110 
ON THE PARTITIONS OF A POLYGON. 
[915 
as before. The calculation is 
r — 10 . r — 9. r — 8 . r — 7 = 
+ 35. r — 9 . r — 8. r — 7 
+ 378. r - 8.r - 7 
+ 1470.r - 7 
+ 1680 
r 4 — 34r 3 + 431r 2 — 2414r + 5040 
+ 35r 3 — 840r 2 + 6685r — 17640 
+ 378r 2 - 5670r + 21168 
+ 1470r - 10290 
+ 1680 
r 4 + r 3 — 31r 2 + 7lr - 
— y* __ 0 ry* mmmm 2 . ff X . v -j- *7 
Four diagonals, G.—We have 
C = C° + 4(7' + 6(7" + 4(7" + 1 (7 V , 
where the coefficients are the terms of (1, l) 4 . We have 
42 
G° = 
C = 
C" = 
C'" = 
C iv = 
— 11.1— 10.r — 9 .r — 8 . r — 7 . r — 6.1— 5 
5040 
1— 10.r — 9.1— 8.1— 7 . r — 6 . r — 5 
720 
r — 9.r — 8.r — 7.r — 6.r — 5 
120 
r — 8 . r — 7 . r — 6 . r — 5 
24 : 
r — 7.r — 6.r — 5 
6 
and thence 
(7 = 
r — 7 . r — Q . r — 5 .r — 4.r — 3.r — 2.r — 1 
5040 
I omit the calculation, as the equation is at once seen to be a particular case of 
a known factorial formula. 
25. We may analyse the partitions of an r-gon into a given number of parts, 
according to the nature of the parts, that is, the numbers of the sides of the 
several component polygons. It is for this purpose convenient to introduce the 
notion of “weight”; say a triangle has the weight 1, then a quadrangle, as divisible 
into two triangles, has the weight 2, a pentagon, as divisible into three triangles, 
has the weight 3,..., and generally an r-gon, as divisible into r —2 triangles, has 
the weight r — 2. It at once follows that, if 
W=w + w\ or = w + w' + w", &c., 
then a polygon of weight W is divisible into two polygons of the weights w, w', 
or into three polygons of the weights w, w, w" respectively; and so on. Thus the 
2-partitions of an 8-gon (weight = 6) are 15, 24, and 33; the 3-partitions are 114,