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Encyclopedia Britannica

IIIIII, 2IIII, 221I, 222, 3111, 321, 33

This article appears in Volume V19, Page 866 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: I27-INV

IIIIII, 2IIII, 221I, 222, 3111, 321, 33 ,
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

SERIES (a Latin word from serere, to join)

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

term

TERM

x", and hence that in the

term

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

FUNCTION

(G. F.) for the number of such partitions. And so in other cases we obtain a generating

function

FUNCTION

.
Thus for the function
1
I x Ix2. -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

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)

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+ rx I-x2+"
which lead to theorems in the

partition

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
rx.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

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

them and so obtain the number of partitions.
Another method is by L. F. A. Arbogast's

rule

RULE

of the last and the last but one; in fact, taking the value of a to be unity, and, understanding this

letter

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)

in each term, the

rule

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 2kI can be made up, and in one way only, with the parts 1, 2, 4, .. 2k-1. A like

formula

is
Ix3 1x9 1x27 1x31 1x81
x.I x x3.Ix3 x9. Ix9 x27.Ix27 x40.Ix' 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

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

. (A. CA.)

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Jesus Christ Saves Ministries, P.O. Box 70696, Pasadena, CA 91117JCSM is a 501(c)(3), non-profit organization. Copyright � 1997-present.

Jesus Christ Saves Ministries, P.O. Box 70696, Pasadena, CA 91117
JCSM is a 501(c)(3), non-profit organization. Copyright � 1997-present.