ON THE PARTITIONS OE A POLYGON. 
107 
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22. Although the foregoing process (which, it will be observed, deals with the 
diagonal-types, without any consideration of the sub-types) is the most simple for the 
determination of the numbers A, B, G, &c., yet it is interesting to give a second 
process. Considering the several cases in order: 
One diagonal, A.—The diagonal has two summits; we must have on each side 
of it one summit, and there remain r — 4 summits which may be distributed between 
'7* ^ 
the two intervals formed by the diagonals. This can be done in — 
have, as before, 
ways, or we 
A = 
r — 3 
Two diagonals, A.—The diagonals have four summits; we must have outside each 
diagonal one summit, and there remain r — 6 summits to be distributed between the 
four intervals formed by the diagonals; this can be done in ——^- r ^ ^ ‘ r —- ways, 
or we have this value for A°. But the two top summits of the diagonals, or the 
two bottom summits, may coalesce; in either case, the diagonals have three summits. 
We must have outside each diagonal one summit, and there remain r— 5 summits 
to be distributed between the three intervals formed by the diagonals; the number 
of ways in which this can be done is 
r — 4 . r — 3 
for 
the 
of 
say this is the value of A'. And we then have A =A° + 2A', 
_r+ l.i— 3 . r — 4 
6 ’ 
as before. The calculation is 
r — 5 + 6 = r+l. 
23. Three diagonals, A.—See No. 15 for the figures of the sub-types. We have 
A =A° + 4,A' + 4.A", 
where the coefficients, 4 and 4, are the number of ways in which A' and A" 
respectively can be derived from A 0 by coalescences of summits. For 4°, the 
diagonals have six summits, and there must be outside of two diagonals one 
summit; there remain r — 8 summits to be distributed between the six intervals 
formed by the diagonals, and we have 
/)0 _r—7.r—6.r — 5 .r— 4 .r — 3 
A ~ 120 • 
For A', the diagonals have five summits, and we must have outside of each of two 
diagonals, one summit; there remain r — 7 summits to be distributed between the 
five intervals formed by the diagonals ; we thus have 
r — 6. r — 5.r — 4. r — 3 
A' = 
24 
14—2