 



When a partition enumerated by GF (I, m, n) is represented graphically by nodes 

 in three dimensions, we see that the nodes form a portion of a parallelepiped of 

 nodes, the sides having I, m, n nodes respectively ; the unoccupied nodes graphically 

 represent another partition of the number Imnw if the former partition be of the 

 number w. Hence, if GF (I, m, n) be F (x), we have (writing Co as short for coefficient), 



Co x" in F (x) equal to Co xt"*-" in F (x) or equal to Co x w in x lma ~F (-). From which 



\x/ 



it appears that F(-J = aj~'*"F (x), and this property may be directly verified in the 



\x/ 



fundamental relation by means of the formulae 



From another point of view we may suppose the nodes of the lattice of m rows 

 and n columns to be all occupied by parts, zero being taken as a part, and then if we 

 diminish each part by I, we obtain a partition of the negative integer (lmn-w) into 

 negative parts 0, -1, -2, ... -1; the effect upon the generating function F (a;) is 



alternatively to substitute - for x or to divide it by x ln *. It will be noted that in this 



cc 



respect L.( <x, m, n) possesses the same property as GF (s, m, n). 

 The numerator function / L + /iL 1 + ... + ^L M has the factor 



which stands as a determined factor of the generating function. 

 Writing 



I, = (l+l) (1+2) ... (l+m+n-1)//, 



I.' = (1-8+1) (l-s+2) ... (I) (l+m+n) (l +m+n+1) ... ((-s+mn) ; 

 // involving (ro-1) (n-1) or /* bracket factors. 



