890 
MAJOR P. A. MACMAHOX ON THE THEORY 
node must be also a blank space node. The only line of route through the reticu¬ 
lation which does not involve an essential (that is a blank space) node is ACB. 
Consequently the graphs of all the combinations having but one part must be alono' 
the line of route ACB. Similaidy in the reticulation of the multipartite number 
Pilh • • • P., 
the graphs of all the one-part combinations will be along the line of route traced out 
by 2^1, oL^ segments, po, ao segments, and so on in succession. The Jc — 1 different 
symbols at disposal may be placed at pleasure at + Pc + • • • +p« — 1 points 
along this line of route. Hence 
diffei’ent one-part combinations are obtainable. 
In other words the generating function for such combinations is 
/q “k — 1) ho “k — 1)^ A3 “k . . . -k (A — 1)” (n = 00 , 
where in a previous notation h,i is the homogeneous product sum, degree n, of the 
quantities 
^1? ^ 3 ? • • • 
72 . Next as to the combinations which have two parts. At any point D of the 
reticulation place a blank space node. All two-part combinations whose graphs pass 
through D must follow the line of route AEDFB, for otherwise an additional 
essential (and blank space) node would be introduced. The whole combination may 
be split up into a one-part combination along the line of route AED, followed first by 
a blank space, and then by a one-part combination along the line of route DFB. All 
the two part combinations whose graphs pass througli the point D are obtained by 
associating every one-parb combination in the reticulation AD with every one-part 
combination in the reticulation DB. 
Hence the whole number of combinations, having two parts, of the multipartite 
number 
is 
where 
P1P3 . . . p„ 
^ _ 1) Aa + • • • + ~ ^ (A _ 1 )+•••■'■ r"" “ 1 
v'l + = Ih ; P'3 + = lh> • • • p'n + p'n = P«. 
Hence, the generating function for two-part combinations is 
{hi + (A — 1) A3 + (A — I)-" A3 “k • • • + (A 1)" * A;,/}". 
n' = CO. 
