Generalization of Scott's identity on permanents
If \(x_{1},x_{2},\dots ,x_{n}\) are the zeroes of a polynomial \(P(x)\) of degree \(n\) and \(y_{1},y_{2},\dots ,y_{m}\) are the zeroes of another polynomial \(Q(y)\) of degree \(m\), the permanent \(\text{per} (1/(x_{i}-y_{j}))_{1\leq i\leq n,1\leq j\leq m}\) is named ``Scott-type'' permanent (the case of \(P(x)=x^{n}-1\) and \(Q(y)=y^{n}+1\) has been considered by R.~F.~Scott in the last century, who conjectured a formula on the permanent). In the book of \textit{H. Minc} [Permanents (1978; Zbl 0401.15005)] there is a list of conjectures on permanents. The Scott conjecture was proved by \textit{H. Minc} [Linear Algebra Appl. 28, 141-153 (1979; Zbl 0417.15004)]. In the present paper, the Scott identity is generalized and reproved not by the use of the explicit form of the roots, but by using new techniques in the algebra of polynomials, due to \textit{A. Lascoux} [Square-ice enumeration. Sémin. Lothar. Comb. 42, 15 pp. (1998; Zbl 0921.05003)]. The main result is: Let \(r<n\) be two relative prime natural numbers, \(\{x_{1},\ldots ,x_{n}\}\) and \(\{y_{1},\ldots ,y_{n}\}\) be the roots of the polynomials \(x^{n}-1\) and \( y^{n}+ay^{r}+b\) respectively and \(A\) be the square matrix \(n\times n\) with the element \(1/(x_{i}-y_{j})\) on the \((i,j)\) position. Then \(\text{per}(A)= {{(-1)^{n+1}}\over {(b+1)^{n}-(-a)^{n}}} \prod_{i=1}^{n}(i-(n-i)b)\). Some corollaries are obtained as interesting particular cases. For example, for the polynomials \(x^{n}-1\) and \(y^{n}+y-1\) one has \(\text{per}(A)=n^{n}\).
scientific article; zbMATH DE number 1462173