THEORY OF THE PARTITION OF NUMBERS. 
631 
“ Of the partitions whose parts are limited in magnitude to p and in number to q , 
there are, 
q + a — c\ p — a 4 - c 
a \ c 
such that the highest part + a and number of parts <£ c.” 
In particular, if 
a + b = p, 
or 
a — c = p — q, 
the number is 
which enumerates the lines of route with p — a left-bends. There is thus a one-to- 
one correspondence between the lines of route passing through the point for which 
(a, b) = (a, p — a), and the lines of route with p — a left-bends, and hence also 
between the partitions under consideration and those which involve parts of p — a 
different kinds. 
Art. 17. A line of route is a graphical representation of a principal composition of 
the bipartite. Such a composition being 
the numbers 
(Mi Wh Mj • • • M»)> 
9A P'2’ ( l2> Vz’ 9.z • • • P* 
are all superior to zero. A positive number q x of the bipart P\q x is adjacent to a 
positive number p 3 of the bipart ptfp, and we may assert of the composition that all 
its contacts are positive-positive (see Art. loc. cit.). 
Each line of route represents a composition with positive-positive contacts, and 
there is a one-to-one correspondence. Hence : 
“ There is a one-to-one correspondence between the compositions, with positive¬ 
positive contacts, of the bipartite pq and the partitions of all unipartite numbers into 
parts limited in magnitude by p and in number by q” 
Further, there are as many such compositions with 5+1 parts as there are such 
partitions which involve s different parts. 
Art. 18. Every line of route through the reticulation of pq may be represented by 
a permutation of the letters in o+S 7 . We have merely to write down the a and /3 
segments as they occur along the line of route to obtain such a permutation. 
The composition 
(Mi Ms Ms • • - Mi 2*+i)> 
