Efficient methods of constructing universal cycles for k-permutations

We present two efficient methods of constructing universal cycles for the set of all $k$-permutations of the $n$-set $\{1,2,\dots ,n\}$ for $n>k\ge 3$. These two methods generate universal cycles for $k$-permutations from pure cycling registers with feedback function $f(x_0,x_1,\dots ,x_{k-1})=x_0$, using the cycle joining method. Here we design two classes of successor rules that build upon a framework proposed by Gabric et al. (2020), each of which produces $n-k$ shift inequivalent universal cycles for $k$-permutations. Each universal cycle for $k$-permutations can be generated in $O\left(n\right)$ time per symbol using $O\left(n\right)$ space.