THEORY OF THE PARTITIONS OF NUMBERS. 



1G3 





and taking each number in succession, in order from the highest to the lowest, write 

 a letter a, /3, or y, according as the number is in the first, second, or third row. Thus 

 beginning with 6 we write down a, then for 5 ft, for 4 y, for 3 a, for 2 a, and, lastly, 

 for 1 ft, thus obtaining 



a , ft, J, , , ft- 



Now underneath the a's write G, 5, 4 in order, under the ft's, 3, 2 in order, and 

 under y 1, in accordance with the rows of the arrangement 



654 



32 



1 



We thus obtain 



I say that 



a ft y a. a ft 



G 3 1 5 4 2. 



631542 



is a permutation subject to the given conditions as defined by the descending orders 



in the arrangement 



654 



32 

 1 



The 16 permutations corresponding to the 16 graph arrangements are 



aaappy 

 654321 



a.a.ftftya, 

 653214 



aaapyp 

 654312 



act.fta.fty 

 G53421 



a,a.fta.yft 

 653412 



a.aftfta.y 

 653241 



aa.ftya.ft 

 653142 



aaftyfta aftyafta aftyaaft aftayfta aftayaft 

 653124 631524 631542 635124 635142 



aftaftya. aftaayft aftaftay aftaafty 

 635214 635412 635241 635421. 



To show that there are no other permutations it is sufficient to prove that one can 

 pass back from a permutation to a graph in a unique manner. 



Thus take the permutation 



635124; 







write a's under 6, 5, 4 ; /3's under 3, 2 ; and y under 1 : 



635124 



the succession 



ctftctyfta. 

 Y 2 



