Flip-sort 与 pop-stack 排序的组合方面

Permutree 排序

Permutree sorting

本文提出 permutree 排序，将栈排序与 $c$-排序推广到由不相交集合 $U,D$ 参数化的排列排序，并用自动机刻画可排序性和约化表达式。

通过区块移动对置换进行排序

Sorting a Permutation by block moves

我们证明了一个较低的和一个上界的数量块移动排序置换。我们把我们的结果与现有的结果进行排序的块换位，并提出了一些开放的问题。

Sorting Permutations: Games, Genomes, and Cycles

Permutation sorting, one of the fundamental steps in pre-processing data for the efficient application of other algorithms, has a long history in mathematical research literature and has numerous applications. Two special-purpose sorting operations are considered in this paper: context directed swap, abbreviated cds, and context directed reversal, abbreviated cdr. These are special cases of sorting operations that were studied in prior work on permutation sorting. Moreover, cds and cdr have been postulated to model molecular sorting events that occur in the genome maintenance program of certai…

The sorting index

We consider a bivariate polynomial that generalizes both the length and reflection length generating functions in a finite Coxeter group. In seeking a combinatorial description of the coefficients, we are led to the study of a new Mahonian statistic, which we call the sorting index. The sorting index of a permutation and its type B and type D analogues have natural combinatorial descriptions which we describe in detail.

循环排序

Circular sorting

本文研究圆上标记点的排序问题，确定了相邻交换排序所需的最大步数，并给出了非相邻交换排序的下界。

Revstack sort, zigzag patterns, descent polynomials of $t$-revstack sortable permutations, and Steingrímsson's sorting conjecture

In this paper we examine the sorting operator $T(LnR)=T(R)T(L)n$. Applying this operator to a permutation is equivalent to passing the permutation reversed through a stack. We prove theorems that characterise $t$-revstack sortability in terms of patterns in a permutation that we call $zigzag$ patterns. Using these theorems we characterise those permutations of length $n$ which are sorted by $t$ applications of $T$ for $t=0,1,2,n-3,n-2,n-1$. We derive expressions for the descent polynomials of these six classes of permutations and use this information to prove Steingrímsson's sorting conjecture…

扩展耐心排序算法的一种几何形式

A Geometric Form for the Extended Patience Sorting Algorithm

耐心排序是一种组合算法，可以被视为Schensted插入算法的迭代，非递归形式。在最近的工作中，作者将耐心排序扩展到对称群和某些组合对象（称为堆配置）之间的完全双射，这些组合对象最自然地定义为广义排列模式和禁止模式避免。这种扩展耐心排序算法非常类似于Robinson-Schensted-Knuth（或RSK）对应，它本身是通过重复应用Schensted插入算法构建的。在这项工作中，我们介绍了一个几何形式的扩展耐心排序算法，在某种意义上是一个自然的对偶算法G。Viennot著名的几何RSK算法。然而，与几何RSK不同的是，来自耐心排序的晶格路径允许相交。因此，我们也给出了一个表征这些格路径的交叉点的桩配置与一个给定的置换下的扩展耐心排序算法。

Peaks are preserved under run-sorting

We study a sorting procedure (run-sorting) on permutations, where runs are rearranged in lexicographic order. We describe a rather surprising bijection on permutations on length $n$, with the property that it sends the set of peak-values to the set of peak-values after run-sorting. We also prove that the expected number of descents in a permutation $σ\in S_{n}$ after run-sorting is equal to $(n-2)/3$. Moreover, we provide a closed form of the exponential generating function introduced by Nabawanda, Rakotondrajao and Bamunoba in 2020, for the number of run-sorted permutations of $[n]$, ($RSP(n)…

受限耐心排序和禁止模式避免

Restricted Patience Sorting and Barred Pattern Avoidance

耐心排序是一种组合算法，可以被视为Schensted插入算法的迭代，非递归形式。在最近的工作中，作者已经表明，耐心排序提供了一个算法描述的排列，避免了禁止（广义）的排列模式$3-\bar{1}-42$。出于这一点和最近制定的几何形式的耐心排序在某些交叉格路径，我们研究了相关的主题限制输入和避免类似的禁止排列模式。一个这样的结果是将患者排序是可逆算法的那些排列表征为同时避免禁止模式$3-\bar{1}-42$和$3-\bar{1}-24$的排列集合。然后，我们列举这个回避集，其中涉及卷积的斐波那契数。

带点模式避免栈的栈排序

Stack-Sorting with Dotted-Pattern-Avoiding Stacks

本文提出带点模式避免映射并扩展模式避免机器，证明了长度为2的带点模式避免映射的经典结果，并给出猜想。

栈排序前像与0-1树

Stack-sorting preimages and 0-1-trees

本文通过定义部分标记的树类，为Colin Defant关于排列类栈排序前像的两个最新计数结果提供简单证明。

确定性 pop-stack-sorting 的新下界

A new lower bound for deterministic pop-stack-sorting

本文研究了确定性 pop-stack-sorting 过程，通过一个有用的引理，为完全排序均匀随机选择的排列所需的排序次数提供了 $\ rac{3}{5}n$ 的新下界。

Stack-Sorting with Consecutive-Pattern-Avoiding Stacks

We introduce consecutive-pattern-avoiding stack-sorting maps $\text{SC}_σ$, which are natural generalizations of West's stack-sorting map $s$ and natural analogues of the classical-pattern-avoiding stack-sorting maps $s_σ$ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns $σ$ such that $\text{Sort}(\text{SC}_σ)$, the set of permutations that are sortable via the map $s\circ\text{SC}_σ$, is a permutation class, and we enumerate the sets $\text{Sort}(\text{SC}_σ)$ for $σ\in\{123,132,321\}$. We also study the maps $\text{SC}_σ$ from a dynamical point of view, char…

排列模式中的若干问题：单峰性、等价性、错排和排序

An assortment of problems in permutation patterns: unimodality, equivalence, derangements, and sorting

本文探讨了排列模式中的单峰性、等价性、错排计数及排序问题，提出四个开放研究方向。

连续避免$S_3$模式的栈排序映射的繁殖数

Fertility Numbers of Consecutive $S_3$ Pattern-Avoiding Stack-Sorting maps

该研究针对连续模式避免栈排序映射$\textrm{SC}_σ$的繁殖数问题，证明所有长度3模式下正整数均为其繁殖数，解决了已有猜想并提出新猜想。

Preimages under the Stack-Sorting Algorithm

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 $t$-stack sortable permutations of length $n$ with exactly $k$ descents. From these bounds, we are able to significantly improve the best known upper bounds for $\displaystyle{\lim_{n\to\infty}\sqrt[n]{W_t(n)}}$ when $t=3$ and $t=4$.

Stack-Sorting for Coxeter Groups

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 …