On a Combinatorial Generation Problem of Knuth

The well-known middle levels conjecture asserts that for every integer $n\geq 1$, all binary strings of length 2(n+1) with exactly n+1 many 0s and 1s can be ordered cyclically so any two consecutive differ in swapping the first bit a complementary at some later position. In his book Art Computer Programming, Knuth raised stronger form this (Problem 56 section 7.2.1.3), which requires sequence positions is swapped each step such an ordering has 2n+1 blocks same length, block obtained by adding s=1 (modulo 2n+1) to previous block. work, we prove Knuth's more general form, allowing arbitrary shifts $s\geq 1$ are coprime 2n+1. We also present algorithm compute ordering, generating new bitstring $\mathcal{O}(n)$ time, using memory total.