well as over the ferromagnetic state, are connected with enumeration of the trajectories made by the walkers moving on the lattice. A relationship is established between the number of trajectories made by a several vicious walkers and the number of paths made by a single random turns walker on a lattice of a dimension equal to the number of the vicious walkers. Differentiation of the generating function, calculated over the superposition of the eigen-states, demonstrates a more complicated combinatorial picture than that of the ferromagnetic case. In particular, the set of the paths made by a single pedestrian is replaced by the set of trajectories made simultaneously by the principal and virtual (both vicious) pedestrians. An estimate is obtained for the number of trajectories made both by the principal and the virtual pedestrians.

Acknowledgement {#acknowledgement .unnumbered}
===============

This paper was partially supported by the Russian Foundation for Basic Research, No. 07-01-00358, and by the Russian Academy of Sciences program ,,Mathematical Methods in Non-Linear Dynamics”.

Appendix {#appendix .unnumbered}
========

Asymptotic expansion for the logarithm of the gamma-function at large $|z|$ and $|\arg z |< \pi$ has the form [@27]: $$\begin{array}{rcl}
\log\Gamma
(z\,+\,\alpha)&=&\displaystyle{\Bigl(z\,+\,\alpha\,-\,\frac12\Bigr) \log
z\,-\,z\,+\,\frac12\,\log(2\pi)}\nonumber\\[0.4cm]
&+&\displaystyle{\sum\limits_{p=1}^n
(-1)^{p+1}\,\frac{B_{p+1}(\alpha)}{p(p+1)}\,z^{-p}\,+\,
\mathcal{O}\Bigl(\frac1{z^{n+1}}\Bigr)}\,,
\end{array}\eqno(A1)$$ where $n= 1, 