118 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



m 1, we obtain by subtraction the following corollary. The number of 

 partitions of n into m parts equals the number of partitions of n into parts 

 one of which is m and the others are ^ m. Sylvester credited the corollary 

 to N. M. Ferrers who communicated to him the following proof. Take 

 any set A composed of 3, 3, 2, 1, written as 



1, 1, 1 

 1, 1, 1 



1, 1 

 1. 



Reading it by columns, we get the set B composed of 4, 3, 2. Similarly, 

 every A in which the number of parts is 4 gives rise to a B in which 4 is a 

 part and every part is =i 4; conversely, every B produces an A. Euler's 

 theorem can be proved by the same diagram. Similarly, the number of 

 partitions of n into m or more parts equals the number of partitions of n 

 into parts the greatest of which is ^ m. If we partition each of i numbers 

 into parts so that the sum of the greatest parts shall not exceed (or be less 

 than) m, the number of ways this can be done is the same as the number of 

 ways these i numbers can be simultaneously partitioned so that the total 

 number of parts shall never exceed (or never be less than) m. 



P. Volpicelli 36 arranged the natural numbers n, n + 1, in a rectangle 

 with k + 1 rows, each with h + 1 numbers, but in reverse order in alternate 

 rows. For example, 



18 19 20 



23 22 21 



24 25 26. 



The successive sums by columns are 65, 66, 67 (of common difference unity) 

 and so always when the number of columns is odd; but, if k + 1 is even, 

 the sum of the numbers in each column is constant, being 



a = {2n + h(k + 1) + k}(k + l)/2, 



and we have special partitions of a. Given a, to find integral solutions 

 n, h, k, we note that h = y/8, where 5 = (k + I) 2 , while 7 and (2a) 2 /5 are 

 integers. Hence seek those divisors of (2a) 2 which are squares 5; for each 

 such 5, we have k and seek integers n for which 7/5 is an integer h. 



Volpicelli 37 expressed n k as a sum of numbers in arithmetical progression. 



* P. Bonialli 38 treated partitions. 



T. P. Kirkman 39 proved that the number of partitions of N into p parts 

 ^ a equals the sum of the number of partitions of N a, N p a, 

 N 2p a, - into p 1 parts =^ 0. The case a = 1 is the last formula 

 of Warburton. 32 He gave an analytic expression for the number (x, k) of 



Atti Accad. Pont. Nuovi Lincei, 6, 1852-3 (1855), 631; 10, 1856-7, 43-51, 122-131; Annali 



di sc. mat. e fis., 8, 1857, 22-27 



87 Atti Accad. Pont. Nuovi Lincei, 6, 1852-3, 104-119. Fregier. 220 

 "Formole algebriche esprimenti il numero delle partizioni di qualunque intero. Progr., 



Clusone, 1855. 

 39 Mem. Lit. Phil. Soc. Manchester, (2), 12, 1855, 129-145. 



