MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBEBB. 91 



Art. 35. The method of calculation establishes that the number N (...) is unaltered 

 by reversal of the order of the numl>ers in the bracket. 



Also that the results are only dependent upon the magnitudes of thr parts in the 

 specification of the assemblage and not ujxm the order of their occurrence. 



General Investigation of a Generating Function. 

 Art. 36. I have shown above that, for numbers specified by 



(K'K 1 -), 



an auxiliary generating function is 



W-. 



for, from its expansion in terms of monomial symmetric functions, the numbers 



can be succeasively calculated. 



For present convenience I take the above generating function to be 



and recall that N(a6p ... ) + N(a+6> f> ...) + N (a, 6+c, ...)+... 

 is equal to the coefficient of symmetric function 



(abc...) 

 in the expansion of 



The above linear function of the numbers 



N(...) . 

 is formed by adding adjacent numbers 



0, 1, 2, 3, ..., k at a time, 



where the numbers a, b, c, ... are k in number. 

 It thus comprises 2*" 1 terms in general 

 Art. 37. Let this linear function be denoted by 



so that if we write = w(abe...).(abc...) t 



From this system of linear relations is determined the set 

 N (a) = C (a), where a = n, 

 N (ah] = C(nb)-C(a+b), where a + b = n, 

 N (abc) = C(o&r) C(a + 6, c) C(a, 6 + c) + C(a + 6+c), where a + 6+c = n, and soon; 



N 



