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



The result is convenient for the calculation of the numbers N mjl p+i from the 

 numbers N mil p. 



We have also the remarkable result that the probability of obtaining a permutation, 

 such that the sum of the first m numbers of the descending specification is p, is 

 independent of n, and has the value 



(jH-l)I m ' l ' 

 whenever p is n I or less. 



Art. 31. From the definition we have in respect of the permutations of n numbers 



N 1 +N 2 +N 3 +... = n! 

 I shall now show that 



2 

 nl 6\ 



the summation being for all values of 



"i, "a, 



such that 



zv = 0, 



'Ssv, = n. 



The theorem is the outcome of the multiplication theorem of Art. 12. 

 Observing that, for all values of s, 



N(l) = 1, 

 we have 



!') N ( 1") N 



*l I f< 2 '. #3 ! 



+ N 

 and generally, for the product 



since sv, = n, 



= a linear function of numbers N 



(i !)" (2 ;-'... ( 



We may write down a similar result for every permutation of the factors of 



and, by addition, obtain 

 n! 



= linear function of numbers N, 



where Si> = 6. 



