33 321 3111 222 221I 2IIII See also:

• 33 321 3111 222 221I 2IIII IIIIII

IIIIII  , where no See also:

• PART

part is greater than 3; and so in See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

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 See also:

• ANALYTICAL

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, See also:

• WRITING
• WRITING (the verbal noun of " to write," O. Eng. writan, to inscribe)

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 See also:

• SERIES (a Latin word from serere, to join)

series x, x2, x3, for which a+3+—y+ =n, then we have in the development of the product a See also:

• TERM

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 See also:

• FUNCTION

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 See also:

• FACTOR (from Lat. facere, to make or do)

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 See also:

• LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

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 See also:

• LEAD
• LEAD (pronounced iced)

lead to theorems in the See also:

• PARTITION

partition of See also:

• NUMBERS, BOOK OF
• NUMBERS, 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 See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

formula shows that except for numbers of the See also:

• FORM (Lat. forma)

form 2(3n2tn) the number of partitions without repetitions into an See also:

• ODD (in middle English odde, from old Norwegian oddi, an angle of a triangle; the old Norwegian oddamann is used of the third man who gives a casting vote in a dispute)

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 See also:

• LEFT

left-See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

hand See also:

• SIDE
• SIDE (mod. Eski Adalia)

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 See also:

• CASE
• CASE, JOHN (d. 1600)

case, having obtained the partitions, we can See also:

• COUNT (Lat. comes, gen. comitis, Fr. comte, Ital. conte, Span. conde)

count them and so obtain the number of partitions . Another method is by L . F . A . See also:

• ARBOGAST (d. 394)

Arbogast's See also:

• RULE

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 See also:

• ANALYSIS (Gr. avci and Meta, to break up into parts)

ANALYSIS . (A .