Balanced tromino growth rate sufficient to improve the lower bound

Determine or establish a growth rate for balanced three-celled trominoes exceeding $8.037012$, so that the tromino route yields a lower bound for $gr(Av(1324))$ better than the bound proved in the paper.

Background

The paper identifies enumeration of three-celled trominoes as the remaining route proposed by BBEP that it does not resolve. A growth rate for balanced trominoes would enter the function G3(τ)G_3(\tau) and could produce an improved lower bound for the Stanley–Wilf limit of Av(1324)Av(1324). The authors state the threshold that the tromino growth rate must exceed in order to improve on the current result.

References

Finally, the third of BBEP's routes remains open. A block of three cells with one connecting cell meets it twice per period exactly as a domino does, so Section~\ref{sec:background}'s argument gives $G_3(\tau) = \sup (3a\log\tau + f(2b-c,b) + 2f(a+c,a))/(3a+b)$ with $f(x,y) = x\log x - y\log y - (x-y)\log(x-y)$, once a growth rate $\tau$ for balanced trominoes is available. Calibration is exact: with two-cell blocks it is BBEP's Theorem 5.1 and returns $81/8$ both at their $(a,b,c) = (14,8,7)$ and at its own free optimum. To improve on what is proved here the tromino route must now exceed $\tau = 8.037012$, where against BBEP's own value it needed only $7.859497$.

— A new lower bound for the growth rate of Av(1324)  (2608.20292 - Norton, 20 Aug 2026) in Section What remains