CHAP, in] PARTITIONS. 103

Hence a = 8, (3 = n®, 7 = 7i3S, 5 = n6£), • • •, where 1, 3, 6, • - • are the successive triangular numbers. From the above series for a, ft • • -, we see that

» w

2 J* ' *

Hence m(M) is also the number of ways m — n can be obtained by addition from 1, ••-,/*. The former recursion formula for mt-(/i) gives

(2) m<"> = (w - M)W + (m - l)^. He stated, as a fact he could not prove,

(3) p(x} = f[ (1 - s*) = 1 - x - x* + x5 + x7 ---- + ( - 1) nz(3n2± n)/2 + - • • ,

and that the reciprocal of the product is 1 + x + 2z2 + 3z3 + 5#4 + • • • , the coefficient of x3 being the number of ways s can be partitioned into equal or distinct parts. As to (3), see Euler1-6, Ch. X, Vol. 1.

Euler,4 in a letter to N. Bernoulli, Nov. 10, 1742, stated the preceding facts on partitions. The answer to the second problem he stated in the following equivalent form: m(M) is the coefficient of nm in the expansion of

Euler5 gave (3) and p(x) = 1 - Pt + P2 - P3 + • • • [see Euler9].

P. R. Boscovich6 gave a method of finding all the partitions of a given number n into integral parts > 0. Write down n units in a line. Replace the last two units by 2, then replace two units by 2, etc. Next, write n — 3 units and 3; replace two units by 2, etc. Then write n — 6 units and two 3's; replace two units by 2, etc. Thus the partitions of 5 are

11111, 1112, 122, 113, 23, 14, 5.

He applied partitions to find any power of a series in x, also in a paper, ibid., 1748. In his third paper, ibid.j 1748, he showed how to list the partitions of n into parts ^ m, by stopping his above process just before a part m+ 1 would be introduced. He applied the rule also to the case when the parts are any assigned numbers. He treated the problem to find ail the ways in which a given integer n can be decomposed in an assigned number m of parts, equal or distinct; but the solution by Hindenburg16 is much more simple and direct. Boscovich attempted in vain to find a formula for the number of partitions. He gave elsewhere7 his rule.

K. F. Hindenburg8 would obtain the partitions of 8 by annexing unity to those of 7, and supplement them with

_ 2222, 224, 233, 26, 35, 44, 8. _

4 Opera postuma, 1, 1862; Corresp. Math. Phys. (ed., Fuss), 2, 1843, 691-700.

5 Letter to d'Alembert, Dec. 30, 1747; Bull. Bibl. Storia Sc. Mat., 19, 1886, 143.

6 Giornale de' Letterati, Rome, 1747. Extract by Trudi,98 pp. 8-10.

7 Archiv der Math, (ed., Hindenburg), 4, 1747, 402.

8 Ibid., 392, and Erste Samml. Combinatorisch-Analyt. Abhand., 1796, 183. Quoted from

G. S. Kliigel's Math. Worterbuch, 1, 1803, 456-60 (508-11, for references).