论文标题
流行操作员在各种晶格上的图像
The Image of the Pop Operator on Various Lattices
论文作者
论文摘要
在$ s_n $上给定的晶格上，将经典的流行堆栈排序映射扩展在$ s_n $上的晶格上，对任何晶格$ m $的定义，地图$ \ mathsf {pop} _ {m} _ {m} _ {m}：m $ to将m $发送给m $，以$ x $ of $ x $ of $ x $ of $ x $ of $ x $和x $覆盖的元素$ x \。 In parallel with the line of studies on the image of the classical pop-stack sorting map, we study $\mathsf{Pop}_{M}(M)$ when $M$ is the weak order of type $B_n$, the Tamari lattice of type $B_n$, the lattice of order ideals of the root poset of type $A_n$, and the lattice of order ideals of the root poset of type $ b_n $。特别是，我们解决了Defant和Williams在生成函数\ Begin {equation*} \ Mathsf {pop}（M; q）= \ sum _ {b \ in \ Mathsf {pop} _ {m} _ {m}（m）} q^Q^{| \ Mathscr {Mathscr {Mathscr {M Mathsccr {M Mathsccr {Mathsscr {M Mathssf {Mathssf {Mathssf {| \ Mathscr {Mathsscr {| M） \ end {equation*}其中$ \ mathscr {u} _ {m}（b）$是覆盖$ b $的$ m $的一组元素。
Extending the classical pop-stack sorting map on the lattice given by the right weak order on $S_n$, Defant defined, for any lattice $M$, a map $\mathsf{Pop}_{M}: M \to M$ that sends an element $x\in M$ to the meet of $x$ and the elements covered by $x$. In parallel with the line of studies on the image of the classical pop-stack sorting map, we study $\mathsf{Pop}_{M}(M)$ when $M$ is the weak order of type $B_n$, the Tamari lattice of type $B_n$, the lattice of order ideals of the root poset of type $A_n$, and the lattice of order ideals of the root poset of type $B_n$. In particular, we settle four conjectures proposed by Defant and Williams on the generating function \begin{equation*} \mathsf{Pop}(M; q) = \sum_{b \in \mathsf{Pop}_{M}(M)} q^{|\mathscr{U}_{M}(b)|}, \end{equation*} where $\mathscr{U}_{M}(b)$ is the set of elements of $M$ that cover $b$.
