THEORY OF THE PARTITION OF NUMBERS. 
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The graph is therefore representative of three multipartite numbers and of two 
partitions of each. 
Art. 45. A multipartite number has two characteristics. It may be r-partite, i.e., 
it may consist of r figures, and its highest figure may be p. A multipartite partition 
has three characteristics. Each part may be r-partite ; the highest figure may be p ; 
the number of parts may be q. If the graph be formed of a multipartite partition 
with characteristics 
r, p, q, 
the five other readings yield partitions with characteristics :— 
p, r, q 
q, r, p 
r, q, p 
p, q, r 
q, p, r. 
The six partitions correspond to the six permutations of the three symbols p, q, r. 
The two partitions which are r-partite appertain to the same multipartite number ; 
similarly for the pairs which are yopartite and ^-partite respectively. Hence the 
three multipartite numbers involved correspond to the three pairs of permutations so 
formed that in any pair the commencing symbol of each permutation is the same. 
Art. 46. The consideration of graphs formed with a given number of nodes now 
leads to the theorem : “ The enumeration of the graphically regularised r-partite 
partitions, into q parts and having p for the highest figure, gives the same number 
for each of the six ways in which the numbers p, q, r may be permuted.” 
Also the theorem :— 
“ The enumeration of the graphically regularised partitions which are at most 
r-partite, into q or fewer parts, the highest figure not exceeding p, gives the same 
number for each of the six ways in which the numbers p, q, r may be permuted.” 
The first theorem is concerned with fixed values of p, q, and r; the second with 
restricted values of these numbers. It is also clear that we may fix one or two of 
the numbers and leave the remaining two or one restricted. 
Observe that this six-fold conjugation obtains even though equalities exist between 
the numbers p, q, r ; they must be regarded always as different numbers. Some¬ 
times, as we shall see, the correspondence is less than six-fold, but this does not 
depend solely upon the assignment of the numbers p , q, r. 
If we regard the multipartite number appertaining to a partition and not merely 
the total content, we find that the partitions occur in pairs. 
