74 4. Particular Determinants
The coefficient ofar, 2 n,1≤r≤(2n−1), in Pfnis found by putting
(is,js)=(r, 2 n) for any value ofs. Chooses= 1. Then, the coefficient is
∑
σrai 2 j 2 ai 3 j 3 ···ainjn,
where
σr= sgn
{
12 34...(2n−1) 2n
r 2 ni 2 j 2 ... in jn
}
2 n
= sgn
{
1234 ...(2n−1) 2n
ri 2 j 2 i 3 ... jn 2 n
}
2 n
= sgn
{
1234 ...(2n−1)
ri 2 j 2 i 3 ... jn
}
2 n− 1
(4.3.18)
=(−1)
r+1
sgn
{
1234 ...(r−1)r(r+1)...(2n−1)
i 2 j 2 i 3 j 3 ... r ... jn
}
2 n− 1
,r> 1
=(−1)
r+1
sgn
{
1234 ...(r−1)(r+1)...(2n−1)
i 2 j 2 i 3 j 3 ... ... ... jn
}
2 n− 2
,r> 1.
From (4.3.18),
σ 1 = sgn
{
1234 ... (2n−1)
1 i 2 j 2 i 3 ... jn
}
2 n− 1
= sgn
{
234 ... (2n−1)
i 2 j 2 i 3 ... jn
}
2 n− 2
.
Hence,
Pfn=
2 n− 1
∑
r=1
(−1)
r+1
ar, 2 nPf
(n)
r , (4.3.19)
where
Pf
(n)
r
=
∑
sgn
{
1234 ···(r−1)(r+1)···(2n−2) (2n−1)
i 2 j 2 i 3 j 3 ··· ··· ··· in jn
}
2 n− 2
×ai 2 j 2 ai 3 j 3 ···ainjn, 1 <r≤ 2 n− 1 , (4.3.20)
which is a Pfaffian of order (n−1) in which no element contains the row
parameterror the column parameter 2n. In particular,
Pf
(n)
2 n− 1
=
∑
sgn
{
1234 ··· (2n−3) (2n−2)
i 2 j 2 i 3 j 3 ··· in jn
}
2 n− 2
=Pfn− 1. (4.3.21)
Thus, a Pfaffian of orderncan be expressed as a linear combination of
(2n−1) Pfaffians of order (n−1).