The minimal superpermutation length conjecture

Ashlock and Tillotson posed the conjecture in 1993: the shortest length should be $\sum_{k=1}^n k!$. It is now known to be false, although the underlying optimization problem remains open for $n \ge 6$.

Known results

• The minimum was established for $n \le 4$; the recursive construction is optimal there (Johnston, 2013).
• Chaffin’s exhaustive search established the $n=5$ minimum as $153=\sum_{k=1}^5 k!$ and found exactly $8$ examples (2014).
• Houston’s construction gives length $872$ for $n=6$, below the conjectured $873$ (2014).

August 2014 counterexample

Houston’s explicit $n=6$ construction disproves the formula. Its recursive extension yields lengths below $\sum_{k=1}^n k!$ for every $n \ge 6$, but the exact minimum is unknown there; later records include $a(7)\le 5908$, $a(8)\le 46205$, and $a(9)\le 408966$.

Current status (as of August 2026): The conjectured formula is disproved for $n=6$ and therefore for all $n \ge 6$ as a universal claim; the exact minimal lengths remain open for $n \ge 6$.