Related papers: A Decreasing Stack and an Increasing Stack in Seri…

We use a method for determining the number of preimages of any permutation under the stack-sorting map in order to obtain recursive upper bounds for the numbers $W_t(n)$ and $W_t(n,k)$ of $t$-stack sortable permutations of length $n$ and…

The $\sigma$-machine was recently introduced by Cerbai, Claesson and Ferrari as a tool to gain a better insight on the problem of sorting permutations with two stacks in series. It consists of two consecutive stacks, which are restricted in…

We introduce a lifting of West's stack-sorting map $s$ to partition diagrams, which are combinatorial objects indexing bases of partition algebras. Our lifting $\mathscr{S}$ of $s$ is such that $\mathscr{S}$ behaves in the same way as $s$…

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…

We address the problem of the number of permutations that can be sorted by two stacks in series. We do this by first counting all such permutations of length less than 20 exactly, then using a numerical technique to obtain nineteen further…

We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a…

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation $\pi$ to be $k$-pass…

Given a sequence of $n$ numbers and $k$ parallel First-in-First-Out (FIFO) queues, how close can one bring the sequence to sorted order? It is known that $k$ queues suffice to sort the sequence if the Longest Decreasing Subsequence (LDS) of…

We consider the problem of sorting elements on a series of stacks, introduced by Tarjan and Knuth. We improve the asymptotic lower bound for the number of stacks necessary to sort $n$ elements to $0.561 \log_2 n + O(1)$. This is the first…

We investigate permutations in terms of their cycle structure and descent set. To do this, we generalize the classical bijection of Gessel and Reutenauer to deal with permutations that have some ascending and some descending blocks. We then…

We study permutations that are sorted by operators of the form $\mathbf{S} \circ \alpha \circ \mathbf{S}$, where $\mathbf{S}$ is the usual stack sorting operator introduced by D. Knuth and $\alpha$ is any $D_8$-symmetry obtained combining…

We show that the left-greedy algorithm is a better algorithm than the right-greedy algorithm for sorting permutations using t stacks in series when t>1. We also supply a method for constructing some permutations that can be sorted by t…

Machines whose main purpose is to permute and sort data are studied. The sets of permutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets being enumerated by…

We use stack words to find a new, simple proof for the best known upper bound for the number of 3-stack sortable permutations of a given length. This is the first time that stack words are used to obtain such a result.

Let $s$ be West's deterministic stack-sorting map. A well-known result (West) is that any length $n$ permutation can be sorted with $n-1$ iterations of $s.$ In 2020, Defant introduced the notion of highly-sorted permutations -- permutations…

Inspired by a common technique for shuffling a deck of cards on a table without riffling, we formalize the pile shuffle and investigate its capabilities as a sorting device. Our study is novel in that we consider pile shuffle in three…

Using earlier results we prove a formula for the number $W_{(n,k)}$ of 2-stack sortable permutations of length $n$ with $k$ runs, or in other words, $k-1$ descents. This formula will yield the suprising fact that there are as many 2-stack…

We explore the fundamental problem of sorting through the lens of learning-augmented algorithms, where algorithms can leverage possibly erroneous predictions to improve their efficiency. We consider two different settings: In the first…

We describe a new method for finding patterns in permutations that produce a given pattern after the permutation has been passed once through a stack. We use this method to describe West-3-stack-sortable permutations, that is, permutations…

In sorting situations where the final destination of each item is known, it is natural to repeatedly choose items and place them where they belong, allowing the intervening items to shift by one to make room. (In fact, a special case of…