Consider n quantities a₁, a₂, a₃,...a_n.

Every rational integral function of these quantities, which does not alter its value however the n suffixes 1, 2, 3, ... n be permuted, is a rational integral symmetric function of the quantities. If we write (1 +a₁x) (1 +a₂x)...(1 +a_nx) = 1 +a₁x + a₂x² +... +a_nxⁿ, a₁, a₂, ...a_n are called the elementary symmetric functions.

⁠a₁ = a₁ + a₂ +...+a_n = Σa₁
⁠a₂ = a₁a₂ + a₁a₃ +...+a₂a₃ = Σa₁a₂
⁠⋅⁠⋅⁠⋅⁠⋅⁠⋅
⁠a_n = a₁a₂a₃...a_n.
The general monomial symmetric function is
⁠Σa_p1
1  a_p2
2  a_p3
3 ...a_pn
n,
the summation being for all permutations of the indices which result in different terms. The function is written
⁠(p₁p₂p₃...p_n)
for brevity, and repetitions of numbers in the bracket are indicated by exponents, so that (p₁p₁p₂) is written (p_2
1p₂). The weight of the function is the sum of the numbers in the bracket, and the degree the highest of those numbers.

Ex. gr. The elementary functions are denoted by
⁠(l), (l²), (l³), ... (lⁿ⁠),
are all of the first degree, and are of weights 1, 2, 3,...n respectively.
Remark.—In this notation (0) = Σa_0
1 = (n
1); (0²) = Σa_0
1a_0
2 = (n
1);... (0^s) = (n
s), &c. The binomial coefficients appear, in fact, as symmetric functions, and this is frequently of importance.

The order of the numbers in the bracket (p₁p₂ ...p_n) is immaterial; we may therefore always place them, as is most convenient, in descending order of magnitude; the numbers then constitute an ordered partition of the weight w, and the leading number denotes the degree.

The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by h_w it is connected with the elementary functions by the formula
⁠1/1 – a₁x + a₂x² a₃x³ +... = 1 + h₁x + h₂x² + h₃x³ + ...,
which remains true when the symbols a and h are interchanged, as is at once evident by writing –x for x. This proves, also, that in any formula connecting a₁, a₂, a₃ ,... with h₁, h₂, h₃,... the symbols a and h may be interchanged.

⁠Ex. gr. from h₂ = a_2
1 – a₂ we derive a₂ = h_2
1 – h₂.

The function Σa _p1
1⁠a _p2
2⁠ ...a _pn
n⁠ being as above denoted by a partition of the weight, viz. (p₁p₂ ...p_n), it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as
⁠Σa _p1
1⁠a _p3
2⁠a _p2
1⁠a _p4
2⁠... a _{pn–2
n–2⁠} = (p₁p₃)(p₂p₄ ...p_n–2).

The expression just written is in fact a partition of a partition, and to avoid confusion of language will be termed a separation of a partition. A partition is separated into separates so as to produce a separation of the partition by writing down a set of partitions, each separate partition in its own brackets, so that when all the parts of these partitions are reassembled in a single bracket the partition which is separated is reproduced. It is convenient to write the distinct partitions or separates in descending order as regards weight. If the successive weights of the separates w₁, w₂, w₃,... be enclosed in a bracket we obtain a partition of the weight w which appertains to the separated partition. This partition is termed the specification of the separation. The degree of the separation is the sum of the degrees of the component separates. A separation is the symbolic representation of a product of monomial symmetric functions. A partition, (p₁p₁p₁p₂p₂p₃) = (p_3
1p_2
2p₃) can be separated in the manner (p₁p₂) (p₁p₂) (p₁p₃) = (p₁p₂)² (p₁p₃), and we may take the general form of a partition to be (p_w1
1 p_w2
2 p_w3
3  ...) and that of a separation (J₁)^f₁ (J₂)^f₂(J₃)^f₃... when J₁, J₂, J₃... denote the distinct separates involved.

Theorem.— The function symbolized by (n), viz. the sum of the n^th powers of the quantities, is expressible in terms of functions which are symbolized by separations of any partition (n_v1
1 n_v2
2 n_v3
3 ...) of the number n. The expression is—
⁠(–) ^{v₁+v₂v₃+...}(v₁+v₂v₃+...)– 1)!/v₁!+v₂!v₃!+... (n)
⁠=Σ (–) ^{j₁+j₂j₃+...} (j₁+j₂j₃+...)– 1)!/j₁!+j₂!j₃!+...(J₁)^j₁ (J₂)^j₂(J₃)^j₃...,

(J₁)^j₁ (J₂)^j₂(J₃)^j₃... being a separation of (n_v1
1 n_v2
2 n_v3
3 ...) and the summation being in regard to all such separations. For the particular case (n_v1
1 n_v2
2 n_v3
3 ...) = (lⁿ)
(−)ⁿl/n(n) = Σ (–) ^{j₁+j₂j₃+...} (j₁+j₂j₃+...)– 1)!/j₁!+j₂!j₃!+...(l)^j₁ (l²)^j₂(l³)^j₃...

To establish this write— 1 + μX₁ + μ²X₂ + μ³X₃ +... = II
a(l + μa₁x₁ + μ²a_2
1x₂ + μ³a_3
1x₃ + ...),

the product on the right involving a factor for each of the quantities a₁, a₂, a₃..., and μ being arbitrary.

Multiplying out the right-hand side and comparing coefficients
⁠X₁ = (l)x₁,
⁠X₂ = (2)x₂ + (l²)x_2
1,
⁠X₃ = (3)x₃ + (2l)x₂x₁ + (l³)x_3
1,
⁠X₄ = (4)x₄ + (3l)x₃x₁ + (2²)x_2
2+ (2l²)x₂x_2
1 (l⁴)x_3
1,
⁠•⁠•⁠•⁠•⁠•⁠•⁠•

⁠X_m=Σ(m_μ1
1m_μ2
2m_μ3
3 ...)x_μ1
m1...,
the summation being for all partitions of m.

Auxiliary Theorem.—The coefficient of x_λ1
l1x_λ2
l2x_λ3
l3... in the product
X _μ1⁠
m1⁠X _μ2⁠
m1⁠X _μ3⁠
m1⁠.../μ₁!μ₂!μ₃!... is Σ (J₁) ^j₁(J₂)^j₂(J₃)^j₃... where J₁) ^j₁(J₂)^j₂(J₃)^j₃...is a separation of (l_λ1
1l_λ2
2l_λ3
3 ...) of specification (m_μ1
1m_μ2
2m_μ3
3 ...), and the sum is for all such separations.

To establish this observe the result.
⁠1/p!Xp/₃ =Σ (3)^π₁ (2l)^π₂ (1³)^π₃/π₁!π₂!π₃!xπ₁/3xπ₂/2 xπ₂+3π₃/1⁠
and remark that (3)^π₁(2I)^π₂(I³)^π₃ is a separation of (3^π₁2^π₂1^π₂+3^π₃) of specification (3^p). A similar remark may be made in respect of

1/μ₁!Xμ₁
m₁, 1/μ₂!Xμ₂
m₂, 1/μ₃!Xμ₃
m₃, ...

and therefore of the product of those expressions. Hence the theorem.

Now
log (1+μX₁ +μ²X₂+μ³X₃ +...)
=Σ log (1+μα₁+μ²α^2
1+μ³α^3
1+...) whence, expanding by the exponential and multinomial theorems, a comparison of the coefficients of μⁿ gives

(n)Σ (−)^{ν₁+ν₂+ν₃+...−1} (ν₁+ν₂+ν₃+...−1)!/ν!₁+ν₂!+ν₃!+...${\displaystyle x_{n_{1}}^{\nu _{1}}x_{n_{2}}^{\nu _{2}}x_{n_{3}}^{\nu _{3}}...}$

= Σ ν₁+ν₂+ν₃+...1 (111+112+Y3+... - 1) !Xv1Xv2Xv3 Y1!Y2!1,3!... n1 n 2 n 3 �� � and, by the auxiliary theorem, any term ${\displaystyle {\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}...}$ on the right-hand side is such that the coefficient of ${\displaystyle {\text{x}}_{n_{1}}^{\nu _{1}}{\text{x}}_{n_{2}}^{\nu _{2}}{\text{x}}_{n_{3}}^{\nu _{3}}...}$ in

(J1)11(J2)12(J3)j3�.� jj!j2!j3!..� where since(m1 1 m2 2 m3 3 ...) is the specification of (J1)j1(J2)j2(J3)j3..., � l +�2+/23+��� =ii +j2+j3+���� Comparison of the coefficients of x:14243... therefore yields the result (-) V1+v2+v3+... (P i +Y2+t' +...-1)! () n VI!Y2!P3!...

) j1+j2+j3+..� (J1+ j2 +j3+...-1)!/T1)?1(J2)72 (J 3)/3..., j11j2!j3!... ?.1 for the expression of Za n in terms of products of symmetric functions symbolized by separations of ( n 1 1n 2 2n 3 3) Let (n) a, (n) x, (n) X denote the sums of the n th powers of quantities whose elementary symmetric functions are a₁, a₂, a₃,...; x₁, x₂, x₃,..; X₁, X₂, X₃.... respectively: then the result arrived at above from the logarithmic expansion may be written (n)_a(n)_x = (n)x,

exhibiting (n) $ as an invariant of the transformation given by the expressions of X₁, X₂, X₃... in terms of x₁, x₂, x₃,....

The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = s_r.

Theorem of Reciprocity.—If
X1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ... = ...+O(s i s 2 s 3 ...)xl1x12x13...+..., where 0 is a numerical coefficient, then also O ?2 0.3 P1 P2 P3 Al A2 A3 +.

X,1X82>$3...=...+8(m m m ...)x 11 x 12 x13......

1 2 3

We have found above that the coefficient of (x 1 1 x 12 x 13...) i n the product XmiXm2X m3 ... is �1!�2!�3!

'1 +� �.(11+j2+j3+... -1)!

(1)/1(12) 2(13)73....

(J1)ji(J2)72(J3)13��� jl!j2!j3!...

the sum being for all separations of l₁l₂l₃ ...) which have the specification (m41 m2 2 m3 3 ...). We can multiply out this expression so as to obtain a series of monomials of the form 9(sl is2 2 s3 3 ...). It can be shown that the number 0 enumerates distributions of a certain nature defined by the partitions (m₁,m₂...), (sT1s°2...), 1212 an = a 1 a 2 a 3 ... an.