Hamiltonian cycles for the square of the augmentation graphs and Gray\n codes for restricted permutations and ascent sequences

In this paper, we construct a listing for the vertices of the augmentation\ngraph of given size, and as a consequence, we obtain a Hamiltonian cycle for\nthe square of the augmentation graph of given size. As applications, we have a\nGray code for the $132$-$312$ avoiding permutations of given length such that\ntwo successive permutations differ by at most $2$ adjacent transpositions. Also\nwe obtain Gray codes of strong distance $2$ for the $001$ avoiding ascent\nsequences and the $010$ avoiding ascent sequences of given length.\n