A Hamilton Path for the Sigma-Tau Problem

Nijenhuis and Wilf asked the following question in their Combinatorial Algorithms textbook from 1975: Can the permutations of {1, 2, …, n} be ordered so that each permutation is transformed into the next by applying either the operation σ, a rotation to the left, or τ, a transposition of the first two symbols? Knuth rated the challenge of finding a cyclic solution for odd n (cycles do not exist for even n > 2) at 48/50 in The Art of Computer Programming, which makes it Volume 4's hardest open problem since the ‘middle levels’ problem was solved by Mütze. In this paper we solve the 40 year-old question by Nijenhuis and Wilf, by providing a simple successor rule to generate each successive permutation. We also present insights into how our solution can be modified to find a Hamilton cycle for odd n.