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arXiv · 2609.13305

Rationality and Quasipolynomiality of Restricted Rectangle Partitions

Mubin Shaikh
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Abstract

This paper proves Conjecture 5.10 of Gajdzica, Visser, and Zakarczemny on restricted partitions of a rectangle. For every fixed positive integer k, the generating function for feasible multisets of bars of lengths at most k tiling a 2 x n rectangle has denominator dividing the product of (1 - x^j) for j = 1,...,k. Its coefficients are eventually quasipolynomial of degree k - 1 and period dividing the least common multiple of 1,...,k. The key observation is that appending a two-bar slab makes feasibility upward closed within each parity class of multiplicities. Dickson's lemma and finite inclusion-exclusion then give the precise denominator.

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Mubin Shaikh. 2026-09-10. Rationality and Quasipolynomiality of Restricted Rectangle Partitions. https://arxiv.org/abs/2609.13305

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