OF THE COMPOSITIONS OP NUMBERS. 
897 
(00 00 II), (00 01 10), (00 10 01), (00 II 00), (01 00 10), (10 00 01), 
(MIo"^), (lo'^'^), (ll'^OO), 
in order of correspondence with the above written combinations, of order 3, of the 
unipartite number 3. 
84. The process is j^erfectly general. Every tree of altitude k is representative 
alike of a combination of order k of a unipartite number, zeros excluded, and of a 
composition of a unitary multipartite, zeros not excluded, and each combination or 
composition of the nature considered is uniquely represented by a tree. 
The number of compositions of the multipartite 1"“^, zeros not excluded, into 
k or fewer parts is 
in-i _p 2«-i 
a number which also represents the number of the aggregate of the combinations of 
the number n, of orders 1, 2, . . . k, zero parts not excluded. 
The interesting fact here brought to light is the connection between the unipartite 
numbers and the unitary multipartite numbers. 
85. We have seen that expresses the number of combinations of order k 
possessed by the unipartite number m. Each combination involves a certain number 
of the k different symbols and we may inquire the number of combinations which 
involve exactly p out of the k symbols. It is clear that one combination can be formed 
which involves any one symbol and none of the others; hence, k combinations involve 
but a single symbol. Two out of k symbols may be selected in different ways ; 
for each such selection we must take the number of combinations of the number m, 
of order 2, and subtract the number of these in which but a single symbol appears. 
Hence, the number of combinations involving exactly two symbols is 
Similarly three out of k symbols may be selected in 
ways, and for each selection 
we take the whole number of combinations of order 3 and subtract those of them 
which involve exactly two symbols or exactly one. 
Hence we arrive at the number 
In this way it is easy to see that the number of combinations involving exactly p 
symbols is 
MDCCCXCIEI.— A 5 Y 
