Rationality and Quasipolynomiality of Restricted Rectangle Partitions

This preprint presents a proof of Conjecture 5.10 of Gajdzica, Visser, and Zakarczemny on restricted rectangle partitions.

For every fixed positive integer k, let f_k(n) count multisets of bars of sizes 1 × j, with 1 ≤ j ≤ k, that can tile a 2 × n rectangle. Rotations are allowed, and different geometric arrangements of the same multiset are counted only once. The manuscript establishes that the generating function of f_k(n) has an integer polynomial numerator and denominator dividing the product of (1 − x^j) for j = 1, …, k. Consequently, f_k(n) is eventually quasipolynomial of degree exactly k − 1, with quasiperiod dividing lcm(1, …, k).

The proof combines a two-bar slab construction with parity classes of multiplicities, Dickson’s lemma, and finite inclusion–exclusion. The argument is self-contained and applies to every fixed k ≥ 1.

Supplementary material includes the LaTeX source and a Python verification script with instructions. The computations provide finite exact checks supporting reproducibility; they do not replace the universal proof.

Version 1.0. This is a preprint and has not undergone independent peer review.

Original conjecture: Gajdzica, Visser, and Zakarczemny, “Rectangle Partitions Generalizing Integer Partitions,” DOI: 10.1007/s00026-026-00826-w.