If}*it~ 



PROCEEDINGS 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 



March 1, 1847. 



On the Partitions of Numbers, on Combinations, and on Permu- 

 tations. By Henry Warburton, M.P., F.R.S., F.G.S., Member of 

 the Senate of the University of London ; formerly of Trinity Col- 

 lege, A.M. 



The use made by Waring of the Partitions of numbers in develo- 

 ping the power of a polynome, induced the author to seek for some 

 general and ready method of determining in how many different 

 ways a given number can be resolved into a given number of parts. 

 On his communicating the method described in article 5 of Section 

 I. of this abstract, to Professor De Morgan, in the autumn of 1846, 

 that gentleman intimated a wish that the author would turn his 

 attention also to Combinations ; and such was the origin of the re- 

 searches which form the subject of the 2nd and 3rd sections. 



I. On the Partitions of Numbers. 



1. Let [N, p»~] denote how many different ways there are of re- 

 solving the integer N into p integral parts, none less than ij. Then 



CN,^] = [N± P a,^^] (i.) 



2. Such of the p-partitions of N as contain y as a part, and no 

 part less than t\, are obtained by resolving N— rj into p — 1 parts not 

 less than ij, and by adding y, as a pth part, to every such (p — 1)- 

 partition. That is, 



[N,^]-[N,p] = [N->j,p-l]. . . . (II.) 



3. In (II.), substitute tj + 1, ij + 2, &c. successively for rj. The 

 sum of the results is 



[N,^]-[N,^ + , +1 ]=Sj[N-,-^-l]. . (III.) 



Z ti 



In this expression, when = I*f _ \—rj, the term [N,p„_p_|_i] 

 vanishes, and the formula then becomes analogous to one published 



7n\ ~~ 



* 1 1 — l is employed to avoid the long phrase, "the integer nearest to 



UN 23 190 



N 

 and not exceeding — ." ^-r^^ 



P 



No. IV. — Proceedings of the Cambridge Phil. Soc 



