On a combinatorial generation problem of Knuth

TL;DR

This paper proves Knuth's stronger conjecture on a cyclic ordering of binary strings with a specific swapping pattern, generalizing previous results and providing an efficient algorithm for generating such orderings.

Contribution

It generalizes Knuth's conjecture to arbitrary coprime shifts and presents an efficient algorithm for generating the orderings.

Findings

Proved Knuth's conjecture for arbitrary coprime shifts s.

Developed an O(n) time algorithm for generating the orderings.

Established a method for cyclically ordering binary strings with specific swap patterns.

Abstract

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 that any two consecutive strings differ in swapping the first bit with a complementary bit at some later position. In his book `The Art of Computer Programming Vol. 4A' Knuth raised a stronger form of this conjecture (Problem 56 in Chapter 7, Section 2.1.3), which requires that the sequence of positions with which the first bit is swapped in each step of such an ordering has $2n+1$ blocks of the same length, and each block is obtained by adding $s=1$ (modulo $2n+1$) to the previous block. In this work, we prove Knuth's conjecture in a more general form, allowing for arbitrary shifts $s\geq 1$ that are coprime to $2n+1$. We also present an algorithm to compute this ordering, generating each new bitstring in $\mathcal{O}(n)$ time, using $\mathcal{O}(n)$ memory in total.