Permutation classes need not have proper growth rates

Jay Pantone; Vincent Vatter

Home/Archive/arXiv

arXiv · 2610.30369

Permutation classes need not have proper growth rates

Jay Pantone·Vincent Vatter

View original source ↗Read source document ↗Export metadata ↓

Abstract

By the Marcus-Tardos Theorem (formerly the Stanley-Wilf Conjecture), every permutation class other than the class of all permutations has a finite upper growth rate, meaning that the $n$th root of the number of permutations of length $n$ in the class has a finite limit superior. It has long been conjectured that this limit superior is in fact always a proper limit. We disprove this by constructing a permutation class whose upper and lower growth rates differ. The construction uses grid classes with cubic row-column graphs of increasing girth, obtained from Haar graphs of finite 2-groups. The corresponding hereditary property of ordered graphs also disproves an analogous 2006 conjecture of Balogh, Bollobás, and Morris. Along the way, we show that every proper permutation class with a D-finite generating function has a proper growth rate. We also establish refined enumerative bounds for grid classes, which lead to a short proof of Albert and Vatter's generalization of Bevan's Theorem on the growth rates of grid classes.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jay Pantone, Vincent Vatter. 2026-10-21. Permutation classes need not have proper growth rates. https://arxiv.org/abs/2610.30369

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Cluster scattering diagrams of acyclic affine type

We give an explicit construction of the cluster scattering diagram for any acyclic exchange matrix of affine type. We show that the corresponding cluster scattering fan coincides both with the mutation fan and with a fan constructed in the almost-positive roots model.

math.CO↗

arXiv2026

Embeddability of hieroglyphs in the torus and the intersection form

We characterize the hieroglyphs that embed in the torus, where a hieroglyph is a one-vertex graph with a rotation system. We prove equivalent criteria in terms of four forbidden hieroglyphs, the structure of the loop graph, reductions to three terminal forms, and the rank of the interlacement matrix. The proof uses the intersection form on the first homology of a disc with bands over $\mathbb{F}_2$ and the equality between its rank and twice the genus. We describe the effect of reductions and the role of boundary components. Counting the cycles of the boundary permutation yields a linear-time recognition algorithm for an indexed representation of the input.

math.CO↗

arXiv2026

Locked Polyomino Tilings

A locked $t$-omino tiling is a grid tiling by $t$-ominoes such that, if you remove any pair of tiles, the only way to fill in the remaining $2t$ grid cells with $t$-ominoes is to use the same two tiles in the exact same configuration as before. We exclude degenerate cases where there is only one tiling overall due to small dimensions. It is a classic (and straightforward) result that finite grids do not admit locked 2-omino tilings. In this paper, we construct explicit locked $t$-omino tilings for $t\geq3$ on square and rectangular grids of various dimensions. We show that locked 3- and 4-omino tilings exist on finite square grids of arbitrarily large size. The result for 4-omino tilings in particular is remarkable because they are so rare and difficult to construct: Only a single tiling is known to exist on any grid up to size $40\times40$. We also find locked 5- and 6-omino tilings, and provide a computer-assisted proof that locked 7-omino tilings do not exist on any rectangular grid. In a weighted version of the problem where vertices of the grid may have weights from the set $\{1,2\}$ that count toward the total tile size, we demonstrate the existence of locked tilings on arbitrarily large square weighted grids with only 6 tiles. Finally, we construct locked $t$-omino tilings of the infinite grid for arbitrarily large $t$. Locked $t$-omino tilings arise as obstructions to widely used political redistricting algorithms in a model of redistricting where the underlying census geography is a grid graph. Most prominent is the ReCom Markov chain, which takes a random walk on the space of redistricting plans by iteratively merging and splitting pairs of districts (tiles) at a time. Locked $t$-omino tilings are isolated states in the state space of ReCom. The constructions in this paper are counterexamples to the meta-conjecture that ReCom is irreducible on graphs of practical interest.

math.CO↗