Meeting Covered Elements in $\nu$-Tamari Lattices

DOI10.1016/J.AAM.2021.102303arXiv2104.03890MaRDI QIDQ6364844

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Publication date: 8 April 2021

Abstract: For each complete meet-semilattice $M$, we define an operator $mathsf{Pop}_M:MoM$ by [mathsf{Pop}_M(x)=�igwedge({yin M:ylessdot x}cup{x}).] When $M$ is the right weak order on a symmetric group, $mathsf{Pop}_M$ is the pop-stack-sorting map. We prove some general properties of these operators, including a theorem that describes how they interact with certain lattice congruences. We then specialize our attention to the dynamics of $mathsf{Pop}_{extTam(u)}$, where $extTam(u)$ is the $u$-Tamari lattice. We determine the maximum size of a forward orbit of $mathsf{Pop}_{extTam(u)}$. When $extTam(u)$ is the $n^{e}xtth$ $m$-Tamari lattice, this maximum forward orbit size is $m+n-1$; in this case, we prove that the number of forward orbits of size $m+n-1$ is [frac{1}{n-1}�inom{(m+1)(n-2)+m-1}{n-2}.] Motivated by the recent investigation of the pop-stack-sorting map, we define a lattice path $muinextTam(u)$ to be $t$-$mathsfPop$-sortable if $mathsf{Pop}_{extTam(u)}^t(mu)=u$. We enumerate $1$-$mathsfPop$-sortable lattice paths in $extTam(u)$ for arbitrary $u$. We also give a recursive method to generate $2$-$mathsfPop$-sortable lattice paths in $extTam(u)$ for arbitrary $u$; this allows us to enumerate $2$-$mathsfPop$-sortable lattice paths in a large variety of $u$-Tamari lattices that includes the $m$-Tamari lattices.

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