exchanging the corresponding nonzero rows of r. If r' is derived from r by such a row
transposition, then it follows that
C'(m, r') = -C'(m, r).
Thus, C'(m,r) establishes the value (to within a sign alternation) of the coefficients of k!
distinct monomial terms in the expansion of -I P| I = I P I I P5-I |. An identical
result applies to C'o(m,r) and the expansion of P' = I P(mA)' -I'l -
The collection of monomial terms corresponding to this coefficient identity can be
written as the product of C'(m,r) (or of C'0(m,r)) and a polynomial function of the form
defined in Eq. C.12 of Appendix C. That is, the collection of terms is a quasi-alternating
polynomial function in the array of variables (2LP(i I n) 1).
In addition to the preceding result, the following identity applies to C'(m,r) and the
expansion of -I Pji -1| = I P II Pn 1. For any n e K, transposition of columns m and
n in the combinatorial determinant of Eq. 9.27 is equivalent to representing the value of
C'(n,r') where r' is derived from r by exchanging r, with rj = 0. That is
n K => C'(n,r') = -C'(m,r).
Thus, the identical quasi-alternating function, evaluated in the new set of variables gener-
ated by replacing each P(i I n) with the corresponding P(i I m), is included in the expan-
sion of -I P- II = P- I P- II. Collectively, these results account for (k + 1)! of the
coefficients required for representation of the stationary distribution.
Another class of coefficient identities derives from transpositions of the columns of
r (i.e. transpositions of i,j e S). Let m' be derived from m by setting m(j)' = m(i),
m(i)' = m(j), n,' from n, by setting ni(j)' = n,(i), n,(i)' = nO(j), etc. Then, if r' is derived
from r by transposition of rows m with m', n, with ni', etc. followed by transposition of
columns i and j, it follows from Eq. 9.27 that
C'(m',r') = C'(m, r).
An identical result applies to C'o(mA', r').