Kløve's generalized formula for permutation-code sphere polynomials

Let $d$ be a positive integer and $x$ a real number. Define the $d\times 2d$ matrix $A_{d,x}=(a_{i,j})$ by

$$a_{i,j}=\begin{cases} x,&1\leqslant j\leqslant d+1-i,\\ 1,&d+2-i\leqslant j\leqslant d+i,\\ 0,&d+i+1\leqslant j\leqslant 2d. \end{cases}$$

Let

$$R_d=\{(\rho_1,\rho_2,\ldots,\rho_d)\mid 1\leqslant\rho_i\leqslant d+i,01\leqslant i\leqslant d,\ \text{and}\ \rho_r\neq\rho_s\text{ for }r\neq s\},$$

and define

$$\Omega_d(x)=\sum_{\rho\in R_d}a_{1,\rho_1}a_{2,\rho_2}\cdots a_{d,\rho_d}.$$

Kløve's generalized conjecture. For every positive integer $d$,

$$\Omega_d(x)=\sum_{m=0}^d{d\choose m}(m+1)^d(x-1)^{d-m}.$$

Kløve introduced this function to obtain bounds for the sizes of spheres in permutation codes under the Chebychev distance. The original conjecture is the specialization $x=2$; the paper proves this generalized identity, settling the conjecture positively.