I posted this question to the seq-fan mailing list yesterday. I got some replies, which are mentioned below. In Peter Winkler's very nice new book "Math Puzzles" he mentions an unsolved problem, which is related to a possibly new sequence: a(n) = number of ways to cycle through all the subsets of size n and n+1 of a set of size 2n+1 by adding or deleting one element at a time The conjecture is that a(n) is always > 0. a(1) is 2, because the two cycles are 1 13 3 23 2 12 1 ... and its reverse. Yesterday Edwin Clark <eclark@math.usf.edu> wrote to say: a(n) = 2b(n) where b(n) is the number of Hamiltonian cycles in the so-called bipartite Kneser graph H(2n+1,n) whose vertices are the subsets of size n and n+1 of a set of size 2n+1. A is adjacent to B iff one set is contained in the other. Using the Hamiltonian cycle finder in the program Groups & Graphs, I got b(2) = 24. Then when I tried the case n = 3, I got the reply: "Too much output. Logging of Hamiltonian cycles is being terminated. 10382 cycles were found so far." So all I can say is b(3) >=10382. So all I know at the moment is that the sequences a(n) and b(n) begin 2, 48, >=20764 1, 24, >=10382 Also, Peter Winkler says: I'm delighted to see this sequence in your sights. My old colleague Bob Roth (laker@mathcs.emory.edu) once computed some terms, you might ask him if he still has the results after 25 years or so! So far I have not heard back from Bob Roth. Can anyone supply more terms? Neil