超 $(b,t)$ 元分拆的偏序集是分配格
Posets of Hyper $(b,t)$-ary Partitions are Distributive Lattices
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中文总结 AI 辅助
该研究推广了超二进制分拆偏序集为分配格的结果，证明超 $(b,t)$ 元分拆（部分为 $b$ 的幂，重数至多 $t\ge b$）的偏序集也是分配格，其并不可约元素同构于栅栏偏序集。
中文摘要 AI 辅助
按细分排序的整数分拆偏序集，曾被 Birkhoff 和 Ziegler 批评其结构行为难以控制。然而，Propp、McConville 和 Sagan 最近的工作表明，此类超二进制分拆的偏序集是分配的，其并不可约元素的偏序集同构于已被充分研究的栅栏偏序集类。我们将此结果推广到超 $(b,t)$ 元分拆，其中所有部分都是某个正整数 $b$ 的幂，且重数限制为 $t\ge b$。然后，我们证明超 $(b,t)$ 元分拆的偏序集确实也是分配格。
英文摘要
Posets of integer partitions, ordered by refinement, were criticized for unruly structural behavior by both Birkhoff and Ziegler. However, recent work of Propp, McConville, and Sagan demonstrated that such posets of hyperbinary partitions are distributive, with posets of join irreducible elements isomorphic to the well-studied class of fence posets. We provide a generalization of this result for hyper $(b,t)$-ary partitions, where all parts are powers of some positive integer $b$, and multiplicities are restricted to $t\ge b$. Then, we show that posets of hyper $(b,t)$-ary partitions are indeed also distributive lattices.
发表机构
- Wellesley College（韦尔斯利学院）
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