CHAP, ill] PARTITIONS. 143 



E. Meissel 135 gave the formulas of Weihrauch 74 for n = 3, 4, 5 and noted 

 that a synthesis of these cases gives 



p , ff . 1 df n+l (pP + m) 

 f n (pP + m) - f n (m) 



provided the final term of the derivative be omitted. 



P. A. MacMahon 136 called a partition perfect if it contains one and only 

 one partition of every lower integer; sub-perfect if, when each part is 

 taken positive or negative (but not both), it is possible to compose every 

 lower number in only one way. Thus, 3 + 1 is sub-perfect since 2 = 3 1, 

 3 = 3, 4 = 3 + 1. Any factorization 



leads to the perfect partition (XV 1 - ) of u; then 



u + 1 = (1 + l)(m + !)-, w+l = (Z + l)X, X = (m 



Formulas involving the number of partitions of u are given. For sub- 

 perfect partitions, use 



&P, i = x ~ pq ~ or (p ~ 1)9 + + yr q + 1 + & + 



instead of </>, and divisors of 2u + 1 instead of those of u + 1. 

 E. Catalan 137 noted that 



log (1 + x + x z + ) = - log (1 - a;) = x + - + + 



Developing each exponential, we get Jacobi's result (Jour, fur Math., 22, 

 1841, 372-4) 



d r(o + i)r(6 + i)r(c + !) 



where the summation extends over all solutions ^ of 



a + 26 4- 3c + = n. 



Since the denominator equals 1 2 a 2 4 6 26 3 6 3c , we see 

 that if n is partitioned in all ways into parts a, j8, 7, belonging to pro- 

 gressions with the differences 1, 2, 3, , the sum of the fractions l/(a/?7 ) 

 is unity. 



W. J. C. Sharp stated and H. W. Lloyd Tanner 1370 proved that, if P n 

 or Q n be the number of partitions of n without or with repetitions, then 



Q n = P n + Pn-*Ql + P n-lQ* + '", _ 



136 Uber die Anzahl der Darstellungen einer gegebenen Zahl A durch die Form A = 2p n i n , 

 in welcher die p gegebene, unter sich verschiedene Primzahlen sind, Progr. Kiel, 1886. 

 His /_! has been changed to / to conform to Weihrauch's notation. 



136 Quar. Jour. Math., 21, 1886, 367-373. 



137 Mem. Soc. Roy. Sc. de Liege, (2), 13, 1886, 314-8 (= Melanges Math. II). 

 137a Math. Quest. Educ. Times, 45, 1886, 123. 



