%I #26 Aug 17 2015 04:50:35

%S 1,1,1,1,1,10,123,1492723

%N Number of order-preserving Hamiltonian paths in the n-cube (Gray codes); see the comments for the precise definition of order-preserving.

%C An order-preserving Hamiltonian path in the n-cube is a listing S_1,...,S_N of all N:=2^n many subsets of [n]:={1,2,...,n}, such that if S_j is a subset of S_i then j <= i+1. For the counting we ignore paths that differ only by renaming elements of the ground set (=automorphisms of the n-cube), i.e., without loss of generality every such path starts as follows: S_1={}, S_2={1}, S_3={1,2}, S_4={2}, S_5={2,3}, S_6={3}, S_7={3,4}, S_8={4},..., S_{2n-2}={n-1}, S_{2n-1}={n-1,n}, S_{2n}={n} (after visiting the set {n}, there are multiple ways to proceed).

%C It is shown in [Felsner, Trotter 95] that an order-preserving Hamiltonian path is level-accurate in the following sense: After visiting a set of size k, the path will never visit a set of size (k-2) (*).

%C For odd n we will have S_N={1,2,...,n} (i.e., |S_N|=n) and for even n we will have |S_N|=n-1.

%C Hamiltonian paths that have property (*) have been constructed in [Savage, Winkler 95] for all n (but these paths are not order-preserving).

%C For n=8,9,10 we know that a(n)>=1. It is unknown whether a(n)>=1 for n>=11 (i.e., it is not known whether such order-preserving paths exist). Some partial results have been obtained in [Biro, Howard 09].

%H C. Biro and D. Howard, <a href="http://dx.doi.org/10.1007/s11083-009-9107-y">The first three levels of an order preserving Hamiltonian path in the subset lattice</a>, Order, 26(2):101-107, 2009.

%H S. Felsner and W. Trotter, <a href="http://dx.doi.org/10.1016/0012-365X(94)00283-O">Colorings of diagrams of interval orders and alpha-sequences of sets</a>, Discrete Math., 144(1-3):23-31, 1995.

%H C. Savage and P. Winkler, <a href="http://dx.doi.org/10.1016/0097-3165(95)90091-8">Monotone Gray codes and the middle levels problem</a>, J. Combin. Theory Ser. A, 70(2):230-248, 1995.

%e For n=4 the a(4)=1 solution is S_1={}, S_2={1}, S_3={1,2}, S_4={2}, S_5={2,3}, S_6={3}, S_7={3,4}, S_8={4}, S_9={2,4}, S_10={1,2,4}, S_11={1,4}, S_12={1,3,4}, S_13={1,3}, S_14={1,2,3}, S_15={1,2,3,4}, S_16={2,3,4}.

%Y Cf. A003042, A003043, A006069, A006070, A066037, A091302, A159344.

%K nonn,hard,more

%O 0,6

%A _Torsten Muetze_, Jul 06 2015