论文标题
Tamari Lattices上的流行堆栈操作员
The Pop-stack-sorting Operator on Tamari Lattices
论文作者
论文摘要
由对称组上的流行堆栈分类地图激励，防御者定义了操作员$ \ mathsf {pop} _m：m \ to m $ to m $ to m $ to m $ to m $ for $ \ mathsf {pop} _m（pop} _m（x）_m（x）= \ big bigwedge（ \ {x \}）。$$本文涉及$ \ mathsf {pop} _ {\ mathrm {tam} _n} $的动态，其中$ \ mathrm {tam} _n $是$ n $ n $ n $ - th $ n $ - thtamari lattice。 我们说一个元素$ x \ in \ mathrm {tam} _n $ is $ t $ - $ \ $ \ mathsf {pop} $ - 如果$ \ mathsf {pop} _mm^t（x）$是最小元素，我们是最小元素$ \ mathrm {tam} _n $。我们找到了生成函数的明确公式$ \ sum_ {n \ ge 1} h_t（n）z^n $，并验证防御者的猜想是合理的。我们进一步证明，$ \ mathsf {pop} _ {\ mathrm {tam} _n} $的图像的大小是motzkin number $ m_n $，解决了辩护人和威廉姆斯的猜想。
Motivated by the pop-stack-sorting map on the symmetric groups, Defant defined an operator $\mathsf{Pop}_M : M \to M$ for each complete meet-semilattice $M$ by $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}).$$ This paper concerns the dynamics of $\mathsf{Pop}_{\mathrm{Tam}_n}$, where $\mathrm{Tam}_n$ is the $n$-th Tamari lattice. We say an element $x\in \mathrm{Tam}_n$ is $t$-$\mathsf{Pop}$-sortable if $\mathsf{Pop}_M^t (x)$ is the minimal element and we let $h_t(n)$ denote the number of $t$-$\mathsf{Pop}$-sortable elements in $\mathrm{Tam}_n$. We find an explicit formula for the generating function $\sum_{n\ge 1}h_t(n)z^n$ and verify Defant's conjecture that it is rational. We furthermore prove that the size of the image of $\mathsf{Pop}_{\mathrm{Tam}_n}$ is the Motzkin number $M_n$, settling a conjecture of Defant and Williams.
