MAJOE P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 359 
to enumerate by the coefficient of those partitions of for which 
“h “3 "h “5 “1“ • • • = '^1 
^*2 “h “f" ®6 4" • “ • — '^2* 
This enumerating function, since it involves Xi and x^, is one connected also 
with the partitions of bipartite numbers. In general when k sets are taken, we 
have a theorem of Z:-partite partitions. When h = i, we have at once a real 
generating function for unipartites and an enumerating function for ^-partites, for, 
from the latter point of view, the number unity which appears as the coefficient of 
XpX?. .. X^ shows that the multipartite number 
ttia, . . . 
can be partitioned in one way only into the parts 
1 
0 
1 
1 
0 . 
1 
1 
1 . 
1 
1 
1 .... 1 
there being ^ figures in each part. 
A.rt. 70 . We may now enquire into the partitions of all numbers 
a2, “3, . . • “i, 
subject to the given conditional relations and also to the linear equations 
+ • • • + 'Hi^i — ^ 
+ /^'2«2 + • • • + =■ 
+ /x^*>a2 4- ... 4- = w'*’. 
To illustrate the method, it suffices to take s = 2, and then we have to perform 
the summation 
2X1‘“-Xr’. . . 
The fl function reduced is 
1 - Xl‘Yf-. 1 - X^'X^^Y1''Y?'= ... I - Xl-Xf^ . . . XrYi'Yr^ . . . Yf ’ 
