915] 
ON THE PARTITIONS OF A POLYGON. 
101 
13. The reasoning is perfectly general; and applying it successively to the 
partitions into two parts, three parts, &c., we have 
V 3 =U 3 , 
V 3 = 2U 1 U 2 , 
V 4 = 2U 1 U 3 + TJ£, 
V 5 = 2U 1 U i + 2U 2 U 3 , 
where any function V is related to the corresponding function TJ as above. The 
value of U-y is obviously 
U 1 = x 3 + a? + x 5 + ... = 
1 — x 
and hence the several functions U and V have the values above written down; the 
general term of TJk is 
[r + k — 2] fc_1 [r — 3p -1 
[&p -1 [A; — l]* -1 
x r ; 
and the number of partitions of the r-gon into k parts is equal to the coefficient 
of x r in this general term. 
14. In the investigations which next follow, I consider, without using the method 
of generating functions, the problem of the j>artition of the r-gon into 2, 3, 4 or 5 
parts; it will be convenient to state the results as follows: 
Number of Partitions. 
2 parts, ~ A, 
Zi 
r 
3 parts, ^ 2A, 
4 parts, ^ (3^4 + 2B), 
5 parts, 3 (4A -f 8B + 2C); 
O 
where the capital letters refer to different “ diagonal-types,” thus: 
2 parts, 3 parts, 4 parts, 
or 1 diagonal: or 2 diagonals: or 3 diagonals: 5 parts, or 4 diagonals: 
A A A B A B C 
viz. if, in a polygon divided into k parts by means of k — 1 diagonals, we delete 
all the sides of the polygon, leaving only the diagonals, then these will present them 
selves under distinct forms, which are what I call “ diagonal-types ”; for instance, when 
k = 4, 
there are the two types A and B shown in the above diagram for four parts.