660 
MAJOR P. A. MACMAHON ON THE) 
Art. 53. In attempting to establish these results, it is easy to construct a 
generating function which contains implicitly the complete solution of the problems. 
The problem itself may be enunciated in another manner which has points of great 
interest. A two-dimensional graph of Sylvester may be supposed formed by pushing 
a number of cubes into a hat rectangular corner yOx in such wise that the arrange¬ 
ment is immovable under the action of forces applied in the directions xO, yO. 
It is clear that the number of such arrangements of n cubes is the number of two- 
dimensional graphs of n, or the number of partitions of the unipartite number n. 
Similarly, we may push a number of cubes into a three-dimensional rectangular 
corner, piling of cubes permissible, and such that the arrangement is immovable for 
forces applied in the three directions xO, y 0, zO. The enumeration of these arrange¬ 
ments is the same as that in the problem under discussion. 
Art. 54. First consider arrangements limited in the manner (/; m ; n) = (2 ; 1 ; oo ). 
We have such a graph as 
2 
2 
O 
Lj 
1 
1 , 
obtained by writing a column of nodes, and over it another column of nodes, not 
exceeding the former in number. 
We may take, as the generating function, 
_1_ 
(1 — ax) (1 — x/a) ’ 
in which we are only concerned with that portion of the expansion which is integral 
as regards a. The function is, in fact, redundant since it involves terms which are 
superfluous, and we obtain the reduced or condensed generating function by putting 
a equal to unity in the portion we retain. 
Since 
ax 
1 — 
a 
