S(k + 1)
C'(m, r) (-)N'-k-1 2 Eq. 9.27
"M) (M) (M (M)
m, rj n,, rJ n2,, r nk,,rJ
M M M M
m~r-J K j nirJ n2j '" CkJ
m 1, n rit n2l k riik
The special case r = 0 k k = 0 K= {m} of this result is C'(m, ), the coefficient of
the single order 0 (constant) monomial term previously represented by Eq. 9.8.
Eq. 9.26 and 9.27 represent an exact expression of the value of P-i | =
I P-II PI il, and with Propositions 6.7 and 6.8 constitute an exact representation of
the components of the stationary distributions of the two and three-operator algorithms.
The utility of the representation depends upon the succinct representation of the compo-
nents of the array [P(i I n)] as rational functions of the algorithm parameters and objective
function (Eq. 4.2, 4.11, 4.22 and 4.24) and the ability to extract useful relationships
between the C'(m,r)'s from the general form represented by Eq. 9.27. Section 9.5 below
points out some key coefficient identities related to the latter task, but first the results of
this section are extended to include the limiting case represented by Propositions 7.5 and
7.6.
9.4 The Limiting Case a -> 0'
The matrices introduced in Section 7.4 for use in evaluating the stationary distribu-
tion zero mutation probability limit, P(m)' and Pi ', are very similar in form to P and
PtA. Specifically, P(m)' is a row stochastic matrix and PE is generated from it by