受限矩形划分的有理性与拟多项式性
Rationality and Quasipolynomiality of Restricted Rectangle Partitions
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中文总结 AI 辅助
本文证明了关于矩形受限划分的猜想，通过附加双条板使可行性向上封闭，结合Dickson引理和容斥原理，得出生成函数分母及系数拟多项式性质。
中文摘要 AI 辅助
本文证明了Gajdzica、Visser和Zakarczemny关于矩形受限划分的猜想5.10。对于每个固定的正整数k，铺砌2×n矩形的长度至多为k的可行多集条的生成函数的分母整除(1-x^j)对j=1,...,k的乘积。其系数最终为次数k-1的拟多项式，周期整除1,...,k的最小公倍数。关键观察是，附加一个双条板使得在每个多重性奇偶类内可行性向上封闭。Dickson引理和有限容斥原理随后给出精确的分母。
英文摘要
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.
