143
for all permutations a = (oq, ,2,..., -m,) of the integers (1,2,--.-,m). The set of all m! such
permutations can be placed in one-to-one correspondence with a subset of the set of all n!
= (mk)! permutations of (1,2, -, n).
If the quasi-symmetric polynomial y(x) = l(x,, x2,, Xm) includes the monomial
PU PN2 Pl( P21 P22 P2k P Pm2 Pk
ax 11 X12 .X1k )X21X 22X2k) '"m\ Xm k)
among its terms, then it includes also the monomial
( P P2 PlkY P21 P22 P2k PM( Pm2 Pmk
a(X 1xX2 ..X a, XqlX4a 2 I X4a k .. XalXa,2 .. Xak
where a = (a,, a ,"-, cm) is an arbitrary permutation of the integers (1,2, --.,m). If for a
given
S= (P, P2,, Pm) = (PIPl2, Plk, P21, P22 P2k, Pml, Pm2, Pmk)
the sum of all distinct monomials of this form is designated by
S P11 P12 Plk'( P21 P22 P2 Pml P2 Pmk E C 10
p(x) x ,x ,x C2x...x x (,xIX.2 .. %..x ( ".I',X *X2... %k), Eq. C.10
then p(x) is quasi-symmetric, and further, the arbitrary quasi-symmetric polynomial f((x)
can be written as a linear combination of a finite number of such polynomials. That is
(x) = ap(x).
P
Eq. C.9 and C.10 are analogous to Eq. C.5 and C.6. The counterparts of Eq. C.7 and
Eq. C.8 can be developed by straightforward extension of these results. Thus, let y be a
polynomial function of n = mk scalar variables xi, 1 < i < m, 1 j < k and let y be denoted
y(xi, x2, ,xm) where x is a k-component vector composed from the scalars xij, 1 j 5 k.
Then, y is quasi-alternating if
T(, x,--, ) = -(x, x%, --,x) Eq. C.11
for every odd permutation ot = (a,,, ,---, am) of the integers (1,2, --,m). The set of all
m!/2 such permutations can be placed in one-to-one correspondence with a subset of the
set of all n!/2 = (mk)!/2 odd permutations of (1,2, .--, n).