CHAP, ill] PARTITIONS. 145 



are of one kind, q of another kind, etc. The general problem of combinatory 

 analysis is to enumerate, under various imposed conditions, the distributions 

 of the n objects amongst the m parcels specified by 



(Pi?!---), Pi + ffi + " = m, 



when the arrangement of the objects in a parcel is immaterial, and when the 

 arrangement is material. The solution is effected by identities between 

 symmetric functions. To pass to the special case of partitions of n into m 

 parts, consider the distributions of n similar objects (n) into m similar 

 parcels (m), it being allowed to place more than one object in a parcel. In 

 the partitions of multipartite numbers, we distribute objects (pqr- ) into 

 parcels (m). 



G. Platner 142 found for r ^ 6 the number <f>(r, n) or \f/(r, n) of ways of 

 forming a sum n or a sum == n from r terms of 1, 2, 3, . For r = 2, 

 the result is q + x I or g 2 + (x l)g, respectively, if n = 2q + x, 

 x < 2. In the second paper, he expressed the results as functions of n. 

 For example, the number of pairs with the sum n is (n k)/2, k = 2 or 1 

 according as n is even or odd; the number of pairs with a sum ^ n is 

 (n 2 2n + l)/4, 1 = or 1 according as n is even or odd. For r = 3, 4, 5, 6 

 the formulas involve a parameter with listed values for the least positive 

 residues of n modulo 6, 12, 60, 60, respectively. It is proved that 



f(r, n + r) = f(r, n) + /(r - 1, n), f = <f> or $. 



[[All the results for < are due to De Morgan, 28 Herschel, 33 Kirkman, 39 etc. ; 

 while the results for \f/ follow readily from those for <.]] 



Schubert 143 noted that 10m Pfennige can be made up of 1, 2, 5 and 10 

 Pfennige coins in 1 + lOwi + 19w 2 + 10m 3 ways, if m = (7), and treated 

 two similar problems. 



G. Chrystal 144 collected theorems on partitions and introduced various 

 notations. 



Bellens and Verniory 145 found the number of sets of solutions of 

 x-\-y-\-z = n-{-2, x, y, z chosen from 1, , n, by grouping the solutions 

 corresponding to a fixed x, and separating the cases n = 0, , 5 (mod 6). 



M. F. Daniels 146 obtained the results of Weihrauch 74 another way. 



P. A. MacMahon 147 enumerated the perfect and sub-perfect partitions. 

 For example, if a is a prime, there are 2"" 1 perfect partitions of a a 1. If 

 a, b, are primes, a a b ff ... 1 has as many perfect partitions as the multi- 

 partite number (a, (3, ) possesses compositions (partitions with attention 

 to order). 



S. Tebay 147a found the number of ways s is a sum of i distinct integers, 

 also when each part is =i q. 



142 Rendiconti R. 1st. Lombardo di Sc. Let., (2), 21, 1888, 690-5, 702-8. 



143 Mitt. Math. Gesell. Hamburg, 1, 1889, 269 Cf. d'Ocagne 224 of Ch. II. 



144 Algebra, 2, 1889, 527-537; ed. 2, vol. 2, 1900, 555-565. 



145 Mathesis, 9, 1889, 125-7. 



146 Lineaire Congruences, Diss., Amsterdam, 1890, 120-135. 



147 Messenger Math., 20, 1891, 103-119. Cf. MacMahon. 136 

 147a Math. Quest. Educ. Times, 56, 1892, 34-37. 



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