An arXiv preprint reports an unconditional lower bound of 10.412263 for the growth rate of permutations that avoid the pattern 1324. The result lifts the previous rigorous lower bound from 10.271012 to 10.412263. It does not determine the exact growth rate of the full class.
At the center of the paper is a counting problem that had remained unresolved: can a staircase construction be relaxed in both directions, then counted rigorously without admitting 1324? The authors answer yes for the family built by the new construction, producing the stronger lower bound.
A harder count
The proof starts with an alternating staircase made from dominoes and single connecting cells. Local interleaving rules govern neighboring pieces and are intended to guarantee that the resulting permutations avoid 1324.
The new relaxation changes the rules in both directions. Leaves remain unrestricted, while non-leaves are confined to the gaps between consecutive skew components. The paper shows that this local non-leaf gap rule still preserves 1324 avoidance.
That creates a joint count for horizontal and vertical interleavings. For a fixed number of connecting-cell components, a Harris correlation argument lower-bounds the joint weighted count by the product of the two marginal expectations. The expected joint interleaving count is therefore at least the product of the two separate expected counts, with the generating-function cost specified by the proof.
The strip structure is handled through an equitable profile, which records the per-point densities of empty strips, two-leaf strips and three-leaf strips. Those densities are determined by the leaf density alpha and the empty-strip density beta, giving the profile used in the count.
How the count becomes a bound
The transfer step connects this constructed-family count to the class-wide growth rate. Under the stated saddle and profile conditions, a positive exponent at the chosen evaluation point puts that point outside the generating function's convergence radius and yields a lower bound on the growth rate.
At the selected point, the numerical certificate is . Here, Phi is the exponent used in the transfer argument, z0 is the selected evaluation point and kappa is the component-count scale parameter. The positive certificate meets the condition used for the transfer, but it supports a lower bound rather than an exact value for the class's growth rate.
What remains unresolved
Several internal checks support the calculation. When the two-direction construction is reduced to one direction, it reproduces the earlier published and algebraic values to the precision supported by the rounded parameter check. Two internal control readings are exact and recover the earlier unrelaxed construction.
The numerical evaluation fixes the parameter gamma at 1 by using balanced dominoes. The paper leaves open whether an unbalanced family could improve the exponent away from gamma=1.
The supplied manuscript is arXiv version 3, dated 4 Sep 2026. It also states that versions 1 and 2 reported 10.617 and 10.466425, but those constructions were withdrawn as unproved and are not used in the current construction.
The result is therefore a measured advance rather than a final answer: the two-direction relaxation is counted rigorously, and the resulting construction improves the unconditional lower bound. The exact growth rate remains unknown, as does whether changing the balance of the dominoes can push the exponent higher.
Paper data and sources
Original title: A new lower bound for the growth rate of Av(1324)
Authors: Charles C. Norton
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text