start at the $l$-th site and end up at the $j$-th site. Indeed, let us introduce the notation $\mathcal D^K_{\lambda}$ for the operator of differentiation of $K$-th order with respect to $\lambda$ at the point $\lambda=0$. The application of $\mathcal D^K_{\lambda}$ to the average  leads to the answer: $$\mathcal D^K_{\lambda}\bigl[F_{j;l}({\lambda})\bigr]=
\langle\Uparrow|\sigma_{j}^{+}(-H_0)^K\sigma_{l}^{-}|\Uparrow\rangle=
\sum_{n_1,\dots,n_{K-1}}\Delta^{(+)}_{jn_{K-1}}\dots
\Delta^{(+)}_{n_2n_1}\Delta^{(+)}_{n_1l}.
\label{19}$$ The right hand side of  coincides with the entry at the crossing of the $j$-th row and the $l$-th column of the matrix given by the product of $K$ copies of the hopping matrix . Each matrix in this product corresponds to a transition between the two nearest sites of the lattice. After multiplication by $2^{K}$ (this is due to the accepted normalization of the matrix ), the right-hand side of  gives the number of the trajectories that consist of $K$ steps and are connecting the $l$-th and $j$-th sites. Let us denote this number by $|P_K(l\rightarrow j)|$.

Let $|P_K (l_1, \dots, l_N\rightarrow j_1, \dots, j_N)|$ be a number of trajectories consisting of $K$ links made by $N$ vicious walkers in the random turns model. Here, the initial and final positions of the walkers on the sites are given respectively by the sequences $l_1 > l_2 