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



gives every composition of the number 



n-2c*. 

 Hence, since N (1) = 1, 



Art. 17. By varying the order of the factors, on the sinister of the multiplication 

 formula, a variety of interesting results present themselves ; thus 



Ul 



where after a m , on the dexter, occurs every composition of 



ntap; 



and the portion 



...a 



includes every composition of 



which terminates with a number not less than a^ 

 Hence, for such a summation, 



a formula which is independent of p. 

 Art. 18. In particular from 



{N(l)}- 2a - 1 N(a 1 2 ... m l) 

 we obtain 



SN (...a\a 2 ...a m l) = ^ JS(a,a,...a m l) ; 



wherein the summation is for every composition of 



n a a ... a m 1 



which terminates with a number not less than c^. 

 E.g., for n = G, , = 1, a, = 1, 



N (41 2 ) + N (131 2 )+N (2 2 1 2 ) + N (1 8 21 3 ) = |JN (211). 



10 + 26 + 35 + 19 = 6.5.3 

 Art. 19. As another example of the power of the theorem, let 



(the numbers a,, o 8 ...a OTj , b ly h a ...b mi being given) denote a summation in regard to 

 compositions of _^ _^, 



