Locked Polyomino Tilings

TL;DR

This work studies locked $t$-omino tilings as obstructions to ReCom-based redistricting on grid graphs, formulating the problem in terms of the metagraph $\mathcal{M}(G,t)$ of $t$-tile partitions. It establishes domino tiling connectivity, demonstrates abundant locked tilings for $t=3$ and, surprisingly, rare but existent locked tilings for $t=4$ and $t=5$, and constructs explicit infinite-families of locked tilings for $t=4$ on arbitrarily large finite grids and for arbitrarily large $t$ on infinite grids. An enumeration algorithm leveraging cell-type incompatibilities enables exhaustive search for small grids, with results showing extreme rarity of higher-$t$ locked tilings. The paper also extends the locked-tiling phenomenon to weighted grids and infinite toroidal geometries, revealing fundamental limitations to reversible recombination dynamics in grid-based redistricting models.

Abstract

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 $t \geq 3$ on grids of various dimensions. Most notably, we show that locked 3- and 4-omino tilings exist on finite square grids of arbitrarily large size, and locked $t$-omino tilings of the infinite grid exist for arbitrarily large $t$. 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 $40 \times 40$. 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. 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.

Figures (19)

• Figure 1: The metagraph $\mathcal{M}(G(3, 3), 3)$. Each vertex is a partition of the $3 \times 3$ grid graph into connected subgraphs, each containing 3 vertices. Note that this graph is connected, as is claimed by Conjecture \ref{['cnjGridKEquals3']}.
• Figure 2: A locked 3-omino tiling of $G(6, 6)$. In total, there are 50 such tilings.
• Figure 3: A locked 3-omino tiling of $G(12, 12)$ obtained from copying and flipping the tiling from Figure \ref{['fig3Omino6']}.
• Figure 4: A generic step of the algorithm to transition to the all-horizontal domino tiling.
• Figure 5: Steps required to make all 3-ominoes horizontal in a $2 \times n$ grid.

Theorems & Definitions (15)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• proof