THEORY OP THE PARTITION OF NUMBERS. 
645 
S a 
« 
°32 
Permutation. 
Composition. 
Partition. 
Number partitioned. 
1 
1 
aya/3y/3 
(101 111 010) 
(To 2 T) 
3T 
1 
1 
/3ya~y/3 
(Oil 201 010) 
(TT 21) 
22 
1 
1 
ay/3'ya 
(101 021 100) 
(To IT) 
22 
1 
1 
ayayfi- 
(101 101 020) 
(Id 20) 
30 
1 
1 
fiyfiyor 
(011 011 200) 
(01 02) 
03 
1 
1 
ya~/3y/3 
(001 211 010) 
(do 21) 
TT 
1 
1 
ay/3ya/3 
(101 011 no) 
(To IT) 
2d 
1 
1 
yoryfi 2 
(001 201 020) 
(do 20) 
20 
1 
1 
y/8 2 ya 3 
(001 021 200) 
(do 02) 
02 
1 
1 
y/3ya.~/3 
(001 011 210) 
(oo oT) 
oT 
2 
0 
fi~yoLyu 
(021 101 100) 
(02 12) 
TT 
2 
0 
fiyafiya 
(Oil 111 100) 
(oT 12) 
13 
2 
0 
yafi~yu 
(001 121 Too) 
(do 12) 
12 
2 
0 
fiyayufi 
(Oil 101 L10) 
(01 TT) 
12 
2 
0 
ya/3yrx./3 
(001 111 110) 
(od IT) 
II 
2 
0 
yaya/3 3 
(001 101 120) 
(oo To) 
To 
0 
2 
ory/3y/3 
(201 011 010) 
(20 2l) 
TT 
There are 
36 partitions. 
The first two columns show the nature of the permutation in regard to ya and y/3 
contacts and the nature of the composition in regard to positive-positive and positive- 
zero contacts. The partitions are into two parts, zero not excluded, and have regard 
to bipartite numbers extending from 44 to 00. They are doubly regularised by 
ascending magnitude, and the figures of the parts do not exceed 2, 2 the first two 
figures of the tripartite. 
If we write down the partitions of 4 into two parts, zeros not excluded, limited 
not to exceed 2 in magnitude, viz. :— 
22 , 12 , 02 , 11 , 01 , 00 , 
the ascending order of part magnitude being adhered to, we can obtain one of the 
36 partitions by combining any one of these partitions with itself or any other of the 6. 
Thus the fourth of the above partitions is obtained by combining the unipartite 
partitions 
02 , 22 , 
and from any two unipartite partitions 
ab, cd , 
