Frankl’s weighted insertion-graph conjecture

Prove Frankl’s Conjecture 8 that the weighted walk count in the insertion graph for 1324-avoiding permutations never exceeds the corresponding unweighted walk count.

Background

Frankl’s approach encodes 1324-avoiding permutations by walks in an insertion graph and assigns edge weights based on proportions of source classes having specified structural properties. The paper reports that the conjectured comparison has been verified computationally for walks of bounded length and agrees with exact enumeration data through size 50, but no proof is supplied. Establishing the conjecture would validate Frankl’s conditional lower bound of 10.418.

References

Frankl in's insertion graph is quotiented on the classes of $132$-avoiders of a given length with a given number of non-right-to-left maxima, each edge weighted by the fraction of its source class that has one. His Conjecture 8 is that the weighted walk count never exceeds the unweighted one, and his Corollary 9 is the bound $10.418$ granted that. It is verified for walks of at most fifteen steps at every cutoff, and along $n = k$ against the exact counts of Conway, Guttmann and Zinn-Justin through $n = 50$, where the weighted count runs about thirty per cent below the true one. Only a proof is missing.

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