33 321 3111 222 221I 2IIII

IIIIII  , where no part is greater than 3; and so in general we have the theorem, the number of partitions of n into not more than k parts is equal to the number of partitions of n with no part greater than k . We have for the number of partitions an analytical theory depending on generating functions; thus for the partitions of a number n with the parts 1, 2, 3, 4, 5, &c., without repetitions, writing down the product I +x. r +x2 . I +x3 . I +x4 ... , = I +x+x2 -2X3 ... +Nx^+ ... , it is clear that, if xa, x13, xY, . are terms of the series x, x2, x3, for which a+3+—y+ =n, then we have in the development of the product a term x", and hence that in the term Nx" of the product the coefficient N is equal to the number of partitions of n with the parts 1, 2, 3, ... , without repetitions; or say that the product is the generating function (G . F.) for the number of such partitions . And so in other cases we obtain a generating function . Thus for the function 1 I — x I—x2 . -x3 .

. . =+x+2x2+.,+Nx"+,,,, observing that any

factor 1/r —x1 is=l+x1+x21+ ... , we see that in the term Nx" the coefficient is equal to the number of partitions of n, with the parts 1, 2, 3, . . , with repetitions . Introducing another letter z, and considering the function I+xz.I+x2z.I+x3z . . .,=l+z(x+x2+ . .)...+Nx"zk+, we see that in the term Nx"zk of the development the coefficient N is equal to the number of partitions of n into k parts, with the parts 1, 2, 3, 4, , without repetitions . And similarly, considering the function - xZ . I - I x3Z . .' =I+z(x+x2+..)...+Nx"zk+ .. Ix22.- we see that in the term Nx"zk of the development the coefficient N is equal to the number of partitions of n into k parts, with the parts I, 2, 3, 4, , with repetitions . We have such analytical formulae as I-xZ.I-x22.I-x3Z .

I +I — , x+ r—x I-x2+" which

lead to theorems in the partition of numbers . A remarkable theorem is I -x . I -x2 . I -x3 . I -x4 . = I -x-x2+x5+x7-x12-x15+ ... , where the only terms are those with an exponent z (3n2 tn), and for each such pair of terms the coefficient is The formula shows that except for numbers of the form 2(3n2tn) the number of partitions without repetitions into an odd number of parts is equal to the number of partitions without repetitions into an even number of parts, whereas for the excepted numbers these numbers differ by unity . Thus for the number II, which is not an excepted number, the two sets of partitions are II, 821, 731, 641, 632, 542 10.1,92, 83, 74, 65, 5321, in each set 6 . We have r—x.1+x.I+x2.I+x4.1+x2 ... =1; or, as this may be written, 1+x . I +x2 . I +x4 .

I +x8 ... = I I x, = I +x+x2-{-x3+ ... , showing that a number n can always be made up, and in one way only, with the parts 1, 2, 4, 8, . The product on the

left-hand side may be taken to k terms only, thus if k=4, we have 18 I +x.I +x2 . I +x4.I +.',C8,= _ 1I x '=i+x+x2...+x15, number or of the successive numbers 1, 2, 3, &c . And of course in any case, having obtained the partitions, we can count them and so obtain the number of partitions . Another method is by L . F . A . Arbogast's rule of the last and the last but one; in fact, taking the value of a to be unity, and, understanding this letter in each term, the rule gives b; c, b1 d, bc, b3; e, bd, c2, b2c, b4, &c., which, if b, c, d, e, &c., denote I, 2, 3, 4, &c., respectively, are the partitions of 1, 2, 3, 4, &c., respectively . An important notion is that of conjugate partitions . zx 22x2 11 that is, any number from t to 15 eau be made up, and in one way only, with the parts 1, 2, 4, 8; and similarly any number from 1 to 2k—I can be made up, and in one way only, with the parts 1, 2, 4, ..

2k-1 . A like formula is I—x3 1—x9 1—x27 1—x31 1—x81 x.I— x x3.I—x3 x9 . I—x9 x27.I—x27 x40.I—x' that is, x'+I+x.x 3+I+x3.x-9+I+x9.x'7+I+x27 =x-40+x-39 . +I+x ... +x39+x90, showing that any number from -40 to +40 can be made up, and that in one way only, with the parts 1, 3, 9, 27 taken positively or negatively; and so in general any number from -s(3k-I) to +(3k-1) can be made up. and that in one way only, with the parts t, 9, . . 3k-' taken positively or negatively . See further COMBINATORIAL

ANALYSIS . (A .