Minimum length of a superpermutation

The problem asks for the shortest string containing every permutation of $n$ symbols as a contiguous substring for each $n>5$. The original factorial-sum conjecture was disproved in 2014, but exact minima beyond $n=5$ remain undetermined.

Known results

• Exact values are known for $n\le 5$; in particular, $s(5)=153$, with $8$ inequivalent minimizers (Chaffin, 2014).
• Houston, Pantone, and Vatter established the lower bound $s(n)\ge n!+(n-1)!+(n-2)!+n-3$ (2018).
• Greg Egan gave the general upper bound $s(n)\le n!+(n-1)!+(n-2)!+(n-3)!+n-3$ (2018).

2019 constructions

For $n=7$, Egan produced a superpermutation of length $5906$, while the best reported lower bound is $5884$. A length-$872$ example is known for $n=6$; a reported search found none of length $871$, but its proof was explicitly unfinished and the exact value remains unverified.

Current status (as of August 2026): Exact minima are settled only for $n\le 5$; for every $n\ge 6$, including $n=6$, only bounds and constructions are publicly recorded, so the problem remains open.