OF THE COMPOSITIONS OF NUMBERS. 
841 
Inversely conjugate, 
Self inverse. 
(4111) 
(7) 
(3211) 
(313) 
(2221) 
(232) 
(2131) 
(21112) 
(1312) 
(151) 
(1222) 
(12121) 
(1123) 
(11311) 
(1114) 
(1111111) 
The general form of an inversely conjugate composition is 
. . . ag + 2 . 1“^-“ . + 2 . 1"^'" . + 2 . r*"' 
in its form prepared for conjugation. 
7. The compositions of a number, m, give rise to the compositions of m + 1 by 
rules somewhat similar to those in Arbogast’s method of derivations. The rules are 
obvious as soon as stated. 
Each composition of the number m gives rise to two compositions of the number 
m + 1. 
I. By prefixing the part unity. 
II. By increasing the magnitude of the first part by unity. 
All the compositions thus obtained are necessarily distinct. As an example see 
the subjoined scheme for passing from the compositions of 3 to those of 4. 
Ill 12 21 3. 
1111 211 112 22 121 31 13 4. 
If the conjugates of these two lines be taken, the result is the same as the two 
• fl BS fc 
lines inverted. We have the theorem, easily proved : ‘‘ The conjugate of the 
second 
derivative of a composition is the 
second 
first 
derivative of the conjugate composition.” 
8. The theory of the compositions of numbers is closely connected with the theory 
of the perfect partitions of numbers.* The connection is between the compositions 
of all multipartite numbers and the perfect partitions of unipartite numbers. The 
enumeration of the compositions of a single multipartite number enumerates also the 
perfect partitions of an infinite number of unipartite numbers. There is, moreover, 
* “ The Theory of Perfect Partitions of Numbers and the Compositions of Multipartite Numbers,” 
The Author, ‘ Messenger of Mathematics.’ New Series, No. 235. November, 1890. 
MDCCCXCIII.—A. 5 P 
