THEORY OF THE PARTITION OF NUMBERS. 
G67 
where 
and also 
X. -f /x, 
X' + X" + ff, 
X, X' X",_ 
are in descending order of magnitude. 
A line of the graph has a certain weight 2X + /x. Any number of lines may be 
identical and consequently of the same weight, but no two different lines may have 
the same weight in the same graph. Let us form a graph, beginning at the lowest 
line, taken to be of weight unity, and proceeding upwards through every superior 
weight. We will find that such a graph may have a variety of forms. Construct the 
subjoined scheme of graph lines. 
/2_21 
1/ C 
2 5 
2 3 /2 3 1 
,21 3 / 21 3 
2 3 
2 3 1 3 / 
/2 4 
/2 3 1 / 2 3 1 3 
2 3 1 3 /2 3 1 4 
2 4 1 
2 3 1 3 
2 3 1 5 
_2 4 1 3 
2 3 1 4 
2 3 1 6 
21 4 
21 s / 
21 6 
21 7 
21 8 
l 4 l 5 
l 6 
l 7 
l 8 
l 9 
l 10 
In each column every graph line has the same weight. In each line every graph 
line has the same number of twos. From any graph line, say 2 A P (/x > 0), of 
weight 2X + p,, we can pass to a graph line of weight 2X fi- /x + 1 in two ways ; 
viz., by taking 2 A 1' X + 1 by horizontal progression or 2 A+1 l fX ~ 1 by diagonal progression. 
From 2 A we can only pass to 2 A 1 by horizontal progression. In accordance with these 
laws we can form a graph consisting of graph lines of all weights, from unity 
upwards, in a definite number of ways, depending upon the weight of the highest 
graph line. For example, we can select the graph whose successive lines are 
1 , 2 , 21 , 21 3 , 2 * 1 , 2 3 1 3 , 2 3 1 , 2 4 ’, 2 4 1 , 2 4 1 3 . . . . 
The progression from graph line to graph line is either horizontal or diagonal, 
which we can denote by A and B respectively. Then the graph may be denoted by 
BAABABBAA. 
The specification of the selected graph may be taken to be a collection of line 
graphs, each of which is reached by diagonal progression, and which proceeds by 
horizontal progression. 
Thus, in the particular case before us, the specification is 
2 , 2 3 1 , 2 4 . 
4 Q 2 
