MAJOR P. A M \.M.\IION ON THE COMPOSITIONS OF NUMBERS. 1,7 



There is no other method of ordering a collection of numbers which is of ^ru 

 application. 



We see that the same collection of numbers gives rise to only one partition, but, by 

 permutation, to more than one composition. 



Art. 2. Both partitions and compositions have an appropriate graphical represen- 

 tation. That of a partition was first given by FERKKI:S. and the notion was elaborated 

 by SYLVESTKK during the time he was at the Johns Hopkins University in Baltimore, 

 U.S.A. It consisted merely iti writing a row of nodes, or units, corresponding to each 

 number (or part) of the partition, the left-hand nodes of the rows being placed in a 



vertical line. Thus 



774211 



is denoted by 



Art. 3. A trial will show that this method is not suited to compositions. One 

 method, effective for certain purposes, was given by the author.* To indicate it, 



consider the composition 



142 



of the number 7. 



* . . . * --- . 



We take seven segments on a line, and place nodes, *, so that the line is divided 

 off into 1, 4 and 2 segments respectively in order. The conjugate composition is 

 reached from this by suppressing the existing nodes and placing nodes at the points 

 of division which are free from nodes. 



Thus 



. . * * * . * . 



denotes the composition 21121 







Art. 4. There is a more illuminating mode of representation which is here given, it 

 is believed, for the first time ; it is akin to the method of FERRERS, and enables 

 methods of research which SYLVESTER'S exertions have made familiar. 



It consists in taking rows of nodes in order and placing the left-hand node of any 

 row vertically beneath the right-hand node of the previous row. 



Thus 



142 

 is denoted by 



* " Memoir on the Theory of the Compositions of Numters," ' Phil. Trans. Roy. Soo.,' 1893. 



ic 2 



