denotes the indicated mixed partial derivative of (w p I ). Further, in computing the deter-
minant of this sub-array, the order in which the row/column indices are drawn from KI- in
the sub-array construction is immaterial because any transposition of the order introduces
exactly one row transposition and one column transposition into the sub-array, so both
the magnitude and the algebraic sign of its determinant are preserved. Thus, the most
general form of I F|-I K can be expressed as
PO(m m) P(nI) ) P (n21 .) P (nk m)
Sm n, ) P(n, I n) P(n2 n,) ... PT(nk In,)
m ( n) P((,nn Inn) )
P-(m n2P 1r 2 2) 2 n2) ..P k I n
IP = .... Eq. 9.18
P((m n) Pn, I) PkI(n 2 k) .. P nk In)
From Eq. 4.15 and 4.25, it follows that each nonzero element in row v E K,- of P
is composed of a combinatorial coefficient and an order M 1 rjl| product of the P(i I v)'s.
The general form of the element in column w e KI of row v is given by
M iE S Hj P(i I v)_w-i')- Vi e S:w(i) > r(i,v)
P(l(wIv)= w H(w(i))-r(i,v))!i ,
ie S
0 otherwise
which can be rewritten as
Eq. 9.19
w(i)!.x (n[r(i, v)!]P(i Iv)
(M-iS S Y Vi e S:w(i) r(i, v)
P (w | v)= Fw r(i,v)!(w(i)- r(i,v))!
ie 0 otherwise
0 otherwise
Further, by noting that the factor