THEORY OF THE PARTITION OF NUMBERS. 
659 
where the notation ( l;m;n ) is employed to represent the generating function of 
partitions whose graphs are limited in height, breadth, and length by /, m. n 
respectively. 
Similarly we shall find :— 
(3 
; co 
; oo 
) 
= (1 
— 
X ) 
Mi 
- X 2 )- 
'M(i 
-a; 3 )(l -x^) 
-3 
(4 
; °o 
; 
) 
= (1 
— 
x)~ 
Ml 
0\ _ 
~ x ) 
Mi' 
— a ,3 ) _3 [(1 — x‘ 
Ml" 
- x h ) 
...]M 
(M 
oo \ 
; co 
) 
= (1 
— 
x)~ 
Mi 
— X 2 )' 
- 2 
. (i _ x i-i )-(*- 
■ 1} C(1 
— a: 
0(i — x 
: ' +1 ) • • • ]M 
(M 
; i ; 
CO 
) 
= (1 
— 
x)~ 
Mi 
— X 1 ) 
■1 # 
. {1 -af)"\ 
(M 
; 2 ; 
00 
) 
= (1 
— 
x)~ 
Mi 
-- X 2 )~ 
’Mi 
- x 3 )- 2 ... (1 
— af)~ 
■Ml 
— x l+1 )“ 
-1 
5 
(M 
3 ; 
00 
) 
= (1 
— 
x)~ 
‘Mi 
— x 2 )~ 
Mi 
— ad) -3 . . . (1 • 
— af)~ 
■Ml 
— a/ +1 ) _ 
i 
+ 
to 
1 
V—* 
(M 
m ; 
CO 
) 
= (1 
— 
x)~ 
Mi 
— x 2 )~ 
-2 
. (1 — x m ~ l )~^- 
-B X 
[(1- 
- x m ) . . 
1 
Si, 
V— 
1_I 
X 
(1 
— 
x l+l 
■ ^+2)- (m -2) < _ _ 
(1- 
jqI + 1)1 
if m be not greater than l ; 
with an equivalent form 
(l ; m ; co ) = (1 — x) 1 (1 — ar) 2 ... (1 — x l l ) l ~ l X [(1 — x l ) . . . (1 — x m )~\' 
and finally 
X (1 — x w+1 )-6“ 1 ) (1 — x m + 3 ) 
if m be greater than l ; 
- 6 - 2 ) 
(1 — x l+m ~ 1 )- 1 , 
(l ; m ; n) = —-. —- — . . . v —:-gyMy- 
v ' 1 — x (1 — x 3 ) 3 (1 — x l l y 1 
X 
X 
"1 - x n+l 1 — 
✓yW + £ + 1 
1 — x n+m 
l 
_ 1 - x l ■ 1 - 
- * ; + 1 ‘ ‘ * 
1 — x m 
(1 —x n + m + l ) l ~ l 
(1 — x n+m+ 
2 y -2 
^ _ gAi + l -J m — 1 
1 _ a-m+iy-i 
(1 — x m+2 
y-2 • * * 
1 — X l+M ~ l ’ 
a result which can be shown to be symmetrical in l, m and n, as ought, of course, to 
be the case. 
This expression for (l ; m ; n) can be exhibited in a more suggestive form, viz. : — 
Writing 1 — x s = (s) 
n . . \ _ ( 11 + 1) ( n + 2). (I -f m + n — 1) 
( ’ 5 } (1) (2) . (I + m ~ 1) 
X 
(n + 2) (n + 3) . (I + m + n — 2) (n -f 3) (n + 4).(/ + m + n — 3) 
( 2 ) 
(3) 
(1 + rn - 2 ) 
X 
(3) (4) . (l + m- 3) 
X . . . . to l factors or m factors, according as m or l is the greater. 
4 P 2 
