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Séminaire Lotharingien de Combinatoire, B43g (2000), 25 pp.

# Guo-Niu Han and Christian Krattenthaler

#
Rectangular Scott-type Permanents

**Abstract.**
Let
*x*_{1},*x*_{2},...,*x*_{n}
be the zeroes of a polynomial *P*(*x*) of degree *n* and
*y*_{1},*y*_{2},...,*y*_{m}
be the zeroes of another polynomial *Q*(*y*) of degree *m*.
Our object of study is the permanent
per(1/(*x*_{i}-*y*_{j}))_{1<=i<=n,
1<=j<=m}, here named ``Scott-type" permanent, the case of
*P*(*x*)=*x*^{n}-1 and
*Q*(*y*)=*y*^{n}+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*)=*x*^{n}-1 and
*Q*(*y*)=*y*^{2n}+*y*^{n}+1
then the corresponding Scott-type permanent is equal to
(-1)^{n+1}*n*!.

Received: January 23, 2000; Accepted: March 19, 2000.

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