915] 
93 
915. 
ON THE PARTITIONS OF A POLYGON. 
[From the Proceedings of the London Mathematical Society, vol. xxn. (1891), 
pp. 237—262. Read March 12, 1891.] 
1. The partitions are made by non-intersecting diagonals; the problems which 
have been successively considered are (1) to find the number of partitions of an 
r-gon into triangles, (2) to find the number of partitions of an r-gon into k parts, 
and (3) to find the number of partitions of an r-gon into p-gons, r of the form 
n(p— 2) + 2. 
Problem (1) is a particular case of (2); and it is also a particular case of (3); 
but the problems (2) and (3) are outside each other; for problem (3) a very elegant 
solution, which will be here reproduced, is given in the paper, H. M. Taylor and 
R. C. Rowe, “ Note on a Geometrical Theorem,” Proc. Lond. Math. Soc., t. xm. (1882), 
pp. 102—106, and this same paper gives the history of the solution of (1). 
The solution of (2) is given in the memoir, Kirkman “ On the ^-partitions of 
the r-gon and r-ace,” Phil. Trans., t. cxlvii. (for 1857), p. 225; viz. he there gives 
for the number of partitions of the r-gon into k parts (or, what is the same thing, 
by means of k — 1 non-intersecting diagonals) the expression 
[r + k — 2p -1 [r — 3R -1 
[&]*- 1 [k - I]*“ 1 ’ 
but there is no complete demonstration of this result. 
If k = r— 2, we have the solution of the problem (1); viz. the number of partitions 
of the r-gon into triangles is 
= [2r — 4] r_s -7- [r — 2] r-3 . 
The present paper relates chiefly to the foregoing problem (2), the determination