论文标题
栅栏分布晶格的部分等级对称性
Partial rank symmetry of distributive lattices for fences
论文作者
论文摘要
与任何构图beta =（a，b，...）相关 x_1 <x_2 <... <x_ {a+1}> x_ {a+2}> ...> x_ {a+b+1} <x_ {a+b+2} <... f（beta）所有低阶理想的分布晶格L（beta）在集群algebras的理论中很重要。此外，其等级生成函数r（q; beta）用于定义有理数的Q-Analogues。 Oguz和Ravichandran最近表明，其系数满足了交织条件，证明了McConville，Smyth和Sagan的猜想，这又意味着先前的Morier-Genoud和Ovsienko R（Q; Beta）的猜想是不合同的。我们表明，当beta具有奇数零件时，多项式也是部分对称的：k k的f（beta）理想数量等于k的过滤器数量，当k低于某个值时。我们的证明是完全的培养基。 Oguz和Ravichandran还引入了围栏的圆形版本，并使用代数技术证明了这种Poset的分布晶格是对称的。我们也给出了这一结果的徒证明。我们以这项工作提出的一些问题和猜想结尾。
Associated with any composition beta=(a,b,...) is a corresponding fence poset F(beta) whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... The distributive lattice L(beta) of all lower order ideals of F(beta) is important in the theory of cluster algebras. In addition, its rank generating function r(q;beta) is used to define q-analogues of rational numbers. Oguz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a conjecture of McConville, Smyth and Sagan, which in turn implies a previous conjecture of Morier-Genoud and Ovsienko that r(q;beta) is unimodal. We show that, when beta has an odd number of parts, then the polynomial is also partially symmetric: the number of ideals of F(beta) of size k equals the number of filters of size k, when k is below a certain value. Our proof is completely bijective. Oguz and Ravichandran also introduced a circular version of fences and proved, using algebraic techniques, that the distributive lattice for such a poset is rank symmetric. We give a bijective proof of this result as well. We end with some questions and conjectures raised by this work.
