Rationality and Quasipolynomiality of Restricted Rectangle Partitions

RESUMO

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.