Explicit Constructions of Universal Cycles on Partial Permutations

A k-partial permutation out of n elements is a k-tuple (p1, p2, . . . , pk) with k distinct elements and pi ∈ [n] = {1, 2, . . . , n}, i = 1, 2, . . . , k. Let (p1, p2 . . . , pn) be a full permutation of size n, where the elements are distinct and pi ∈ [n], i = 1, 2, . . . , n. Then we say the k-partial permutation (p1, p2, . . . , pk) is induced from the full permutation. A universal cycle on k-partial permutations is a cycle (a1, a2, . . . , aN ), with length N = ( n k ) k!, elements ai ∈ [n], i = 1, 2, . . . , N , where each k-partial permutation appears as a subsequence in this cycle exactly once. The main contribution of the paper is the first explicit construction of universal cycles on k-partial permutations for arbitrary 1 ≤ k < n.