Proof of a conjecture of Kl{o}ve on permutation codes under the Chebychev distance

Let $d$ be a positive integer and $x$ a real number. Let $A_{d, x}$ be a $d\times 2d$ matrix with its entries $$ a_{i,j}=\left\{ \begin{array}{ll} x\ \ & \mbox{for} \ 1\leqslant j\leqslant d+1-i, 1\ \ & \mbox{for} \ d+2-i\leqslant j\leqslant d+i, 0\ \ & \mbox{for} \ d+1+i\leqslant j\leqslant 2d. \end{array} \right. $$ Further, let $R_d$ be a set of sequences of integers as follows: $$R_d=\{(\rho_1, \rho_2,\ldots, \rho_d)|1\leqslant \rho_i\leqslant d+i, 1\leqslant i \leqslant d,\ \mbox{and}\ \rho_r\neq \rho_s\ \mbox{for}\ r\neq s\}.$$ and define $$\Omega_d(x)=\sum_{\rho\in R_d}a_{1,\rho_1}a_{2, \rho_2}\ldots a_{d,\rho_d}.$$ In order to give a better bound on the size of spheres of permutation codes under the Chebychev distance, Kl{\o}ve introduced the above function and conjectured that $$\Omega_d(x)=\sum_{m=0}^d{d\choose m}(m+1)^d(x-1)^{d-m}.$$ In this paper, we settle down this conjecture positively.