Shorthand Universal Cycles for Permutations

Abstract

The set of permutations of 〈n〉={1,…,n} in one-line notation is Π(n). The shorthand encoding of a ₁⋯a _n ∈Π(n) is a ₁⋯a _n−1. A shorthand universal cycle for permutations (SP-cycle) is a circular string of length n! whose substrings of length n−1 are the shorthand encodings of Π(n). When an SP-cycle is decoded, the order of Π(n) is a Gray code in which successive permutations differ by the prefix-rotation σ _i =(1 2 ⋯ i) for i∈{n−1,n}. Thus, SP-cycles can be represented by n! bits. We investigate SP-cycles with maximum and minimum ‘weight’ (number of σ _n−1s in the Gray code). An SP-cycle n a n b⋯n z is ‘periodic’ if its ‘sub-permutations’ a,b,…,z equal Π(n−1). We prove that periodic min-weight SP-cycles correspond to spanning trees of the (n−1)-permutohedron. We provide two constructions: B(n) and C(n). In B(n) the spanning trees use ‘half-hunts’ from bell-ringing, and in C(n) the sub-permutations use cool-lex order by Williams (SODA, 987–996, 2009). Algorithmic results are: (1) memoryless decoding of B(n) and C(n), (2) O((n−1)!)-time generation of B(n) and C(n) using sub-permutations, (3) loopless generation of B(n)’s binary representation n bits at a time, and (4) O(n+ν(n))-time ranking of B(n)’s permutations where ν(n) is the cost of computing a permutation’s inversion vector. Results (1)–(4) improve on those for the previous SP-cycle construction D(n) by Ruskey and Williams (ACM Trans. Algorithms 6(3):Art. 45, 2010), which we characterize here using ‘recycling’.