The associated monomial term in the Taylor's series expansion of-_ p -I =i
IP-II -I|Pij-II is given by
C(m,jr') x (P(i, I n,)- 1/2').
In subsequent paragraphs, I1 rTiJ = 0 and 11 rill < M for n # m, which together imply
II I < (N' 1) x M, are shown to be suitable upper bounds on the order of differentiation
with respect to the P(i I n)'s when computing C(m,r). Thus, the Taylor's series terminates
at finite order, as indeed it must since -| P-I is a polynomial function of the P(i I n)'s,
and as noted earlier, the highest order monomial terms are order (N' 1) x M.
These upper bounds on the order of differentiation (i.e. upper bounds on II ral ),
along with the lower bound of 0 on every component of r imposed by the requirement
that r be a nonnegative integer array, can be represented to advantage in terms of a set
related to S'. Let S', which is completely determined by the parameters L and M, be
momentarily represented by S'(M) (that is, let its dependence on M be explicitly indi-
cated) and let the set S" be defined as the set union of all S'(k) for 0 < k < M. That is
S"= S'(0)uS'(I) .--. S'(M- 1)uS'(M)
M
= u S'(k).
k=O
The above constraints on the rows of are then equivalent to requiring that every row ofr
be drawn from S", with the additional requirement that row m be the specific element
ri = 0 S". Since the S'(k) for distinct k are disjoint, it follows that N" = card(S") is the
sum of the N'(k) = card[S'(k)], and consequently from Eq. 4.1 and an elementary recur-
sion on the binomial coefficient that
N"=M+2_M+N
N" = M ) = M +
This result is precisely that supplied by Eq. 5.1, accompanying the state space enumer-
ation empirical results tabulated in Section 5.2.