MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 107 



Hence 



E.g., D 4 M=< 



where the compositions of 4 have been taken in the order 



40, 31, 13, 22. 



In general T) / / / ?/> h I 



where / <> 



is a composition of a into a or fewer parts. 



It is to be noted that in forming the compositions zeros are parts, so that, for 



iastance ' 400, 040, 004 



count as different compositions. 

 If the operand be 



since i\ / \ i 



*J a (ct fl ) = unless a 1, 



we need only attend to the compositions composed of units and zeros. 

 Thus 



It is easy to show that 

 from which 



and, particularly, 



N(4^ )1 = 



7 4+2+1 



from the table. 



Similar formulae can be established at pleasure. 



The Conjugate Law. 



Art. 65. It has been seen (Art. G) that, when the numbers permuted are specified by 



I", 



where 



(pq...), 



denote conjugate compositions. 

 We write the theorem 



N (^---)a-) 



and we may inquire into the existence of an analogous theorem when the numbers 

 permuted have any other specification. 



P 2 



