random walks is a classical problem both for combinatorics and statistical physics. The problem of enumeration of the paths made by the, so-called, *vicious* walkers on the one-dimensional lattice has been formulated and investigated in details by Fisher [@1]. It is supposed that any two vicious walkers, provided both arrive at the same lattice site, annihilate not only one another but all other walkers as well. The problem mentioned still continues to attract considerable attention both of physicists and mathematicians [@2]–[@9]. Closely related problems arise also in the studies of the self-organized criticality [@10], domain walls [@11], and polymers [@12]. In paper [@13] a random walks of the annihilating particles on a ring was considered. In paper [@14] a random turns walks on a semi-axes with a possible creation of the particles at the origin was studied.

It has been shown in [@15], [@16] that the correlation functions, obtained as an averages over the ferromagnetic state of the $XX$ Heisenberg chain, can be used for enumeration of the paths of random walks of vicious walkers. In the present paper the averages of special type are investigated both for the case of ferromagnetic state and for superposition of the eigen-states of 