The essential task of representing the stationary distribution solution consists of
evaluating the determinants required to express the results of Propositions 6.7-8 and their
limiting counterparts in Propositions 7.5-6. The development proceeds by examining the
three distinct cases which arise from applying three different sets of constraints on the
value of the mutation probability parameter. The special case pm = 1/2 <= a = 1 is
examined in Section 9.2. It leads to a very simple (trivial) result that is of no particular
interest in its own right, but is fundamental to the mechanism employed in Section 9.3 in
developing the more general case 0 < a 1. The approach pursued in Section 9.3
involves expanding -I PE- I- = IP-II IPj-II as a multivariate Taylor's series in the
N' x N array of conditional probabilities [P(i I n)] for i E S and n e S' (defined by Eq.
4.14 for the two-operator algorithm and by Eq. 4.22 for its three-operator counterpart)
about the point corresponding to a = 1. The product of that effort is an expression for the
coefficient of the general term of the series as a determinant with combinatorial elements.
The case pm -) 0+ <- a -) 0' is examined in Section 9.4. The methodology developed in
Section 9.3 extends with very little modification to represent the a --- 0 limiting behav-
ior of I Pj I. Section 9.5 concludes by pointing out some significant identities which
exist among the Taylor's series coefficients and the connection of those identities to the
algebra of symmetric and alternating polynomials. Its purpose is to provide a foundation
for extending the stationary distribution representation work begun here.
9,2 The Limiting Case a= 1
As pointed out in Section 6, the determinants required for expressing the value of
the stationary distribution components by Propositions 6.7-8 are the characteristic poly-
nomials of the Pm matrices evaluated at X = 1. The coefficients of the characteristic poly-
nomial of any square matrix X with finite dimensions can be expressed in terms of the
principal minors of IXI (i.e. minors generated from I XI by deleting combinations of rows
and columns with the same indices). For example, the characteristic polynomial of P can
be expressed as