CHAP, ill] PABTITIONS. 143

E. Meissel135 gave the formulas of Weihrauch74 for n = 3, 4, 5 and noted that a synthesis of these cases gives

provided the final term of the derivative be omitted.

P. A. MacMahon136 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- • •) of u] then

u + 1 - (I + l)(m + 1). - -, u + 1 = (1+ 1)X, X = (w +!)/£, • • ••

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

instead of <£, and divisors of 2u + 1 instead of those of u + 1. E. Catalan137 noted that

log (1 + x + x* + • • •) = - log (1 - *) = x + £ + ~ +

Developing each exponential, we get Jacobi's result (Jour, fur Math., 22, 1841, 372-4)

- T(a + l)r(6 + l)r(c + !)••• where the summation extends over all solutions ^ 0 of a + 2b + 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, 0, 7, • • • belonging to progressions with the differences 1, 2, 3, • • • , the sum of the fractions lf(apy • • •) is unity.

W. J. C. Sharp stated and H. W. Lloyd Tanner137* proved that, if P« or Qn be the number of partitions of n without or with repetitions, then

Qn = Pn + P«~2Qi + P«-4Q2 H ---- ,

184 tlber die Anzahl der Darstellungen einer gegebenen Zahl A durch die Form A —

in welcher die p gegebene, unter sich verschiedene Primzahlen sind, Progr. Kiel, 1886. His /n-i has been changed to /» to conform to Weihrauch'a notation.

1M Quar. Jour. Math., 21, 1886, 367-373.

l« Mem. Soc. Roy. Sc. de Liege, (2), 13, 1886, 314-8 <- Melanges Math. II).

"7a Math. Quest. Educ. Times, 45, 1886, 123.