118 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 



It may be verified, for example, that the complete coefficient of 







(1 + 20X+48X 2 +20X 3 +X 4 ), 



which agrees with a previous result. 



From a previous result also the coefficient of 





fm + p 1\ fm+ql\ /m + r 1\ 

 ( f )( I )( r ) 



/m+p-2\ /m + q-2\ im+r-2\ 

 \ 1 A P )( <! )( r ) 



+ 1\ /m+p-3\ im+q-3\ /m-f>-3\ 

 2 A P )( 9 )( r ) 



where n = p + q + r. 



Art. 81. Observe that the generating function is a symmetric function of a, yS, y, 

 verifying a previous conclusion that an N m number is not altered by any interchange 

 of the letters a, ft, y. 



When the numbers p, q, r are equal, that is when the objects are specified by the 

 partition ,,. 



we can establish a symmetrical property of the numbers N. 



For coefficient 



X"- l (a.p y y in ( 

 is, by writing 



for X and Xa, Xy6, Xy for a, ft, y, 

 X 



equal to coefficient of 



X 3p - ra+l (y) p in (X 



equal to coefficient of 



^-"^(oLfty)" in 



equal to coefficient of 



in (a + X/3 + Xy)* (a 



Art. 82. Hence M _ M 



N m JNjjp-m+S 



and the numbers N range from > T xr 



INi to lN2p+i, 



showing that 2p+l is the maximum number of parts in the descending specification, 

 when the objects are specified by the partition 



