CHAP. Ill] PARTITIONS. 157 



which for k = 1 becomes 



Rm ' n = n\ (m - l 



The number R m , n of ways of separating m like objects into n groups is the 

 number of partitions of m into n parts > 0. Let k = m n. Then 



k,} (k ^ ri), R m , = R k ,j (k < H) . 



j=l J=l 



There are as many partitions of m as partitions of 2m into m parts. Recur- 

 sion formulas are found for the number N of ways of separating into n 

 groups m = I + ai + + ah objects 192 " of which I are distinct, but one 

 is repeated i times, and the last a h times. Thus if the objects are a, a, a, 

 b, b, c, d, then I = 4, a l = 2, 2 = 1. There are N factorizations into n 

 positive integral factors of a number which is a product of m primes not 

 necessarily distinct. 



Minetola 193 proved by use of (2n + l)(2n' + 1) = 2k + 1, etc., that if 

 2k + 1 is decomposed into a product of h primes, the h 1 equations 



Inn' + 2n = k, 2 2 ninjni' + 2Zftini + Sni = k, 



admit R h ,2, Rh,s, sets of positive integral solutions, respectively. 



P. A. MacMahon 194 used the example of a permutation 3, 1 | 4 | 5, 2 of 

 the first five integers separated into compartments with the numbers in each 

 arranged in descending order; the succession of numbers 2, 1, 2 giving 

 the size of the compartments is a composition of 5. He found the number 

 N(a, b, - ) of permutations of 1, -, n having as the descending specification 

 (corresponding to 2, 1, 2 in the example) a given composition (a, b, ) 

 of n. He proved that 



( 



\ 



N(at a s )N(a s+l a s+t } 



t*l -f- T" ttg 



- - a s -i, a s + a s+i , a s+2 , -, a s+t ) 



and similar formulas. He found the number of permutations of 1, -, n 

 whose descending specifications contain a given number of integers. He 

 treated the analogous problems for permutations of numbers not all dif- 

 ferent, and problems on packs of cards. The number of permutations of 

 eft af with descending specifications of m parts is the coefficient of 

 \m-i a pi . . . oft m tj^ reciprocal of 



1 - Zai + (1 - X)2 ai a 2 - (1 - X 2 )2a ia .a 3 + -. 

 His 154 study of this generating function is continued here. 



192a Giornale di Mat., 47, 1909, 43-54, for the number of combinations of these m objects n 

 at a time. 



193 Ibid., 47, 1909, 305-320. 



194 Phil. Trans. Roy. Soc. London, 207, A, 1908, 65-134. Abstract, Proc. Roy. Soc., 78, 



1907, 459-60. 



