arXiv subjects
Publications and source records attributed to A. M. Kamenetskii.
Combinatorial formulas expressing cyclic rook polynomials and cyclic permanents of rectangular matrices in terms of expansions along rows are presented
In this paper we give the absolutely new proof of a conjecture of R.F.Scott(1881) on the permanent of a Cauchy matrix $\ls \frac{1}{x_i-y_j} \rs_{1 \leqslant i,j \leqslant n},$ where $x_1, ..., x_n$ and $y_1, ..., y_n$ are the distinct roots of the polynomials $x^n-1$ and $y^n +1,$ respectively. The simple formula is given for the permanent of the Cauchy matrix $A= \ls \frac{1}{x_i-y_j} \rs_{1 \leqslant i,j \leqslant n},$ where $x_1, ..., x_n$ and $y_1, ..., y_n$ are the distinct roots of the polynomials $x^n+a$ and $y^n +b$, respectively: \begin{gather*} \per (A) =\frac{n}{(b-a)^n} \prod_{k=1}^{n-1}[nb-k(b-a)] = =\begin{cases} %\begin{eqnarray} (-1)^{\frac{n-1}{2}} \cfrac{n}{(b-a)^n} \prod\limits_{k=1}^{\frac{n-1}{2}}[-na-k(b-a)][nb-k(b-a)], \mbox{if $n \equiv 1 (\mod 2)$,} \cfrac{n}{2} \cdot \cfrac{n(a+b)}{(b-a)^n} \prod\limits_{k=1}^{\frac{n}{2}-1}[na+k(b-a)][nb+k(a-b)], \mbox{if $n\equiv 0(\mod 2)$}. %\end{eqnarray} \end{cases} \end{gather*} from which the corrected formula of R.F.Scott follows instantly. Proof follows from obtained by the author a formula for the determinant of an arbitrary of the Hadamard degree $m$ of a Cauchy matrix $A$ and Borchard's theorem.
In this paper we give a generalization created by the author of the theory of rook polynomials and permanents of circulants, Toeplits matrices and their submatrices.