Combinatorics and Graph Theory

A Fixed-Port Recursive Construction for Superpermutations

Authors: Yang Liu
Comments: 18 Pages.

We present a bottom-up recursive construction for superpermutations on n symbols, based on a fixed-port invariant of length n−2 and a layer-wise merging procedure where overlap decreases from n−2 down to 1. Starting from (n−1)! directed cycles as base units, the construction iteratively groups units by port matching, closes each group into a cycle, and cuts it to restore fixed-length ports. A key structural property—the self-port property—emerges after the first layer: every merged unit has identical head and tail ports, which greatly simplifies subsequent matching. Reverse-cycle pairs remain separated at every layer due to their fixed maximum overlap of 1, ensuring exactly two units survive to the final step. The construction yields explicit linear superpermutations of length Σu2096u208cu2081u207f k! for all n, matching the classic upper bound and the known minimum for n ≤ 5. The same framework naturally produces three variants—linear, directed cycle, and undirected cycle—each with a closed-form length. While not improving the n=6 record of 872, the construction is fully explicit, manually verifiable, and offers a novel invariant-based perspective that unifies all three variants within a single combinatorial framework.
Category: Combinatorics and Graph Theory