896 
MAJOR P. A. MACMAHOI^ OX THE THEORY 
which for the multipartite 1'' ^ is 
k"' 
This expression also gives [ante) the number of combinations of order k of the 
unipartite number 
zeros excluded. 
11 , 
83. The correspondence may be shown by reference to the “ Theory of Trees.” 
The trees to be considered are of altitude k and have n terminal knots. 
For simplicity take n — k = ?,. 
The trees are 
which, as shown above, represent combinations of order 3 of the unipartite number 3. 
These are 
3 21 12 111 2|1 ll2 11|1 1|11 1|1|1, 
the number of parts (observe that blank space symbols are between adjacent parts) 
being equal to the number of bifurcations in the second row from the top, increased 
by unity. 
Now these trees may be interpreted so as to represent the compositions of the 
multipartite (1^) into three parts, zero parts not excluded, by attending to the con¬ 
nection between the twm bifurcations of each tree and the two inter-terminal-knot 
spaces. 
In any row of knots in a tree we have or we have not bifurcations, and each 
bifurcation communicates either Avith the first or with the second inter-terminal-knot 
space. 
Beginning with the row marked A, if we find no bifurcation aa^c write 00; if we 
find a bifurcation communicating with the first space but not Avith the second we 
write 10 ; if with the second space and not with the first 01 ; if there be two bifur¬ 
cations, Avhich necessarily in the present case communicate Avith both spaces, we 
write 11 ; and proceeding in the same way Avith the row^s B and C in succession we 
will have finally written down three bipartite parts constituting a composition of the 
multipartite 11. 
We thus obtain, beginning Avith the left hand tree, 
