MAJOR P. A. MA.MAHON ON THE COMPOSITIONS OF NUMBERS. 119 



Art. 83. In general, when there are / different letters, 



the number of permutations of 



which have descending specifications containing m parts, is the coefficient of 



in the development of 



or of 



This is the general generating function of the numbers 



N.. 



Art. 84. Since it is symmetrical in regard to 



a,, a,, ..., a t , 



the value of N w is not affected by permutation of the letters 



i, s , ... , a t . 



Art. 85. It can be shown also, as in the simpler case, that when 



/>i = P*= = /'* = />. 



the coefficient of \- 



A 



is equal to the coefficient of 



-- 



so that -T _ -j 



S* m i>(*-i 



the numbers N range from M ,., 



IS, tO JN^-i 



and (k\)p+l is the maximum number of parts in a descending specification. 



SECTION 10. 

 Art. 86. The generating function 



now presents itself for examination. 

 Introducing the elementary functions 



a,, a,, 



