PreprintCC BY 4.0arXiv
Rationality and Quasipolynomiality of Restricted Rectangle Partitions
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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BibTeX
@misc{shaikh2026rationality,
title = {{Rationality and Quasipolynomiality of Restricted Rectangle Partitions}},
author = {Mubin Shaikh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13305},
eprint = {2609.13305},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13305},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rationality and Quasipolynomiality of Restricted Rectangle Partitions
AU - Mubin Shaikh
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.13305
UR - https://arxiv.org/abs/2609.13305
AB - 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.
ER -