CHAP, ill] PARTITIONS. 161 



that n = HI 1 + u* 3 + u 3 5 + is solvable in integers such that 

 4 i= Ui ^ u z = u s ^ 0. A generalization is noted. 



S. Minetola 211 investigated the number R(t; i, , a p ; n) of ways of 

 separating into n groups m = t + ai + + a p objects of which t are not 

 repeated, while p further objects, distinct from each other and from the 

 preceding t, are repeated i, , a p times, respectively. After finding 

 recursion formulas for R, he proved theorems on the maximum value of R 

 when m and n vary, but so that m n remains constant. Finally, he 

 studied R(l', m; ri), so that one object is taken single and another is 

 repeated m times. It is the coefficient of x m+1 in 



x n t 1 (1 - z) 2 (l - x 2 ) (1 - z 3 ) . (1 - re"- 1 ) } . 

 Its recursion formula is 



R(l- } m; ri) = R(l; m 1; n 1) + R(l; m n + 1; ri). 



G. Scorza 212 evaluated sums of reciprocals of products, each summation 

 extended over all the partitions of a given arbitrary integer. 



G. Candido 213 noted that a m is a sum of a consecutive odd integers. 

 For m = 3, this was also proved by J. W. N. le Heux. 214 Cf. Fregier. 22 " 



G. Csorba 215 stated that all questions concerning partitions can be 

 reduced to a single one, viz., the question of the number of ways A can be 

 obtained from 0,1, , a n by addition, repetitions allowed. Cayley 44 

 had expressed this number of partitions of A in the form 



c (A) + Aci(A) + A 2 c 2 (A) + - + A-^Cn-iCA), 



where d(A) is a periodic function of A; but essentially proved only the 

 existence of such a representation. Csorba gave for d(A) an explicit 

 formula involving Bernoullian numbers and the g.c.d. d of all the a's except 

 0*11 i a i m > an d involving summations extended over all solutions 4 of 

 the congruence 2'^i'a^^ = A (mod d). 



*Csorba 216 treated multiple partition. 



P. A. MacMahon 217 has given an extended account of the theory of 

 partitions as a branch of combinatory analysis. A small part of Vol. I and 

 nearly the whole of Vol. II are taken up with theories more or less connected 

 with the partitions of numbers. The theory is investigated from the 

 standpoint of a new definition of- a partition. A partition is defined 

 as a set of positive integers on, a z , , <x n , whose sum is n, such that 

 OLI ^ 2 = = ot n - The importation of linear Diophantine inequalities 

 leads to a syzygetic theory and thence to the determination of ground 

 forms connected by various orders of syzygies as in the theory of algebraic 

 invariants. A generalization is made by considering one or more general 



211 Periodico di Mat., 29, 1913, 67-82. 



212 Rendiconti Circolo Mat. Palermo, 36, 1913, 163-170. 



213 Suppl. al Periodico di Mat., 17, 1914, 116-7. 



214 Wiskundig Tijdschrift, 12, 1915-6, 97-8. 



215 Math. Annalen, 75, 1914, 545-568. 



216 Math, es termes e>tesito (Hungarian Acad. Sc.), 32, 1914, 565-601. 



217 Combinatory Analysis, Cambridge, I, 1915; II, 1916. 



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