composition includes an even number of transpositions, then every such composition
includes an even number of transpositions. A permutation is designated odd or even
depending upon whether its decomposition into transpositions yields an odd or even num-
ber of factors respectively. If n > 1, then exactly n!/2 odd and n!/2 even permutations
exist [MoSt64].
A polynomial yin n variables possessing the property that
( X, x2,..., x) = --(x, x,, *,, x,) Eq. C.7
for every odd permutation a = (a,, o,, cn) of its argument list is an alternating polyno-
mial. It follows that any sum or difference of alternating polynomials is an alternating
polynomial, that the product of a symmetric polynomial and an alternating polynomial is
an alternating polynomial and that the product of any odd number of alternating polyno-
mials is an alternating polynomial. The product of any even number of alternating poly-
nomials is symmetric.
If the alternating polynomial y(x) = YN(x, x2, ., x) includes the monomial
Pl P2 PD
ax, x2 ***Xn
among its terms, then it includes also the monomial
(-s() axP'x X.x
where a = (a,,, -,- -, On) is an arbitrary permutation of the integers (1,2, ., n) and where
s(a) is the number of transpositions in the permutation a. If for a given p = (p,,p2,'", Pn)
the sum of all distinct monomials of this form is designated by
Pp(x)= (-1)('xxP1 P2*x'P, Eq. C.8
then p(x) is alternating, and further, the arbitrary alternating polynomial '(x) can be writ-
ten as a linear combination of a finite number of such polynomials. That is
S= ap app(x).
p