MAJOR P. A. MA.MMIOX >N THK COMPOSITIONS OF NUMHKl;- ; 



Art. 6. We can now formulate the question : Ot the permutations of the first n 

 numbers, how many have a descending specification denoted hy a given composition 

 of the number n ? Whatever the answer, it is clear that the same answer must, in 

 general, be given for three other compositions, viz., the three others associated with 

 the zig-zag graph. In fact, from 



314592768 of specification 211221, 

 we derive 867295413 2241; 



and from these two by changing the number m into nm+ 1, 



796518342 of specification 1422, 



243815679 122112 



and so forth. 



In two cases there are two associated compositions instead of four, viz. : 



(i) When the composition reads the same as its inverse (that is the same from 



left to right as from right to left), 

 (ii) When the conjugate and the inverse are identical, as in 221, whose 



conjugate is 122. 

 *The numlier of self-inverse compositions of an even numl>er 2m and of an uneven 



number 2w+l is 



& . 



The numl>er of inverse-conjugate compositions of an uneven numlier 2w-t 1 is 



2". 



Hence, in the present theory, the number of different numbers that appear in the 

 case of an even number 2m is, since the whole number of coiujxisitions is 2*""', 



.2" + i(2* '-2-), 

 = 2"-*(2"- I + 1); 

 and, in the case of an uneven number 2i+l, 



viz., it is 2-*+2 1/8( "~ 4> 



according as n is even or uneven. 



* See " Memoir on the Theory of the Compositions of Nurnlxjrs," ' Phil. Trans. Roy. Soc.,' 1893, 



