The number of distinct coefficients whose values are generated in this fashion from
C'(m, r) depends upon both the number and form of the nonzero columns in r. If the num-
ber of nonzero columns is p, then exchanging any of the p nonzero columns with any of
the N p zero columns generates the coefficient of a distinct monomial term. Exchanging
a nonzero column with another nonzero column having a different column sum also gen-
erates a distinct coefficient. However, exchanging a nonzero column with another non-
zero column having identical column sum may or may not generate a distinct coefficient,
depending upon the distribution of the nonzero entries in the two columns, because it is
possible for the transformation described above to translate one column into the other. A
lower bound on the number of distinct coefficients thus generated is
(N.
The collection of monomial terms corresponding to this coefficient identity can be
written as the product of C'(m,r) (or of C'o(m,r)) and a polynomial function of the form
defined in Eq. C.10 of Appendix C. That is, the collection of terms is a quasi-symmetric
polynomial function in the array of variables (2LP(i I n)- 1).
These coefficient identities and their connection to the quasi-symmetric and quasi-
alternating polynomials of Appendix C offer a promising mechanism for extending the
stationary distribution representation work begun here. Examination of the general form
(2LP(i I n) 1) reveals that it is zero mean in the sense that
Y (2'P(i I n) 1) = 0.
ie S
This property, along with the common form of the elements in the conditional probability
array [P(i I n)], suggests that the symmetric and alternating polynomial forms required for
evaluation of Propositions 6.7-6.8 or 6.5-6.6 may admit to large scale simplifications, and
ultimately yield a tractable, explicit closed form expression for the stationary distribution
components.