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.
For each complete meet-semilattice $M$, we define an operator $\mathsf{Pop}_M:M\to M$ by \[\mathsf{Pop}_M(x)=\bigwedge(\{y\in M:y\lessdot x\}\cup\{x\}).\] 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 $\mathsf{Pop}_{\text{Tam}(ν)}$, where $\text{Tam}(ν)$ is the $ν$-Tamari lattice. We determine the maximum size of a forward orbit of $\mathsf{Pop}_{\te…
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 sett…
We consider the set of permutations that are sorted after two passes through a pop stack. We characterize these permutations in terms of forbidden patterns (classical and barred) and enumerate them according to the ascent statistic. Then we show these permutations to be in bijection with a special family of polyominoes. As a consequence, the permutations sortable by this machine are shown to have the same enumeration as three classical permutation classes.
Let $W$ be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator $\mathsf{Pop}:W\to W$ to be the map that fixes the identity element and sends each nonidentity element $w$ to the meet of the elements covered by $w$ in the right weak order. When $W$ is the symmetric group $S_n$, $\mathsf{Pop}$ coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that \[\sup\limits_{w\in W}\left|O_{\mathsf{Pop}}(w)\right|=h,\] where $h$ is the Coxeter number of $W$ (with $h=\infty$ if $W$ is infinite) and $O_f(w)$ denotes…
We study sorting by queues that can rearrange their content by applying permutations from a predefined set. These new sorting devices are called shuffle queues and we investigate those of them corresponding to sets of permutations defining some well-known shuffling methods. If $\mathbb{Q}_Σ$ is the shuffle queue corresponding to the shuffling method $Σ$, then we find a number of surprising results related to two natural variations of shuffle queues denoted by $\mathbb{Q}_Σ^{\prime}$ and $\mathbb{Q}_Σ^{\textsf{pop}}$. These require the entire content of the device to be unloaded after a permuta…
Given a complex simple Lie algebra $\mathfrak g$ and a dominant weight $λ$, let $\mathcal B_λ$ be the crystal poset associated to the irreducible representation of $\mathfrak g$ with highest weight $λ$. In the first part of the article, we introduce the \emph{crystal pop-stack sorting operator} $\mathsf{Pop}_{\lozenge}\colon\mathcal B_λ\to\mathcal B_λ$, a noninvertible operator whose definition extends that of the pop-stack sorting map and the recently-introduced Coxeter pop-stack sorting operators. Every forward orbit of $\mathsf{Pop}_{\lozenge}$ contains the minimal element of $\mathcal B_λ$…
Pattern avoiding machines were recently introduced by Claesson, Ferrari and the current author to gain a better understanding of the classical $2$-stacksort problem. In this paper we generalize these devices by allowing permutations with repeated elements, also known as Cayley permutations. The main result is a description of those patterns such that the corresponding set of sortable permutations is a class. We also show a new involution on the set of Cayley permutations, obtained by regarding a pattern-avoiding stack as an operator. Finally, we analyze two generalizations of pop-stack sorting…
Pop-Stack Sorting is an algorithm that takes a permutation as an input and sorts its elements. It consists of several steps. At one step, the algorithm reads the permutation it has to process from left to right and reverses each of its maximal decreasing subsequences of consecutive elements. It terminates at the first step that outputs the identity permutation. In this note, we answer a question of Defant on the running time of Pop-Stack Sorting on the uniform random permutation $σ_n$. More precisely, we show that there is a constant $c > 0.5$ such that asymptotically almost surely, the algori…
Given an essential semilattice congruence $\equiv$ on the left weak order of a Coxeter group $W$, we define the Coxeter stack-sorting operator ${\bf S}_\equiv:W\to W$ by ${\bf S}_\equiv(w)=w\left(π_\downarrow^\equiv(w)\right)^{-1}$, where $π_\downarrow^\equiv(w)$ is the unique minimal element of the congruence class of $\equiv$ containing $w$. When $\equiv$ is the sylvester congruence on the symmetric group $S_n$, the operator ${\bf S}_\equiv$ is West's stack-sorting map. When $\equiv$ is the descent congruence on $S_n$, the operator ${\bf S}_\equiv$ is the pop-stack-sorting map. We establish …
A fork stack is a generalised stack which allows pushes and pops of several items at a time. We consider the problem of determining which input streams can be sorted using a single forkstack, or dually, which permutations of a fixed input stream can be produced using a single forkstack. An algorithm is given to solve the sorting problem and the minimal unsortable sequences are found. The results are extended to fork stacks where there are bounds on how many items can be pushed and popped at one time. In this context we also establish how to enumerate the collection of sortable sequences.
Given a finite irreducible Coxeter group $W$ with a fixed Coxeter element $c$, we define the Coxeter pop-tsack torsing operator $\mathsf{Pop}_T:W\to W$ by $\mathsf{Pop}_T(w)=w\cdotπ_T(w)^{-1}$, where $π_T(w)$ is the join in the noncrossing partition lattice $\mathrm{NC}(w,c)$ of the set of reflections lying weakly below $w$ in the absolute order. This definition serves as a "Bessis dual" version of the first author's notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack-sorting map on symmetric groups. We show that if $W$ is coincidental or of type $D$, then…
We study sorting machines consisting of a stack and a pop stack in series, with or without a queue between them. While there are, a priori, four such machines, only two are essentially different: a pop stack followed directly by a stack, and a pop stack followed by a queue and then by a stack. In the former case, we obtain complete answers for the basis and enumeration of the sortable permutations. In the latter case, we present several conjectures.
For a finite irreducible Coxeter group $(W,S)$ with a fixed Coxeter element $c$ and set of reflections $T$, Defant and Williams define a pop-tsack torsing operation $\mathrm{Popt}\colon W \to W$ given by $\mathrm{Popt}(w) = w \cdot π_T(w)^{-1}$ where $π_T(w) = \bigvee_{t \leq_{T}w, \ t \in T}^{NC(w,c)}t$ is the join of all reflections lying below $w$ in the absolute order in the non-crossing partition lattice $NC(w,c)$. This is a "dual" notion of the pop-stack sorting operator $\mathrm{Pops}$ introduced by Defant as a way to generalize the pop-stack sorting operator on $\mathfrak{S}_n$ to gene…
We introduce semidistrim lattices, a simultaneous generalization of semidistributive and trim lattices that preserves many of their common properties. We prove that the elements of a semidistrim lattice correspond to the independent sets in an associated graph called the Galois graph, that products and intervals of semidistrim lattices are semidistrim, and that the order complex of a semidistrim lattice is either contractible or homotopy equivalent to a sphere. Semidistrim lattices have a natural rowmotion operator, which simultaneously generalizes Barnard's $\overlineκ$ map on semidistributiv…
In an exercise in the first volume of his famous series of books, Knuth considered sorting permutations by passing them through a stack. Many variations of this exercise have since been considered, including allowing multiple passes through the stack and using different data structures. We are concerned with a variation using pop-stacks that was introduced by Avis and Newborn in 1981. Let $P_k(x)$ be the generating function for the permutations sortable by $k$ passes through a pop-stack. The generating function $P_2(x)$ was recently given by Pudwell and Smith (the case $k=1$ being trivial). We…
We consider a stack sorting algorithm where only the appropriate output values are popped from the stack and then any remaining entries in the stack are run through the stack in reverse order. We identify the basis for the $2$-reverse pass sortable permutations and give computational results for some classes with larger maximal rev-tier. We also show all classes of $(t+1)$-reverse pass sortable permutations are finitely based. Additionally, a new Entringer family consisting of maximal rev-tier permutations of length $n$ was discovered along with a bijection between this family and the collecti…
