

A259839


Number of orderpreserving Hamiltonian paths in the ncube (Gray codes); see the comments for the precise definition of orderpreserving.


0




OFFSET

0,6


COMMENTS

An orderpreserving Hamiltonian path in the ncube 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 ncube), 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_{2n2}={n1}, S_{2n1}={n1,n}, S_{2n}={n} (after visiting the set {n}, there are multiple ways to proceed).
It is shown in [Felsner, Trotter 95] that an orderpreserving Hamiltonian path is levelaccurate in the following sense: After visiting a set of size k, the path will never visit a set of size (k2) (*).
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=n1.
Hamiltonian paths that have property (*) have been constructed in [Savage, Winkler 95] for all n (but these paths are not orderpreserving).
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 orderpreserving paths exist). Some partial results have been obtained in [Biro, Howard 09].


LINKS

Table of n, a(n) for n=0..7.
C. Biro and D. Howard, The first three levels of an order preserving Hamiltonian path in the subset lattice, Order, 26(2):101107, 2009.
S. Felsner and W. Trotter, Colorings of diagrams of interval orders and alphasequences of sets, Discrete Math., 144(13):2331, 1995.
C. Savage and P. Winkler, Monotone Gray codes and the middle levels problem, J. Combin. Theory Ser. A, 70(2):230248, 1995.


EXAMPLE

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}.


CROSSREFS

Cf. A003042, A003043, A006069, A006070, A066037, A091302, A159344.
Sequence in context: A239760 A296190 A263552 * A005174 A034668 A215854
Adjacent sequences: A259836 A259837 A259838 * A259840 A259841 A259842


KEYWORD

nonn,hard,more


AUTHOR

Torsten Muetze, Jul 06 2015


STATUS

approved



