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.
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