OF THE COMPOSITIONS OF NUMBERS. 
881 
60. The unsymmetrical form gives a direct connection between the numbers of the 
compositions and of the permutations of the letters in the product 
• • • “/t 
In the unsymmetrical product the term ay^a/- . . . attached to some numerical 
coefficient, arises in connection with every permutation of the letters in the term. 
One factor of such coefficient is manifestly 
^Pi—i • 
^ > 
if the letter a^, in a particular permutation, occur q .2 times in the last 
^ 9 ^ -f Pg -f ... + p„ places of the permutation there will be a factor 
further, if the letter a^, in the same permutation, occur qg times in the last 
Vs Ps+i + • • • Pu places of the permutation there will be a factor 
2^/'. 
Hence for the permutation considered there arises a term 
2;ji—l+y2+y3+ • • • 
and the number of compositions must therefore be 
the summation being taken in regard to every permutation of the letters in the 
product 
E.g., to find in this way the number of compositions of the bipartite 22, we have 
the following scheme :— 
% 
2 
ai«3 
1 
aoa^ 
1 
1 
a^ai 
1 
0 
Therefore 
F (22) = 22-1 (22 4- 21 + 21 + 21 + 21 + 2°) = 2G. 
MDCCCXCIII,—A. 5 U 
