Title: Proof of a conjecture of Kløve on permutation codes under the Chebychev distance,
Abstract: 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.