112 
ON THE PARTITIONS OE A POLYGON. 
[915 
partitions 115 is the half of this, = 45. And by the like process it is found that 
the numbers of the partitions 124, 133, 223 are equal to 90, 45, 45 respectively; 
and then, as a verification, we have 
45 + 90 + 45 + 45=225, 
the whole number of the 3-partitions of the 9-gon. 
29. The third column (4 parts) is derived in like manner from the second 
column by aid of the first column; and so in general, each column is derived in 
like manner from the column which immediately precedes it, by aid of the first 
column. And we have for the numbers in each compartment of any column the 
verification that the sum of these numbers is equal to the whole number (for the 
proper values of k and r) of the ^-partitions of the r-gon. 
It might be possible, by an application of the method of generating functions, 
to find a law for the numbers in any compartment of a column of the table; but 
I have not attempted to make this investigation. 
30. In the table in No. 2, the numbers 1, 2, 5, 14, 42, &c., of the diagonal 
line show the number of partitions of the triangle, the quadrangle, the 5-gon, ..., r-gon 
into triangles: viz. these numbers show the number of partitions of the r-gon into 
r — 2 parts, that is, into triangles; and, for the r-gon, writing 
k = r — 2, 
the number is 
_ [2r - 4] r ~ 3 
[r — 2] r ~ 3 
If, as above, taking the weight of the triangle to be 1, we write 
o— 2 = w, 
then the number is 
[2 wp 
[w] w 
viz. this is the expression for the number of partitions of the polygon of weight w, 
or ('w + 2)-gon, into triangles. 
31. The question considered by Taylor and Rowe, in the paper referred to in 
No. 1, is that of the partition of the r-gon into ^-gons, for p, a given number > 3; 
this implies a restriction on the form of r, viz. we must have r — 2 divisible by 
p — 2. In fact, generalizing the definition of w, if we attribute to a p-gon the 
weight 1, and accordingly to a polygon divisible into w p-gons the weight w, then, 
r being the number of summits, we must have 
r = (p — 2) w + 2. 
In particular, if p = 4, so that the r-gon is to be divided into quadrangles, then r is 
necessarily even, and for the values 
w= 1, 2, 3, ..., 
we have 
r =4, 6, 8, ....