If the quasi-alternating polynomial y(x)= (x~,X2, --*',m) includes the monomial
a(x 1 P12. P"X Ik XP21 Px .x P XP Pm Pm2* P
x1 12 X)Ikj X222 A2k *1XmIm2 *Xnk)
among its terms, then it includes also the monomial
4s P11 P12 P* k 2 P21 P i P m1 2 Pmak
a- ( atxaxc2-Xk.a Xl2Xk %Ikj XlaX1a2 %k
where a = (ax,,oX, ---,, m) is an arbitrary permutation of the integers (1,2, -., m) and
where s(a) is the number of transpositions in the permutation a. If for a given
P= (P,P2, ,Pm) 1(P1,P2,"2Plk, P22P2P22, P2k, ,Pml Pm2", Pmk)
the sum of all distinct monomials of this form is designated by
PII P12 Pik'( P2| P22 P Pml Pr2 Pmt
p(x)= 1(-)s) x P",x P2.*x.Pl x ',x Px2' '.x2k x ..x Pm2x***x Pmk Eq. C.12
then p-(x) is quasi-alternating, and further, the arbitrary quasi-alternating polynomial y(x)
can be written as a linear combination of a finite number of such polynomials. That is
y(x) = app(x).
p