A new approach to the middle levels via a Catalan-number system of numeration

Let n = 2k + 1 with 0 < k ∈ Z and let Mk be the subgraph induced by the kand (k + 1)-levels of the n-cube graph. A system of numeration according to which the nonnegative integers are written as restricted growth strings, so called by Arndt and Ruskey, has specifically the k-th Catalan number Ck = (n k ) /n expressed as 10k. This permits a linear ordering of the vertex set of a quotient graph Rk of Mk under a dihedral-group action. The Shields-Savage lifting of a Hamilton path in Rk between certain two distinguished vertices of Rk to a Hamilton cycle placed in a dihedrally symmetric fashion in Mk is presented in terms of the Kierstead-Trotter lexical matchings of Mk, which highlights the relevance of the said system in pursuing the existence, enumeration and sorting (according both to that system and to the lexical matchings) of such Hamilton cycles in Mk. With these tools in hand, we ask whether there are as many such cycles as the 22 Ω(n) ones obtained by Mütze in proving the existence of Hamilton cycles (not necessarily placed in dihedrally symmetric fashion) in Mk.