64 2 
MAJOR P. A. M ACM AH ON - OH THE 
depicted, one line of route for each permutation of the symbols in the product a^/3 q y. 
A. study of these permutations shows the connection with a certain class of bipartite 
partitions. 
Consider a permutation 
a.P'fty 1 apy-iyt . . . a 
which is not the most general permutation, but such that, in regard to any section 
ol^P 1 y* 
of the permutation 
(1.) r k must be superior to zero except when k — s. 
(2.) p k , q k may be either, but not both, zero, except when k = 1. 
The permutation has ya and y(3 contacts, but no /3« contact. 
In the reticulation corresponding thereto, we have lines of route with ya and y(3 
bends but not with /3a bends. All the lines of route with /5a bends are excluded 
from consideration. From the permutation we can form a bipartite partition. 
_>'l_‘2 _1-3 _ 
(['i'll Sh+Qa Pi~^rPz~\~P3> Q'i+S’s+Ss •• -P\ J tP-2^r •• <?i + 9 , 3 + ••■•+?? )> 
which is regularised in the sense that the partitions of the unipartites p, q, that 
appear are each separately regularised. 
The two parts of the bipartite number thus partitioned are 
T \P\ + A {pi + P 2 ) + r 3 (pi + P-2 + Ps) + • • • + r * (lb + P% + • • • + Ps), 
r i9i + r -2 ( ( h + <h) + ('ll + 'h + <h) + • • • + rt (qi + ffs + • • • + p)- 
The associated principal composition is 
(PPIPl PPIPi PI IP'A • • • Ps'lPs). 
As before, consider the contacts 7’ 1 p. 2 , r 2 y> 3 , &c. . . . Looking at the whole of the 
principal compositions, observe that a ya contact in the permutation yields a contact 
r ip/c+i in the composition in which r k and 2 ) &+i are both superior to zero, say a 
positive-positive contact. A y(5 contact yields a positive-zero contact and a /3a 
contact a zero-positive contact. Hence the present correspondence is only concerned 
with compositions which possess positive-positive and positive-zero contacts, and not 
with those which involve contacts of other natures. The bipartite partitions are 
those of all bipartite numbers into biparts whose parts are limited to p and q 
respectively in magnitude and whose biparts are limited to r in number. 
