OF THE COMPOSITION'S OF NUMBERS. 
867 
we take pn + 1 exactly similar graphs of the multipartite number 
and place them similarly with corresponding lines parallel and like points lying on 
straight lines. When these straight lines are drawn the graph is complete. Other 
than the initial and final points A, B there are (^9^ + 1) {p -2 + !)••• [Pn + 1) — 2 
points in the graph. 
There are lines in n different directions passing through or meeting at each point 
(including A and B). Call these directions the direction, the ag, &c., the a,, 
direction. A segment joining two adjacent points is called an a segment when it 
is in the a direction. A step through the reticulation traverses in succession any 
number of a^, . «« segments; but in any one step the segments must be taken 
in the order . . . a,^. In a step the number of steps traversed may be zero 
in any one, any two, &c., any n — 1 of the n different directions. If a step involve 
p\ segments in the a, direction, p'g segments in the ag direction and so on, it may be 
represented by the multipartite number 
7. . p)'n^ 
A succession of steps, the first starting at A and the last terminating at B, is 
represented by a succession of multipartite numbers, constituting a composition of the 
multipartite _ 
PiPAh • • • 
Any composition follows a certain line of route through the reticulation. The 
number of distinct lines of route is the number of permutations of the letters in 
. . . a,/'*, 
and is therefore 
/Pl + 2^2 + • • • + P«\ 
\ lh>Vv’-Pn-i /’ 
employing an obvious extension of notation. 
The graph of a composition is obtained by placing nodes at the points which 
terminate the first, second, &c., and penultimate steps. Essential nodes occur upon 
a line of route whenever the direction at a point changes from to where u > t. 
At this point the contact between the adjacent parts of the composition is such that 
a part terminating with n — u zero elements precedes a part commencing with 2—1 
zero elements. We may speak of this as a contact oi n — u zeros with t — 1 zeros. 
The number of zeros in contact, being 
n — 1 — [u — t), 
may be any number from 0 to n — 2, according to the magnitude o£ u — t; and 
there are ^ + 1 different contacts for which the number of zeros in contact is j". 
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