On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure

Counting descent pairs with prescribed colors in the colored permutation groups

We define new statistics, (c, d)-descents, on the colored permutation groups Z_r \wr S_n and compute the distribution of these statistics on the elements in these groups. We use some combinatorial approaches, recurrences, and generating functions manipulations to obtain our results.

具有指定罗宾逊-斯chensted形状的排列的显式边缘分布

Explicit marginal distributions for permutations with prescribed Robinson-Schensted shape

本文研究了给定形状对排列局部行为的控制，推导了特定形状下σ(i)=j的概率公式，并分析了矩阵中的复杂衍射模式，发现当形状最大部分趋于无穷时，固定点比例趋近于Wallis积分。

On Connection between the Numbers of Permutations and Full Cycles with Some Restrictions on Positions and Up-Down Structure

We discuss both simple and more subtle connections between the numbers of permutations and full cycles with some restrictions,in particular, between the numbers of permutations and full cycles with prescribed up-down structure.

低维整数格壳层内具有指定带符号置换自同构的球面码

Spherical codes with prescribed signed permutation automorphisms inside shells of low-dimensional integer lattices

该研究通过计算机搜索指定自同构的码，计算低维整数格壳层球面码参数的下界，部分构造码改进了已有球面码下界。

Permutations with Extremal number of Fixed Points

We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Desarmenien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics "DEZ" and "lec". Explicit formulas for the maximal case are derived by using symmetric function tools.

Enumeration of permutations by number of cyclic occurrence of peaks and valleys

In this paper, we focus on the enumeration of permutations by number of cyclic occurrence of peaks and valleys. We find several recurrence relations involving the number of permutations with a prescribed number of cyclic peaks, cyclic valleys, fixed points and cycles. Several associated permutation statistics and the corresponding generating functions are also studied. In particular, we establish a connection between cyclic valleys and Pell numbers as well as cyclic peaks and alternating runs.

On the order of arc-stabilisers in arc-transitive graphs with prescribed local group

Let $Γ$ be a connected $G$-arc-transitive graph, let $uv$ be an arc of $Γ$ and let $L$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$. We study the problem of bounding $|G_{uv}|$ in terms of $L$ and the order of $Γ$.

Cycle up-down permutations

A permutation is defined to be cycle-up-down if it is a product of cycles that, when written starting with their smallest element, have an up-down pattern. We prove bijectively and analytically that these permutations are enumerated by the Euler numbers, and we study the distribution of some statistics on them, as well as on up-down permutations, on all permutations, and on a generalization of cycle-up-down permutations. The statistics include the number of cycles of even and odd length, the number of left-to-right minima, and the number of extreme elements.

Permutations with Ascending and Descending Blocks

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 provide the first bijective proofs of some known results. We also solve some problems posed in [3] by Eriksen, Freij, and Wastlund, who study derangements that descend in blocks of prescribed lengths.

计数上-上或下-下排列

Counting up-up-or-down-down permutations

本文回答了Donald Knuth的问题，通过分析奇数长度上-上或下-下排列的最后一个元素，求出了其双变量指数生成函数，刻画了峰和谷均位于奇数位置的排列结构。

Counting permutations by their runs up and down

We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.

A lifting of the Goulden-Jackson cluster method to the Malvenuto-Reutenauer algebra

The Goulden-Jackson cluster method is a powerful tool for counting words by occurrences of prescribed subwords, and was adapted by Elizalde and Noy for counting permutations by occurrences of prescribed consecutive patterns. In this paper, we lift the cluster method for permutations to the Malvenuto-Reutenauer algebra. Upon applying standard homomorphisms, our result specializes to both the cluster method for permutations as well as a q-analogue which keeps track of the inversion number statistic. We construct additional homomorphisms using the theory of shuffle-compatibility, leading to furth…

Restricted permutations by patterns of type $(2,1)$

Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In this paper we study the generating functions for the number of permutations on $n$ letters avoiding a generalized pattern $ab\mn c$ where $(a,b,c)\in S_3$, and containing a prescribed number of occurrences of generalized pattern $cd\mn e$ where $(c,d,e)\in S_3$. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.

Block decomposition of permutations and Schur-positivity

The block number of a permutation is the maximal number of components in its expression as a direct sum. We show that, for $321$-avoiding permutations, the set of left-to-right maxima has the same distribution when the block number is assumed to be $k$ as when the last descent of the inverse is assumed to be at position $n - k$. This result is analogous to the Foata-Schützenberger equi-distribution theorem, and implies that the quasi-symmetric generating function of descent set over $321$-avoiding permutations with a prescribed number of blocks is Schur-positive.

Equidistribution and Sign-Balance on 321-Avoiding Permutations

Let $T_n$ be the set of 321-avoiding permutations of order $n$. Two properties of $T_n$ are proved: (1) The {\em last descent} and {\em last index minus one} statistics are equidistributed over $T_n$, and also over subsets of permutations whose inverse has an (almost) prescribed descent set. An analogous result holds for Dyck paths. (2) The sign-and-last-descent enumerators for $T_{2n}$ and $T_{2n+1}$ are essentially equal to the last-descent enumerator for $T_n$. The proofs use a recursion formula for an appropriate multivariate generating function.

Quadrant marked mesh patterns in alternating permutations

This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns. We study quadrant marked mesh patterns on up-down and down-up permutations, also known as alternating and reverse alternating permutations, respectively. In particular, we refine classic enumeration results of André on alternating permutations by showing that the distribution of the quadrant marked mesh pattern of interest is given by $(\sec(xt))^{1/x}$ on up-down permutations of even length and by $\int_0^t (\sec(xz))^{1+\frac{1}{x}}dz$ on down-up permutations of odd length.

Alternating permutations with restrictions and standard Young tableaux

In this paper, we give bijections between the set of 4123-avoiding down-up alternating permutations of length $2n$ and the set of standard Young tableaux of shape $(n,n,n)$, and between the set of 4123-avoiding down-up alternating permutations of length $2n-1$ and the set of shifted standard Young tableaux of shape $(n+1, n, n-1)$ via an intermediate structure of Yamanouchi words. Moreover, we get the enumeration of 4123-avoiding up-down alternating permutations of even and odd length by presenting bijections between 4123-avoiding up-down alternating permutations and standard Young tableaux.

Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions

We consider a large family of equivalence relations on permutations in Sn that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, subject to the constraint that a third element of a suitable type be in a suitable position. For various instances of the problem, we compute the number of equivalence classes, determine how man…

Caylerian多项式

Caylerian polynomials

该研究提出了Cayley排列上的下降多项式即Caylerian多项式，通过关联Burge词与Burge矩阵推广经典结果，重述双边欧拉多项式γ非负性，并证明指定上升集的Cayley排列可由固定行和的Burge矩阵计数。

仅含奇长度或仅含偶长度循环的排列的下降集分布

Descent set distribution for permutations with cycles of only odd or only even lengths

本文证明对称群中仅含奇长度循环与仅含偶长度循环的排列数等式在下降集层面的细化，并给出奇数阶情形的变体，方法基于特征标生成函数与高阶 Lie 特征标新恒等式。

Quadrant marked mesh patterns in alternating permutations II

This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns initiated in "S. Kitaev and J. Remmel, Quadrant marked mesh patterns, J. Integer Sequences 12, Issue 4 (2012), Article 12.4.7.". We study quadrant marked mesh patterns on up-down and down-up permutations.

错排及具有指定不动点数的其他排列中的逆序统计量

The inversion statistic in derangements and in other permutations with a prescribed number of fixed points

本文研究具有指定不动点数的排列中逆序统计量的期望，给出精确公式，并利用中国餐馆构造证明错排及不同不动点数情形下逆序期望相对一般随机排列的精确偏差。

Enumerating Segmented Patterns in Compositions and Encoding by Restricted Permutations

A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of occurrences of arbitrary segmented partially ordered patterns among compositions of (n) with a prescribed number of parts. These patterns generalize the notions of rises, drops, and levels studied in the literature. We also obtain results enumerating parts with given sizes and locations among compositions and palindromic compositions with a given number of part…

弱EKR群的刻画及具有指定点稳定子的交集密度

Characterization of Weak EKR Groups and Intersection Densities with Prescribed Point Stabilizers

本文刻画弱EKR群的性质，建立相关准则，还解决公开问题，证明对指定有限群H和整数n≥m，存在对应传递置换群的交集密度为n/m。

Counting Descent Pairs with Prescribed Tops and Bottoms

Given sets X and Y of positive integers and a permutation sigma = sigma_1, sigma_2, ..., sigma_n in S_n, an X,Y-descent of sigma is a descent pair sigma_i > sigma_{i+1} whose "top" sigma_i is in X and whose "bottom" sigma_{i+1} is in Y. We give two formulas for the number P_{n,s}^{X,Y} of sigma in S_n with s X,Y-descents. P_{n,s}^{X,Y} is also shown to be a hit number of a certain Ferrers board. This work generalizes results of Kitaev and Remmel on counting descent pairs whose top (or bottom) is equal to 0 mod k.

On weak twins and up-and-down sub-permutations

Two permutations $(x_1,\dots,x_w)$ and $(y_1,\dots,y_w)$ are weakly similar if $x_i<x_{i+1}$ if and only if $y_i<y_{i+1}$ for all $1\leqslant i \leqslant w$. Let $π$ be a permutation of the set $[n]=\{1,2,\dots, n\}$ and let $wt(π)$ denote the largest integer $w$ such that $π$ contains a pair of disjoint weakly similar sub-permutations (called weak twins) of length $w$. Finally, let $wt(n)$ denote the minimum of $wt(π)$ over all permutations $π$ of $[n]$. Clearly, $wt(n)\le n/2$. In this paper we show that $\tfrac n{12}\le wt(n)\le\tfrac n2-Ω(n^{1/3})$. We also study a variant of this problem.…

将 $\u007b1,2,...,n\u007d$ 划分为具有指定阶数的子集的常和划分

Constant Sum Partition of $\{1,2,...,n\}$ Into Subsets With Prescribed Orders

本文研究将集合 $\{1,2,\ldots,n\}$ 划分为指定大小的子集并使其元素和相等的问题，给出存在常和划分的充要条件，并讨论其在图论中的应用及常和划分置换概念。

受限雅可比排列

Restricted Jacobi permutations

本文系统研究雅可比排列中的模式避免问题，完整枚举了避免长度3模式的雅可比排列，并借助二叉树子族、Dyck路径双射与生成函数技巧，对单一模式限制情形给出了关于上升数、从左到右极小值数及末位字母的精细枚举。

具有和不具有指定不动点个数的Ewens抽样分布下随机排列的逆序数

Inversions in Random Permutations Under the Ewens Sampling Distribution With and Without a Prescribed Number of Fixed Points

针对Ewens抽样分布下有无指定不动点个数的随机排列，研究其逆序数统计量，推导逆序数期望、特定索引对逆序概率的精确公式，分析相关量的参数性质与渐近行为。

Counting descents, rises, and levels, with prescribed first element, in words

Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper were extended and generalized in several ways. In this paper, we shall fix a set partition of the natural numbers $N$, $(N_1, ..., N_t)$, and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in $N_i$ over the set of words over the alphabet $[k]$. In particular, we refine and g…

A Combinatorial Proof of the Enumeration of Alternating Permutations with Given Peak Set

Using the correspondence between a cycle up-down permutation and a pair of matchings, we give a combinatorial proof of the enumeration of alternating permutations according to the given peak set.

Stirling permutation codes

The development of the theories of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and presented combinatorial interpretations of the second-order Eulerian polynomials. Recently, there is a growing interest in the properties of Stirling permutations. The motivation of this paper is to develop a general method for finding equidistributed statistics on Stirling permutations. Firstly, we show that the up-down-pair statistic is equidistributed with ascent-plateau statist…

A Bijection Between Weighted Dyck Paths and 1234-avoiding Up-Down Permutations

Three-dimensional Catalan numbers are a variant of the classical (bidimensional) Catalan numbers, that count, among other interesting objects, the standard Young tableaux of shape (n,n,n). In this paper, we present a structural bijection between two three-dimensional Catalan objects: 1234-avoiding up-down permutations, and a class of weighted Dyck paths.

A bijective enumeration of $3$-strip tableaux

Baryshnikov and Romik derived the combinatorial identities for the numbers of the $m$-strip tableaux. This generalized the classical André's theorem for the number of up-down permutations. They asked for a bijective proof for the enumeration of $3$-strip tableaux. In this paper we will provide such a bijective proof. First we count the $3$-strip tableaux by decomposition. Secondly we will apply this "decomposition" idea on the up-down permutations and down-up permutations to enumerate the $3$-strip tableaux bijectively.

Minimal Permutations and 2-Regular Skew Tableaux

Bouvel and Pergola introduced the notion of minimal permutations in the study of the whole genome duplication-random loss model for genome rearrangements. Let $\mathcal{F}_d(n)$ denote the set of minimal permutations of length $n$ with $d$ descents, and let $f_d(n)= |\mathcal{F}_d(n)|$. They derived that $f_{n-2}(n)=2^{n}-(n-1)n-2$ and $f_n(2n)=C_n$, where $C_n$ is the $n$-th Catalan number. Mansour and Yan proved that $f_{n+1}(2n+1)=2^{n-2}nC_{n+1}$. In this paper, we consider the problem of counting minimal permutations in $\mathcal{F}_d(n)$ with a prescribed set of ascents. We show that suc…

Linear and Circular Permutations with Limited Number of Repetitions

The problem of linear and circular permutations of n identical objects in m boxes, where a limit l is imposed on the number of objects in a box, is considered. In the linear case, where the boxes are arranged as a row, two methods of solution are described. The first uses the partition diagram that is modified with prescribed number of zeros, and the second applies a direct combinatorial approach. Subject to constraints set on the relation between n and l, results of the combinatorial approach are used to solve the problem where there are more than one type of objects. These solutions are then…

Bounds on Determinantal Complexity of Two Types of Generalized Permanents

We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing permutations by signed permutations, and the other by replacing permutations by surjective functions with preimages of prescribed sizes.

Gamma-positivity for a Refinement of Median Genocchi Numbers

We study the generating function of descent numbers for the permutations with descent pairs of prescribed parities, the distribution of which turns out to be a refinement of median Genocchi numbers. We prove the $γ$-positivity for the polynomial and derive the generating function for the $γ$-vectors, expressed in the form of continued fraction. We also come up with an artificial statistic that gives a $q$-analogue of the $γ$-positivity for the permutations with descents only allowed from an odd value to an odd value.

A new refinement of Euler numbers on counting alternating permutations

At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer the combinatorial question on some particular relation of Euler numbers proved by Heneghan-Petersen, Power series for up-down min-max permutations, College Math. Journal, Vol. 45, No. 2 (2014), 83-91.

The alternating run polynomials of permutations

In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family of q-alternating run polynomials. Furthermore, we introduce the definition of semi-gamma-positive polynomial and we show the semi-gamma-positivity of the alternating run polynomials of dual Stirling permutations. A connection between the up-down run polynomials of permutations and the alternating run polynomials of dual Stirling permutations is established.

On Degrees in the Hasse Diagram of the Strong Bruhat Order

For a permutation $π$ in the symmetric group $S_n$ let the {\it total degree} be its valency in the Hasse diagram of the strong Bruhat order on $S_n$, and let the {\it down degree} be the number of permutations which are covered by $π$ in the strong Bruhat order. The maxima of the total degree and the down degree and their values at a random permutation are computed. Proofs involve variants of a classical theorem of Turán from extremal graph theory.

交替排列、斯普林格数及避免扁平部分有序模式（POPs）的极大值与极小值统计量分布

Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs

该研究推导了交替排列上多种极值统计量的分布，推广了相关经典结果，验证了关于斯普林格数的猜想，并计数了避免特定扁平部分有序模式的交替排列。

Wilf equivalence relations for consecutive patterns

Two permutations $π$ and $τ$ are c-Wilf equivalent if, for each $n$, the number of permutations in $S_n$ avoiding $π$ as a consecutive pattern (i.e., in adjacent positions) is the same as the number of those avoiding $τ$. In addition, $π$ and $τ$ are strongly c-Wilf equivalent if, for each $n$ and $k$, the number of permutations in $S_n$ containing $k$ occurrences of $π$ as a consecutive pattern is the same as for $τ$. In this paper we introduce a third, more restrictive equivalence relation, defining $π$ and $τ$ to be super-strongly c-Wilf equivalent if the above condition holds for any set o…

Enumerating Permutations by their Run Structure

Motivated by a problem in quantum field theory, we study the up and down structure of circular and linear permutations. In particular, we count the length of the (alternating) runs of permutations by representing them as monomials and find that they can always be decomposed into so-called `atomic' permutations introduced in this work. This decomposition allows us to enumerate the (circular) permutations of a subset of the natural numbers by the length of their runs. Furthermore, we rederive, in an elementary way and using the methods developed here, a result due to Kitaev on the enumeration of…

Plethystic formulas for permutation enumeration

We prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formulas involve certain kinds of plethystic substitutions on quasisymmetric generating functions, and the permutation statistics we consider include the descent number, peak number, left peak number, and the number of up-down runs. We apply these results to cyclic permutations, involutions, and derangements, and more generally, to derive formulas for counting all permutations by the above statistics jointly…

按指定大小的相邻差异数对 k-ary words 进行计数

Counting k-ary words by number of adjacency differences of a prescribed size

本文研究了 k-ary words 上指定大小相邻差异数的分布问题，计算了相应统计量的常生成函数显式公式，并扩展至有限集合分块和绝对差情形，同时提供了相关递推式的组合证明。

A Grammatical Calculus for Peaks and Runs of Permutations

We develop a nonstandard approach to exploring polynomials associated with peaks and runs of permutations. With the aid of a context-free grammar, or a set of substitution rules, one can perform a symbolic calculus, and the computation often becomes rather simple. From a grammar it follows at once a system of ordinary differential equations for the generating functions. Utilizing a certain constant property, it is even possible to deduce a single equation for each generating function. To bring the grammar to a combinatorial setting, we find a labeling scheme for up-down runs of a permutation, …

Toppleable Permutations, Excedances and Acyclic Orientations

Recall that an excedance of a permutation $π$ is any position $i$ such that $π_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. One of our main results is that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Additionally, we show that the above is also the number of acyclic orientations with unique sink (AUSOs) of the complete bipa…

More bijections for Entringer and Arnold families

The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$ alternating permutations (resp. starting with $k$). In this paper, using bijections we first derive the counterparts in {\em André permutations} and {\em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then the counterparts in {\em signed André permutations} and {\em type $B$ increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$.

On a conjecture concerning the shuffle-compatible permutation statistics

The notion of shuffle-compatible permutation statistics was implicit in Stanley's work on P-partitions and was first explicitly studied by Gessel and Zhuang. The aim of this paper is to prove that the triple ${\rm (udr, pk, des)}$ is shuffle-compatible as conjectured by Gessel and Zhuang, where ${\rm udr}$ denotes the number of up-down runs, ${\rm pk}$ denotes the peak number, and ${\rm des}$ denotes the descent number. This is accomplished by establishing an ${\rm (udr, pk, des)}$-preserving bijection in the spirit of Baker-Jarvis and Sagan's bijective proofs of shuffle-compatibility property…

如何反弹你的因果排列

How to bounce your canon permutation

本文研究了一类新的回文下降多项式，通过Dyck路径和排列生成因果排列，并探讨了其多项式性质及组合解释。

A versatile combinatorial approach of studying products of long cycles in symmetric groups

In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which many surprisingly simple formulas have been obtained. Here we combinatorially enumerate the pairs of long cycles whose product has a given cycle-type and separates certain elements, extending several lines of studies, and we obtain general quantitative relations. As consequences, in a unified way, we recover a number of results expecting simple combinatorial pr…

涉及特定排列总数的多个$\ ilde{T}$值的关系

Relations of multiple $\tilde{T}$-values involving the total numbers of certain permutations

本文推广了Kaneko和Tsumura关于多个$\ ilde{T}$值的关系，给出了涉及Entringer数和第一类Dumont排列总数的新关系，并提供了这些排列总数的显式公式。

再探错位排列：统计量与模式回避

Desarrangements revisited: statistics and pattern avoidance

本文重新研究首次上升位置为偶数的错位排列，给出按多种排列统计量计数的生成函数公式，并完整枚举回避长度3模式的错位排列，为 Catalan、Fine、Jacobsthal 和 Fibonacci 数提供了新解释。