Séminaire Lotharingien de Combinatoire, B43g (2000), 25 pp.
Guo-Niu Han and Christian Krattenthaler
Rectangular Scott-type Permanents
Abstract.
Let
x1,x2,...,xn
be the zeroes of a polynomial P(x) of degree n and
y1,y2,...,ym
be the zeroes of another polynomial Q(y) of degree m.
Our object of study is the permanent
per(1/(xi-yj))1<=i<=n,
1<=j<=m, here named ``Scott-type" permanent, the case of
P(x)=xn-1 and
Q(y)=yn+1 having been considered by
R. F. Scott. We present
an efficient approach to determining explicit evaluations of
Scott-type permanents, based on generalizations of
classical theorems by Cauchy and Borchardt, and of a
recent
theorem by Lascoux.
This continues and extends the work initiated by the first
author (``Généralisation
de l'identité de Scott sur les
permanents,'' to appear in Linear Algebra Appl.). Our approach enables
us to provide numerous closed form evaluations of Scott-type
permanents for special choices of the polynomials P(x) and
Q(y), including generalizations of all the results from the above
mentioned paper and of Scott's permanent itself. For example, we prove that
if P(x)=xn-1 and
Q(y)=y2n+yn+1
then the corresponding Scott-type permanent is equal to
(-1)n+1n!.
Received: January 23, 2000; Accepted: March 19, 2000.
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