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EA(psr...), (p, „...)• (pqr...)

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Originally appearing in Volume V06, Page 754 of the 1911 Encyclopedia Britannica.

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EA(psr...), (p, „...)• (pqr...) =hP1h41hr1.... This theorem is of algebraic importance; for consider the See also:

SIMPLE

simple particular See also:

case of the See also:

DISTRIBUTION (Lat. distribuere, to deal out)

distribution of See also:

OBJECTS

objects (43) into parcels (52), and represent objects and parcels by small and See also:

capital letters respectively. One distribution is shown by the See also:

SCHEME (Lat. schema, Gr. oxfjya, figure, form, from the root axe, seen in exeiv, to have, hold, to be of such shape, form, &c.)

scheme AAAAABB aaaabbb wherein an See also:

OBJECT

object denoted by a small 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 is placed in a See also:

PARCEL (Fr. parcelle, Ital. particella, Lat. particela, diminutive of pars, part)

parcel denoted by the capital letter immediately above it. We may interchange small and capital letters and derive from it a distribution of objects (52) into parcels (43); viz.: AAAABBB aaaaabb' The See also:

PROCESS

process is clearly of See also:

general application, and establishes a one-to-one See also:

correspondence between the distribution of objects (pqr ... ) into parcels (pigir1 . . .) and the distribution of objects (pigiri . ) into parcels (pqr . ). It is in fact, in Case I., an intuitive observation that we may either consider an object placed in or attached to a parcel, or a parcel placed in or attached to an object. Analytically we have Theorem.—” The coefficient of symmetric See also:

FUNCTION

function (pqr .) in the development of the product h51h91hr1... is equal to the coefficient of symmetric function (plgirl ..) in the development of the product hphghr .... The problem of Case I. may be considered when the distributions are subject to various restrictions. If the restriction be to the effect that an aggregate of similar parcels is not to contain more than one object of a See also:

KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind, we have clearly to See also:

DEAL

deal with the elementary symmetric functions al, See also:

A2(z)

a2, a3, .

. or (1), (12), (13), .. in lieu of the quantities hi, h2, ha, ... The distribution function has then the value apjasiar1... or (1si) (14,) (1'1)..., and by interchange of object and parcel we arrive at the well-known theorem of symmetry in symmetric functions, which states that the coefficient of symmetric function (pqr . . .) in the development of the product a 1a51a,1... in a See also:

SERIES (a Latin word from serere, to join)

series of monomial symmetric functions, is equal to the coefficient of the function (pigiri .) in the similar development of the product ayagar... . The general result of Case I. may be further analysed with important consequences. Write Xi = (1)xi, X2=(2)x2+(12)x,, X3 = (3)x3+(21)x2x1+(13)x and generally X, =E(7iµy ...)xxxµxy ... the summation being in regard to every See also:

PARTITION

partition of s. Consider the result of the multiplication X3,X4,Xr, ... = EPx 31xa2x sae. To determine the nature of the symmetric f See also:

UNCTION (Lat. unctio, anointing, ungere, unguere, to smear with ointment, to anoint; cf. " ointment," O.Fr. oignement, from oigner, mod. oindre, to anoint)

unction P a few See also:

DEFINITIONS

definitions are necessary. See also:

DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

Definition I.—Of a number n take any partition (X3X2X3 . . . X3) and See also:

SEPARATE

separate it into component partitions thus: (X1T2) (A3x4x5) (A6).•• in any manner.

This may be termed a separation of the partition, the See also:

numbers occurring in the separation being identical with those which occur in the partition. In the theory of symmetric functions the separation denotes the product of symmetric functions F.See also:

AXLE (in Mid. Eng. axel-tre, from O. Norweg. oxull-tre, cognate with the O. Eng. exe or eaxe, and connected with Sansk. dksha, Gr. a wv, and Lat. axis)

aXle-EaX20x'ya5eaA... The portions (X1X2), (X3X4X5), (A6), . are termed separates, and if X1+X2 =p1, X3+X4+X5 =q1, A6 =r1... be in descending See also:

order of magni- tude, the usual arrangement, the separation is said to have a See also:

SPECIES

species denoted by the partition (See also:

PLAIN (O. Fr. plain, from Lat. plenum)

plain. . .) of the number n. Definition II.—If in any distribution of n objects into n parcels (one object in each parcel), we write down a number E, whenever we observe E similar objects in similar parcels we will obtain a See also:

SUCCESSION (Lat. successio, from succedere, to follow after)

succession of numbers y Si, ea, ea, ... , where (e1, Es, E3...) is some partition of n. The distribution is then said to have a See also:

SPECIFICATION (from Med. Lat. specificatio, specificare, to enumerate or mention in detail)

specification denoted by the partition (EIE2E3... Now it is clear that P consists of an aggregate of terms, each of which, to a numerical See also:

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

factor pres, is a separation of the partition (sQis2s°33...) of species 1 2 (Agin...). Further, P is the distribution function of objects into parcels denoted by (plgirl...) subject to the restriction that the distributions have each of them the specification The distribution function. Case I. Case H. (p1 2 3 "lp"2p"a) a partition of n.

Of the whole number of distributions of the n objects, there will be a certain number such that nl parcels each contain pi objects, and in general Ira parcels each contain pa objects, where s = I, 2, 3, .. Consider the product h"1h r2h"3... which can be permuted in P1 P2 P3 m! ways. For each of these ways h'1 h"2h"a... will be a dis- 7rl! 7r2! 7r3!••• P1 P2 P3 tribution function for distributions of the specified type. Hence, regarding all the permutations, the distribution function is m! h"1h"2h"a 7r1! See also:

R2(rt-s2)

r2. 7r3!... P1 P2 P3 ••• and regarding, as well, all the partitions of n into exactly m parts, the desired distribution function is m, h lh 2h a 7r1! 7r2! ital... PI P2 P3"' {171'=m, 17rp=nj, that is, it is the coefficient of xn in (hlx+hzx2+hax3+...)m. The value of A(p"1p"2pa„.)(1m) is the coefficient of(pi1pz2p;3...)xn in 1 2 3 the development of the above expression, and is easily shown to have the value /pl+1n—1\ "l (p2+m—1)'r2 /p3+m—11 "3... Pi P2 Pa (m\ (pl+m -2\ "1 (p2+m—21 "2 (ps+m—21 "a \ 1 J l 1 ` P2 / '. p3 I (nz) (pl+m-3\ "1 (Ps +m-3) "2 (pa+m-31 "a + 2 p1 I \ P2 I ` Pa / -...to m terms.I Observe that when p1= Ps =Ps = ...

=7r1=7r2 =nos. . . = i this expression reduces to the mth divided See also:

DIFFERENCES, CALCULUS OF (Theory of Finite Differences)

differences of on. The expression gives the compositions of the multipartite number pi1pa 2pa 8 ... into m parts. Summing the distribution function from m=I to m=oo and putting x=1, as we may without detriment, we find that the totality of the compositions is given by 1 hlhh3hh h3 . which may be given the See also:

FORM (Lat. forma)

form a1-~+aa "' Adding 1 we bring 1-2(al -azd-a3-...). this to the still more convenient form 1 11-2(al-az+as-...). Let F(pilp22p33...)denote the See also:

TOTAL

total number of compositions of the multipartite pllp;2pe3.... Then 11 120.=1+EF(p)al', and thence F(p) =2P-1. Again 11_ +1 =1+EF(p1p2)aPl13m,andexpanding the See also:

LEFT

left-See also:

hand See also:

side we easily find F(pip2) =2P1+P2—1(P1+P2) !—2PI+P2-2 (A1+p2—1)! 0!p1!p3! 1!(p1-1)!(p2-1)! +2P1+P2-32!03-2 !-a)!2)I-....

We have found that the number of compositions of the multi-partite plpsp3...p. is equal to the coefficient of symmetric function (p1p2p3...p,) or of the single See also:

TERM

term aP1a22a83...aP' in the development according to ascending See also:

POWERS, HIRAM (1805-1873)

powers of the algebraic fraction 1 1 1 — 2 (Mal - Zala2 +f ala2a3 — ... + (—) a+l0.lasa3... aa. This result can be thrown into another suggestive form, for it can be proved that this portion of the See also:

EXPANDED

expanded fraction {1—il(2al+as+...+aa)} 11—is(Ya1+See also:

Las+••.+aa)}...{1—ta(Gal+2a2+...+2as) which is composed entirely of powers of t1a', t2a2, t3a3, ...taaa has the expression 3. 2 1—2(2,11.1—Itlisalas+2611st3al.See also:

ana .. +(—)a+ltlts...tealas...aa), and therefore the coefficient of a 2a21 .aP' in the latter fraction, when tl, t2, &c., are put equal to unity, is equal to the coefficient of the same term in the product 1 (gal+a3+...+aa)Pl(2a1+2a2+...+as)P2...(2a1+2a2+...+2aa)Pe. This result gives a See also:

DIRECT

direct connexion between the number of compositions and the permutations of the letters in the product ai'1a2P2...See also:

APE (Old Eng. ape; Dutch aap; Old Ger. a ffo; Welsh epa; Old Bohemian op; a word of uncertain origin, possibly an imitation of the animal's chatter)

aPe. Selecting any permutation, suppose that the letter a,. occurs q,. times in the last p,+p.+1+•••+pa places of the permutation; the co-efficient in question may be represented by 1E20-+q2+...+qa the summation being for every permutation, and since q1= p1 this may be written 2 P l—1 E 2 g2+q 3+...+q a. Ex. Gr.—For the See also:

BIPARTITE (from the Lat. bi-, two, and partire, to divide)

bipartite z2, pl = p2 2, and we have the following scheme: a2 a2 q2 = 2 al as = 1 as al = 1 al as = 1. as al = 1 a2 a3 al al =0 Hence F(22) =2(22+2+2+2+2+2°) =26. We may regard the fraction 1 i!

• {1—11(2.1+a2+...+ae)} {1—12(2.1+'Las+...+ae)}... {1—1.(22.1+2.2+.. ±2a.) as a redundant generating function, the enumeration of the compositions being given by the coefficient of (t, al) P1(tact) P2... (tacts) Ps. The transformation of the pure generating function into a factorized redundant form supplies the See also:

key to the See also:

SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of a large number of questions in the theory of See also:

ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

ordinary permutations, as will be seen later. [The transformation of the last See also:

section involves The theory a comprehensive theory of Permutations, which it is ofpermuconvenient to discuss shortly here. rations. If X1, X2, X3,... Xn be linear functions given by the matricular relation (X1, X2, ••• Xn = (a11 a12 ... a1n) (x1, x2, ...xn) a21 a22 ... a2n and See also:

ant ... See also:

ann that portion of the algebraic fraction, 1 (1 —s1X1) (1 —s2X2) ... (1 —$nXn) ' which is a function of the products s1x1, s2x2, S3x3...s,,xn only 'is 1 1(1 — allsixl) (1 —a22s2x2) (1 —a33s3x3) ... (1 —annSnxn) ( I where the denominator is in a symbolic form and denotes on expansion 1- 2I au I s1 x1 + Z Ian (122 1 s1 szxlx2- ... + (-) n I aua22a33... an n I slsz... snxl xz... xn, where lank 1aua22l,...lana22,...annl denote the several co-axial minors of the See also:

DETERMINANT

determinant ana22...annl of the See also:

MATRIX

matrix. (For the See also:

PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

proof of this theorem see See also:

MACMAHON, MARIE

MacMahon, " A certain Class of Generating Functions in the Theory of Numbers," Phil.

Trans. R. S. vol. clxxxv. A, 1894). It follows that the co-efficient of xel e2 1 2 denoted by the partition @a', 1.1242...).

End of Article: EA(psr...), (p, „...)• (pqr...)

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