Growth rate of Av(1324)

The problem asks for the exact exponential growth rate of permutations avoiding the pattern $1324$. This remains the last unresolved Stanley–Wilf limit for a pattern of length four.

Known results

• Bevan, Brignall, Elvey Price, and Pantone (2020): $\operatorname{gr}(\operatorname{Av}(1324))\ge 10.271012$.
• The same work gives $\operatorname{gr}(\operatorname{Av}(1324))\le 13.5$.
• An earlier bound was $\operatorname{gr}(\operatorname{Av}(1324))\le 16$; a conditional argument suggested approximately $13.001954$.

August 2026 claimed lower-bound improvement

Charles C. Norton’s preprint, dated August 20, claims the unconditional bound $\operatorname{gr}(\operatorname{Av}(1324))\ge 10.617$, using transfer-operator and concavity arguments. However, the same version also displays only $\operatorname{gr}(\operatorname{Av}(1324))\ge 10.27281380\ldots$, so the claimed improvement has not been independently verified.

Current status (as of August 2026): The exact value is open; the established bounds are $10.271012\le \operatorname{gr}(\operatorname{Av}(1324))\le 13.5$, while the new $10.617$ lower bound remains an unverified claim.