106 
ON THE PARTITIONS OF A POLYGON. 
[915 
the summits to be numbered 1, 2, 3, r in succession (the numbering may begin 
at any one of the r summits); regarding each of these numberings as giving a 
different partition, we should have the factor r. But, in fact, the partitions so 
obtained are not all of them distinct, but we have in each case a system of 
partitions repeated as many times as there are summits of the diagonals, that is, a 
number of times equal to twice the number of the diagonals; and we have thus, 
after the multiplication by r, to divide by the numbers 2, 4, 6, 8, in the four cases 
respectively. 
21. We hence have immediately:— 
Two parts, the number of partitions 
r . r.r — 3 
= 2 i= TT ; 
Three parts, the number of partitions 
r A r+l.r.r — 3 . r — 4 
= 2 A = 3.2.2.1 ’ 
Four parts, the number of partitions 
r + 2.r+l.r.r — 3 . r — 4. r — 5 
4.3.2.3.2.1 
the calculation being 
3 (r 2 + 7r + 2) = 3r 2 + 21r + 6 
+ 2.r — l.r — 2 + 2r 2 — 6r + 4 
5 r 2 + 15r + 10 
5 . r + 1 .r+2 ; 
Five parts, the number of partitions 
= g (4,A + 8B + 2C) = 
r+3.r+2.r+l.r.r — 3.r — 4 . r — 5 . r — 6 
~5.4.3.2.4.3.2.1 
the calculation being 
4 (r 3 4 + 18r 2 + 65r) = 4r 3 + 72r 2 + 260r 
+ 8.r — l.r — 2.r + 7 + 8r 3 + 32r 2 — 152r + 112 
+ 2.r—l.r—2.r — 7 + 2r 3 — 20r 2 + 46r — 28 
= 14r 3 + 84r 2 + 154r + 84 
= 14 (r 3 + 6r 2 + Hr + 6) 
= 14.r + l.r + 2.r + 3. 
To complete the theory, it would be in the first instance necessary to find for 
any given number of diagonals, k — 1, whatever, the number and form of the 
diagonal-types, A, B, G, &c.; this is itself an interesting question in the Theory of 
Partitions, but I have not considered it.