The generating function of two-stack sortable permutations by descents is real-rooted

Fighting fish and two-stack sortable permutations

In 2017, Duchi, Guerrini, Rinaldi and Schaeffer proposed a new family of combinatorial objects called "fighting fish", which are counted by the same formula as more classical objects, such as two-stack sortable permutations and non-separable planar maps. In this article, we explore the bijective aspect of fighting fish by establishing a bijection to two-stack sortable permutations, using a new recursive decomposition of these permutations. With our bijection, we give combinatorial explanations of several results on fighting fish proved previously with generating functions. Using the decomposit…

Describing West-3-stack-sortable permutations with permutation patterns

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 that are sorted by three passes through a stack. We also show how the method can be applied to the bubble-sort operator. The method requires the use of mesh patterns introduced by Brändén and Claesson (2011), as well as a new type of generalized pattern we call a decorated pattern.

132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers

In 1990 West conjectured that there are $2(3n)!/((n+1)!(2n+1)!)$ two-stack sortable permutations on $n$ letters. This conjecture was proved analytically by Zeilberger in 1992. Later, Dulucq, Gire, and Guibert gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on $n$ letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation $τ$ on $k$ letters. In several interesting cases this generating function can be expressed in terms of the generating f…

A Wilf equivalence related to two stack sortable permutations

A permutation is so-called two stack sortable if it (i) avoids the (scattered) pattern 2-3-4-1, and (ii) contains a 3-2-4-1 pattern only as part of a 3-5-2-4-1 pattern. Here we show that the permutations on [n] satisfying condition (ii) alone are equinumerous with the permutations on [n] that avoid the mixed scattered/consecutive pattern 31-4-2. The proof uses a known bijection from 3-2-1-avoiding to 3-1-2-avoiding permutations.

$k$-pop stack sortable permutations and $2$-avoidance

We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $k\in\mathbb N$ the set is characterised by finitely many patterns, answering a question of Claesson and Guðmundsson. Our characterisation demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns. We propose a new notion called \emph{$2$-avoidance}.

Permutations sortable by deques and two stacks in parallel share the same growth rate

Recently Albert and Bousquet-Mélou obtained the solution to the long-standing problem of the enumeration of permutations sortable by two stacks in parallel (2sip). Their solution was expressed in terms of functional equations. E.P. and Guttmann then showed that the equally long-standing problem of the number of permutations sortable by a double ended queue (deque) can be simply related to the solution of the same functional equations. They then conjectured that the radius of convergence of both generating functions is the same, and reduced this conjecture to a series of conjectures of Albert a…

Enumerating permutations sortable by $k$ passes through a pop-stack

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…

Motzkin 数计数以最小项结尾的 2-栈可排序排列

Motzkin Numbers Count 2-Stack-Sortable Permutations Ending in Their Least Entry

本文证明了 Zhang 的猜想：以最小项结尾的 2-栈可排序排列数等于 Motzkin 数，并通过与标准杨表的双射给出证明。

多项式时间枚举以最小元素结尾的t-栈排序排列

Polynomial Time Enumeration of t-Stack-Sortable Permutations Ending in Their Least Entry

研究West栈排序映射s在以最小元素结尾的排列上的行为，引入栈排序表格T_π作为多项式时间算法的关键，用于计数t-栈排序排列。

Passing through a stack $k$ times

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 $π$ to be $k$-pass sortable if $π$ is sortable using $k$ passes through the stack. Permutations that are $1$-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of $2$-pass sortable permutations in terms of their basis. We also show all $k$-pass sortable classes have finite bases by giving bounds on the length of a basis elemen…

可栈排序排列中的非重叠下降与非重叠上升

Non-overlapping descents and ascents in stack-sortable permutations

本文研究可栈排序排列上非重叠下降最大数目（MND）的分布，通过有根平面树双射证明其公式，并构造对合证明 MND 与非重叠上升最大数目（MNA）等分布。

Counting 3-Stack-Sortable Permutations

We prove a "decomposition lemma" that allows us to count preimages of certain sets of permutations under West's stack-sorting map $s$. As a first application, we give a new proof of Zeilberger's formula for the number of 2-stack-sortable permutations in $S_n$. Our proof generalizes, allowing us to find an algebraic equation satisfied by the generating function that counts 2-stack-sortable permutations according to length, number of descents, and number of peaks. The same method yields a recurrence relation for $W_3(n)$, the number of 3-stack-sortable permutations in $S_n$. We compute $W_3(n)$ …

可由两个并行栈排序的排列与四分之一平面游走

Permutations sortable by two stacks in parallel and quarter plane walks

本文通过建立一对函数方程刻画两个并行栈可排序排列的生成函数，解决了悬置40年的计数问题，并将分析归结为四分之一平面回路级数 Q(a,u) 的两个猜想。

Permutations sortable by deques and by two stacks in parallel

Recently Albert and Bousquet-Mélou \cite{AB15} obtained the solution to the long-standing problem of the number of permutations sortable by two stacks in parallel (tsip). Their solution was expressed in terms of functional equations. We show that the equally long-standing problem of the number of permutations sortable by a double-ended queue (deque) can be simply related to the solution of the same functional equations. Subject to plausible, but unproved, conditions, the radius of convergence of both generating functions is the same. Numerical work confirms this conjecture to 10 significant di…

Asymptotics of 3-stack-sortable permutations

We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations. Using this functional equation, we extend the 174-term series to 1000 terms. From this series, we conjecture that the generating function behaves as $$W(t) \sim C_0(1-μ_3 t)^α\cdot \log^β(1-μ_3 t), $$ so that $$[t^n]W(t)=w_n \sim \frac{c_0μ_3^n}{ n^{(α+1)}\cdot \log^λ{n}} ,$$ where $μ_3 = 9.69963634535(30),$ $α= 2.0 \pm 0.25.$ If $α= 2$ exactly, then $λ= -β+1$, and we estimate $β\approx -3.$ If $α$ is not an integer, then $λ=-β$, but we cannot give a usef…

Catalan and Schröder permutations sortable by two restricted stacks

Pattern avoiding machines were introduced recently by Claesson, Cerbai and Ferrari as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns. Some of the obtained results have been further generalized to Cayley permutations by Cerbai, specialized to particular patterns by Defant and Zheng, or considered in the context of functions over the symmetric group by Berlow. In this work we study pattern avoiding machines where the first stack avoids a pair of patterns …

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 $t=4$.

Permutations sortable by two stacks in series

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 coefficients approximately. Analysing these coefficients by a variety of methods we conclude that the OGF behaves as $$S(z) \sim A (1 - μ\cdot z)^γ,$$ where $μ=12.45 \pm 0.15,$ $γ= 1.5 \pm 0.3,$ and $A \approx 0.02$.

Passing through a stack $k$ times with reversals

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…

Methods of computing deque sortable permutations given complete and incomplete information

The problem of determining which permutations can be sorted using certain switchyard networks dates back to Knuth in 1968. In this work, we are interested in permutations which are sortable on a double-ended queue (called a deque), or on two parallel stacks. In 1982, Rosenstiehl and Tarjan presented an O(n) algorithm for testing whether a given permutation was sortable on parallel stacks. In the same paper, they also presented a modification giving O(n) test for sortability on a deque. We demonstrate a slight error in the version of their algorithm for testing deque sortability, and present a …

2-Stack Sorting is polynomial

In this article, we give a polynomial algorithm to decide whether a given permutation $σ$ is sortable with two stacks in series. This is indeed a longstanding open problem which was first introduced by Knuth. He introduced the stack sorting problem as well as permutation patterns which arises naturally when characterizing permutations that can be sorted with one stack. When several stacks in series are considered, few results are known. There are two main different problems. The first one is the complexity of deciding if a permutation is sortable or not, the second one being the characterizati…

Sorting Cayley permutations with pattern-avoiding machines

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…

Polynomial-time sortable stacks of burnt pancakes

Pancake flipping, a famous open problem in computer science, can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible. In that context, a prefix reversal of length k reverses the order of the first k elements of the permutation. The burnt variant of pancake flipping involves permutations of signed integers, and reversals in that case not only reverse the order of elements but also invert their signs. Although three decades have now passed since the first works on these problems, neither their computational complexity nor the maxim…

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…

A Decreasing Stack and an Increasing Stack in Series

We study a sorting machine consisting of two stacks in series where the first stack has the added restriction such that entries in the stack must be in decreasing order from top to bottom. We give the basis of the class of permutations that are sortable by this machine which shows that it is enumerated by the Schröder numbers.

The Pop-stack-sorting Operator on Tamari Lattices

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…

Sorting and preimages of pattern classes

We introduce an algorithm to determine when a sorting operation, such as stack-sort or bubble-sort, outputs a given pattern. The algorithm provides a new proof of the description of West-2-stack-sortable permutations, that is permutations that are completely sorted when passed twice through a stack, in terms of patterns. We also solve the long-standing problem of describing West-3-stack-sortable permutations. This requires a new type of generalized permutation pattern we call a decorated pattern.

Two-stack-sorting with pop stacks

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.

由Schröder数计数的类上的一个双射

A Bijection on Classes Enumerated by the Schröder Numbers

本文针对双栈串联排序机器（首栈递减限制）所生成的排列类，建立了其与Schröder路径间的双射，从而以组合方式解释了该类由Schröder数计数的现象。

受限排列上的奇偶统计量与Catalan–Schett多项式

Parity statistics on restricted permutations and the Catalan--Schett polynomials

受可栈排序排列与Jacobi椭圆函数研究启发，本文构造新双射并细化$321$-避免排列的下降生成函数，解决了Kitaev和Zhang关于非重叠上升的公开问题，给出Catalan–Schett多项式的组合解释。

$σ$-machine的动力学方面

Dynamical aspects of $σ$-machines

本文研究了$σ$-machine的动力学性质，证明了可排序排列通常避免双连模式$ξ$，提供了$ξ$-避免排列的几何分解与直接计数方法，刻画了有效排列，并对$σ$-machine进行了分类。

Comparing algorithms for sorting with t stacks in series

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 stacks in series and from this get a lower bound on the number of permutations of length n that are sortable by t stacks in series. Finally we show that the left-greedy algorithm is neither optimal nor defines a closed class of permutations for t>2.

Edit Distance between Unlabeled Ordered Trees

There exists a bijection between one stack sortable permutations --permutations which avoid the pattern 231-- and planar trees. We define an edit distance between permutations which is coherent with the standard edit distance between trees. This one-to-one correspondence yields a polynomial algorithm for the subpermutation problem for $(231)$ avoiding permutations. Moreover, we obtain the generating function of the edit distance between ordered trees and some special ones. For the general case we show that the mean edit distance between a planar tree and all other planar trees is at least $n/l…

Stack sorting with restricted stacks

The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on the procedure and on the stacks. More precisely, we consider a greedy algorithm where we perform the rightmost legal operation (here "rightmost" refers to the usual representation of stack sorting problems). Moreover, the first stack is required to be $σ$-avoiding, for some permutation $σ$, meaning that, at each step, the elements maintained in the stack av…

Restricted non-separable planar maps and some pattern avoiding permutations

Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimss…

Algorithms for discovering and proving theorems about permutation patterns

We present an algorithm, called BiSC, that describes the patterns avoided by a given set of permutations. It automatically conjectures the statements of known theorems such as the descriptions of stack-sortable (Knuth 1975) and West-2-stack-sortable permutations (West 1990), smooth (Lakshmibai and Sandhya 1990) and forest-like permutations (Bousquet-Melou and Butler 2007), and simsun permutations (Branden and Claesson 2011). The algorithm has also been used to discover new theorems and conjectures related to Young tableaux, Wilf-equivalences and sorting devices. We further give algorithms to p…

Stack Sorting with Increasing and Decreasing Stacks

We introduce a sorting machine consisting of $k+1$ stacks in series: the first $k$ stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes \cite{SM}, which studies the case $k=1$. Here we show that, for $k=2$, the set of sortable permutations is a class with infinite basis, by explicitly finding an antichain of minimal nonsortable permutations. This construction can easily be adapted to each $k \ge 3$. Next we describe an optimal sorting algorithm, again for the case $k=2$. We then analyze two types of le…

使用带旁路的弹出栈对排列进行排序

Sorting permutations using a pop stack with a bypass

本文提出一种带旁路操作的弹出栈排序装置，通过禁止模式刻画可排序排列，并用受限Motzkin路径分析计数序列，同时给出原像计算算法及主类原像的完整描述。

使用队列式和栈式堆的堆洗牌对排列进行排序

Sorting permutations with pile shuffle on queue-like and stack-like piles

本文形式化了堆洗牌的三种变体（队列式、栈式及混合类型），刻画了多轮洗牌的可排序排列并给出最少堆数的高效排序公式，证明了多轮洗牌能力呈指数级增长，并指出发牌者选择使可行性判定成为 NP-Hard 问题。

Decompositions and statistics for beta(1,0)-trees and nonseparable permutations

The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice through a stack (where the letters on the stack have to be in increasing order) was conjectured by West, and later proved by Zeilberger. Goulden and West found a bijection from such permutations to nonseparable planar maps, and later, Jacquard and Schaeffer presented a bijection from these planar maps to certain labeled plane trees, called beta(1,0)-trees. Using…

Actions on permutations and unimodality of descent polynomials

We study a group action on permutations due to Foata and Strehl and use it to prove that the descent generating polynomial of certain sets of permutations has a nonnegative expansion in the basis $\{t^i(1+t)^{n-1-2i}\}_{i=0}^m$, $m=\lfloor (n-1)/2 \rfloor$. This property implies symmetry and unimodality. We prove that the action is invariant under stack-sorting which strengthens recent unimodality results of Bóna. We prove that the generalized permutation patterns $(13-2)$ and $(2-31)$ are invariant under the action and use this to prove unimodality properties for a $q$-analog of the Eulerian …

避免两个长度为3的模式的排列上统计量的联合分布

Joint distributions of statistics over permutations avoiding two patterns of length 3

本文推广了避免两个长度为3的模式的排列上统计量的分布研究，给出了六个及四个统计量的联合分布的显式公式，并提供了五个等分布结果的组合证明。

通过分解引理枚举栈排序前像

Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

本文利用分解引理枚举了West栈排序映射下若干排列类的前像，包括发现不平衡Wilf等价、验证布尔-加泰兰数计数猜想，并给出带下降数限制的显式公式，从而证明涉及加泰兰数与Young格序理想的恒等式。

离散Morse函数与分水岭

Discrete Morse Functions and Watersheds

本文引入与离散Morse函数等价的Morse栈，证明其在正规伪流形上的分水岭唯一确定，可由线性时间折叠算法求得，且为最小生成森林的割。

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。

Two first-order logics of permutations

We consider two orthogonal points of view on finite permutations, seen as pairs of linear orders (corresponding to the usual one line representation of permutations as words) or seen as bijections (corresponding to the algebraic point of view). For each of them, we define a corresponding first-order logical theory, that we call $\mathsf{TOTO}$ (Theory Of Two Orders) and $\mathsf{TOOB}$ (Theory Of One Bijection) respectively. We consider various expressibility questions in these theories. Our main results go in three different direction. First, we prove that, for all $k \ge 1$, the set of $k$-s…

部分Theta函数的首根的组合学

The Combinatorics of the Leading Root of the Partial Theta Function

本文通过有根树与堆叠多联骨牌或Ferrers图的组合解释，给出了部分Theta函数首根系数的显式组合意义，并推导了其渐近公式。