The collected mathematical papers of Arthur Cayley.

186

ON THE NUMBER OF DISTINCT TERMS IN A SYMMETRICAL OR

[580

Consider a determinant (m, n) where n is not =0; for instance, the determinant
above written down, which is (2, 2); this contains terms multiplied by qa, qb, qp', qq'
bb
respectively: where, disregarding signs, the whole factor multiplied by qa is ap'
pq'
which is a determinant (1, 2), and similarly the whole factor multiplied by qb is a
/ aa
determinant (1, 2). But the whole factor multiplied by qp' is the determinant b,
IpU~!
Pq'
which is a determinant (2, 1), and the whole factor multiplied by qq' is also a determinant (2, 1).
Hence, observing that qa, qb, qp', qq' are distinct terms occurring only in the last
line of the determinant, the number of distinct terms is equal to the sum of the
numbers of distinct terms in the several component parts, or we have
<b(2, 2)=2( (1, 2)+ 2(2, 1);
and so in general:
9 (mn, n)= m( (m - 1, n) + n (m, n- 1).
Consider next a completely symmetrical determinant (m, 0); for instance (4, 0), the
determinant
aa aa, ab, ac, ad
bb ba, bb, be, bd
cc ca, cb, cc, cd
dd da, db, dc, dd
( aa
We have first the terms containing dd; the whole factor is bb, which is a
cc
determinant (3, 0); secondly, the terms containing ad. da, or the like combinations,
(aa
bd. db or cd. dc: the whole factor multiplied by ad. da is bb which is a determbb '
inant (2, 0); thirdly, the terms containing ad. db + bd. da, = 2ad.bd; or the like
combinations 2ad. cd or 2bd. cd: the whole factor multiplying the term 2ad. bd is
{ }, which is a determinant (1, 1). Hence observing that ad, bd, cd, =da, db, dc,
ba
and dd are terms occurring only in the last line and column of the original determinant,
it is clear that the number of distinct terms in the original determinant is equal to
the sum of the numbers of distinct terms in the component parts, or that we have
(4, 0)= b (3, 0)+ 3 (2, O)+ 30 (1, 1); and so in general:
m(. n-,
f(r, 0) = (m - 1, 0) + m ( - 2, O) 2 (n,- 3, 1).