A mathematical preprint reports a rigorous lower bound of 10.617 for the growth rate of Av(1324), the class of permutations that avoid the pattern 1324. The theorem says the class's growth rate is at least 10.617; it does not give the exact value.
The result addresses a tightly defined question: can the rigorous lower bound for this class be improved? The paper's route combines a grid-based reduction, several counting constructions and a final interval-certified calculation.
The document is an arXiv version 1 preprint dated 20 August 2026.
A problem of patterns and grids
Av(1324) is not a single permutation but a class: the paper groups together permutations avoiding the 1324 pattern. Its target is the class's exponential growth rate, a quantity that summarizes how quickly the number of eligible permutations grows.
To make that count manageable, the proof uses a staircase representation of the class. The mathematical data in this representation are two-cell gridded dominoes—small arrangements occupying two cells of the grid.
A locality lemma supplies the key reduction. It says that every 1324 occurrence in a staircase-gridded permutation can be reduced to two adjacent cells containing two points each.
That reduction gives the rest of the argument a local set of pieces to track: connecting cells, components, strip profiles and the relationships between neighboring placements. These structures are then used to impose lower bounds on profiles and to count compatible arrangements.
The counting gets more exact
The first major extension relaxes interleaving in both directions. The paper proves the relaxed rule valid when non-leaves lie between consecutive connecting-cell components, while leaves may be placed arbitrarily.
An injection supplies coordinate-wise strip-profile floors, or lower bounds for the relevant strip counts. The proof then aggregates those floors before applying a single tail bound, rather than enforcing all of the profile floors simultaneously.
The proof also adds an algebraic tilt to the domino ensemble and obtains the k-leaf strip densities in closed form. At the selected tilted evaluation, it reports a leaf density α of 0.5711822092916011, a non-empty-strip density η of 0.2507518239003806, an empty-strip density β of 0.1780659668080183, and Λ* of 1.9081642156430817.
The last counting step removes the Harris product factorization. Both neighbours are placed against the same component sequence and counted jointly with an exact transfer operator, which acts on the square of one cell's state space. The paper notes that the matrix is unipotent, so the construction terminates.
From routes to a certified number
The route comparison reports 10.272813 for the injection-profile route, 10.415645 for both-direction aggregated floors, 10.466290 for the Harris route and 10.629601 for the joint-count route. The table states that these values are truncated at six decimal places.
At the selected joint-count vertex, the calculation reports z* = 0.09407691... and 1/z* = 10.6296012.... That is a selected-vertex evaluation; the theorem relies on a conservative interval certificate, so the displayed reciprocal is not presented as the exact growth rate.
Two ideas make the certificate finite. Concavity reduces profile minimization to a finite set of vertices—corner cases that can be checked one by one. The Gibbs variational principle converts the rate infimum, the lowest value the proof must control, into a supremum that can be certified.
The rational interval certificate reports Φ(z0) > 2 × 10−3 at every one of the 80 enumerated vertices, with a separate treatment for the remaining vertices. Interval arithmetic carries numerical results as controlled ranges, helping the argument preserve a positive margin instead of relying on a rounded decimal alone.
The internal controls are similarly mathematical checks. One calculation gives 10.2710129282265 against a comparison value of 10.27101292824530, agreeing to twelve digits; when the relaxation is switched off, the calculation returns exactly 81/8.
A stronger floor, not the final answer
The central result should therefore be read as a certified floor, not a finished census of the class. The theorem establishes gr(Av(1324)) ≥ 10.617, but the result is a lower bound rather than an exact determination.
The main mathematical gap is the relaxed interleaving rule. The paper leaves open the gap between that rule and the exact avoidance condition, so the joint-count construction does not close the underlying problem.
Other choices also mark the boundary of the result. The selected tilted evaluation is not reported as the optimum of the joint exponent, and the proof uses aggregated profile floors before its tail bound rather than simultaneous exact profile control.
The limited takeaway is nevertheless clear: the preprint supplies a stronger rigorous lower bound for a specific combinatorial growth rate, backed by exact counting formulas, control calculations and interval certification. The exact value of gr(Av(1324)) remains open.
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