i MAJOR P. A. MAOMAHON ON THE COMPOSITIONS OF 



aud ' 142771 



by 



This graph is read horizontally ; the conjugate is obtained by reading vertically, 



giving 



21122111112111112, 



or, in brief notation, 



21 2 2 a l 8 21 5 2. 



We may also read the graph horizontally from bottom to top and vertically from 

 right to left, obtaining generally four compositions from the graph. 



The graph is a zig-zag one and will be, without doubt, an important instrument of 

 research. 



PART I. SECTION 1. 



v 



Art. 5. Consider the permutations of the first n integers, and for simplicity 

 take n = 9. 



Writing down a permutation at random, 



31|4|5|92|76|8, 



it is clear that lines can be drawn separating the numbers into compartments in such 

 wise that in each compartment the numbers are in descending order of magnitude. 

 We can then write down a succession of numbers which describe the size of the 

 compartments, proceeding from left to right, and thus arrive at a composition 



211221 

 of the numl>er 9. 



I say that the permutation under examination has a descending specification 



(211221) or (21 2 2 2 1). 

 Similarly, from the ascending character 



^ 



3 | 1459 | 27 | 68 



of the same permutation, I say that the ascending specification is 



(1422) or (142 2 ), 



where it is to be noticed that 142 2 is the composition of 9 which is conjugate to 

 2l a 2 2 l, the composition which specifies the descending character. This is shown by 

 the zig-zag graph 



