Dan Bernstein, Amitai Regev
The Foata bijection $Φ: S_n \to S_n$ is extended to the bijections $Ψ: A_{n+1} \to A_{n+1}$ and $Ψ_q : S_{n+q-1} \to S_{n+q-1}$, where S_m, A_m are the symmetric and the alternating groups. These bijections imply bijective proofs for recent equidistribution theorems, by Regev and Roichman, for A_{n+1} and for S_{n+q-1}.
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We present bijections enumerating (k,m)-trees, k-gon trees, edge labelled (2,1)-trees, and other tree-like structures. Our constructions are based on Foata's (1971) bijection for cycle-free functions, which is simplified here.
In their study of the densest jammed configurations for theater models, Krapivsky and Luck observe that two classes of permutations have the same cardinalities and ask for a bijection between them. In this note we show that the Foata correspondence provides the desired bijection.
Continued fractions using a Laguerre digraph interpretation of the Foata--Zeilberger bijection and its variants
本文用 Laguerre 有向图解释 Foata--Zeilberger 双射以计算排列中的环，从而得到了枚举排列和 D-排列的多变量多项式的 Jacobi 型、Thron 型和 Stieltjes 型连分数，并证明了 Sokal、Zeng 等人提出的连分数猜想。
Let $A_n\subseteq S_n$ denote the alternating and the symmetric groups on $1,...,n$. MacMahaon's theorem, about the equi-distribution of the length and the major indices in $S_n$, has received far reaching refinements and generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and followers. Our main goal is to find analogous statistics and identities for the alternating group $A_{n}$. A new statistic for $S_n$, {\it the delent number}, is introduced. This new statistic is involved with new $S_n$ equi-distribution identities, refining some of the results of Foata-Schutzenberger…
We construct a refined bijection $ϕ$ between alternating permutations and 0-1-2 increasing trees with degree at most 2. It satisfies that the first element of alternating permutation $π$ is equal to the first vertex in $ϕ(π)$ in the postorder.
Partial Alternating Sign Matrix Bijections and Dynamics
本文研究部分交替符号矩阵，建立其与多种组合对象的双射，并借助Wieland回转、toggles与序理想，证明偏序集上的rowmotion和部分完全填充回路构型上的回转等变双射。
We construct weight-preserving bijections between column strict shifted plane partitions with one row and alternating sign trapezoids with exactly one column in the left half that sums to $1$. Amongst other things, they relate the number of $-1$s in the alternating sign trapezoids to certain elements in the column strict shifted plane partitions that generalise the notion of special parts in descending plane partitions. The advantage of these bijections is that they include configurations with $-1$s, which is a feature that many of the bijections in the realm of alternating sign arrays lack.
For an integer $n\geq 1$, we display a poset $Φ_n$ whose maximal chains are in bijection with the $n\times n$ alternating sign matrices. The Hasse diagram $\widehat Φ_n$ is obtained from the $n$-cube by adding some edges. We show that the dihedral group $D_{2n}$ acts on $\widehat Φ_n$ as a group of automorphisms.
We define a bijection that transforms an alternating sign matrix A with one -1 into a pair (N,E) where N is a (so called) ``neutral'' alternating sign matrix (with one -1) and E is an integer. The bijection preserves the classical parameters of Mills, Robbins and Rumsey as well as three new parameters (including E). It translates vertical reflection of A into vertical reflection of N. A hidden symmetry allows the interchange of E with one of the remaining two new parameters. A second bijection transforms (N,E) into a configuration of lattice paths called ``mixed configuration''.
We study the problem of counting alternating permutations avoiding collections of permutation patterns including 132. We construct a bijection between the set S_n(132) of 132-avoiding permutations and the set A_{2n + 1}(132) of alternating, 132-avoiding permutations. For every set p_1, ..., p_k of patterns and certain related patterns q_1, ..., q_k, our bijection restricts to a bijection between S_n(132, p_1, ..., p_k), the set of permutations avoiding 132 and the p_i, and A_{2n + 1}(132, q_1, ..., q_k), the set of alternating permutations avoiding 132 and the q_i. This reduces the enumeration…
We settle a question of Bressoud concerning the existence of an explicit bijection from a class of oriented square-ice graphs to a class of tournaments. We give an algorithm constructing such a bijection.
André proved that the number of alternating permutations on $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. A refinement of André's result was given by Entringer, who proved that counting alternating permutations according to the first element gives rise to Seidel's triangle $(E_{n,k})$ for computing the Euler numbers. In a series of papers, using generating function method and induction, Poupard gave several further combinatorial interpretations for $E_{n,k}$ both in alternating permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretat…
The characterization of an Alternating Sign Matrix using a triplet
本文通过定义三元组并建立交错符号矩阵、六顶点模型与高度函数之间的双射，刻画了交错符号矩阵并揭示了其更多性质。
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 $(S_{n,k})$.
Bijections around Springer numbers
本文构造一系列自然双射，将Snakes、带标签选票路径、rc-不变交替排列和弱递增3维排列这四个由Springer数计数的组合对象联系起来。
We exhibit a bijection between Dyck paths and alternating sign matrices which are determined by their antidiagonal sums.
A sub-problem of the open problem of finding an explicit bijection between alternating sign matrices and totally symmetric self-complementary plane partitions consists in finding an explicit bijection between so-called $(n,k)$ Gog trapezoids and $(n,k)$ Magog trapezoids. A quite involved bijection was found by Biane and Cheballah in the case $k=2$. We give here a simpler bijection for this case.
We give a bijective proof of the MacMahon-type equidistribution over the group of signed even permutations C_2 \wr A_n that was stated in [Bernstein. Electron. J. Combin. 11 (2004) 83]. This is done by generalizing the bijection that was introduced in the bijective proof of the equidistribution over the alternating group A_n in [Bernstein and Regev. Sem. Lothar. Combin. 53 (2005) B53b].
We describe an approach to finding a bijection between Alternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions, which is based on the Schutzenberger involution. In particular we give an explicit bijection between Gog and Magog trapezoids with two diagonals.
Descending plane partitions, alternating sign matrices, and totally symmetric self-complementary plane partitions are equinumerous combinatorial sets for which no explicit bijection is known. In this paper, we isolate a subset of descending plane partitions counted by the Catalan numbers. The proof follows by constructing a generating tree on these descending plane partitions that has the same structure as the generating tree for 231-avoiding permutations. We hope this result will provide insight on the search for a bijection with alternating sign matrices and/or totally symmetric self-complem…
In 1997 Clarke et al. studied a $q$-analogue of Euler's difference table for $n!$ using a key bijection $Ψ$ on symmetric groups. In this paper we extend their results to the wreath product of a cyclic group with the symmetric group. In particular we obtain a new mahonian statistic \emph{fmaf} on wreath products. We also show that Foata and Han's two recent transformations on the symmetric groups provide indeed a factorization of $Ψ$.
A pipe dream perspective on totally symmetric self-complementary plane partitions
本文通过管道梦视角刻画完全对称自补平面划分，建立其与交替符号矩阵的双射，并保持偏序结构。
New bijective proofs pertaining to alternating sign matrices
本文针对 ASM-DPP 双射问题，利用相容性概念构造交替符号矩阵与移位 Gelfand-Tsetlin 模式间更自然的带符号双射，并为含 n+3 个统计量的 ASM 推广的精细化枚举提供双射证明。
Classical and vincular patterns of length three in generalized alternating permutations
该研究计数广义交替排列中避免长度为3的经典与vincular模式的元素个数，通过双射方法得到相关闭式结果，涉及完全k叉树森林、RSK等模型及多种模式类的计数。
Pattern avoidance in alternating sign rectangles I: Extended avoidance
本文为交替符号矩形引入扩展模式避免，对长度三的四个模式建立递推枚举系统，并给出312的猜想证明及213到小Schröder路径的双射。
Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank.
Totally symmetric self-complementary plane partition matrices and related polytopes
本文建立从全对称自补平面分拆到magog矩阵的双射，研究其计数性质，并分析相关多胞体的极点、不等式、顶点与面。
Alternating sign matrices with reflective symmetry and plane partitions: $n+3$ pairs of equivalent statistics and a Cauchy-type identity
本文推广了垂直对称交错符号矩阵与带洞六边形环面平铺的等数性，引入 $n+3$ 个参数并证明多变量生成函数一致，还给出辛特征展开，为构造双射和柯西型恒等式提供关键步骤。
A local bijection between alternating sign matrices and descending plane partitions and a Striker--Fulmek-type $q$-statistic
本文构造了 Behrend--Di Francesco--Zinn-Justin 细化纤维中交错符号矩阵与下降平面分拆之间的局部双射，通过移动边界线实现，并引入了一个保持统计量的 Striker--Fulmek 型 $q$-统计量。
It was shown by Kuperberg that the partition function of the square-ice model related to half-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with half-turn symmetric alternating-sign matrices of odd order. The partition function of the model is expressed via the above mentioned factors. The contributions to the partition function of the states corresponding to the alternating-sign matrices having 1 or -1 as the central entry are found and the related enumerations are obtained.
In 2005 J.L. Waldspurger proved the following theorem: given a finite real reflection group $W$, the closed positive root cone is tiled by the images of the open weight cone under the action of the linear transformations $id-w$. Shortly thereafter E. Meinrencken extended the result to affine Weyl groups. P.V. Bibikov and V.S. Zhgoon then gave a uniform proof for a discrete reflection group acting on a simply-connected space of constant curvature. In this paper we show that the Waldspurger and Meinrenken theorems of type A give a new perspective on the combinatorics of the symmetric group. In p…
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbin…
Polynomial Expressions for the Dimensions of the Representations of Symmetric Groups and Restricted Standard Young Tableaux
本文研究对称群表示维数多项式的二项式展开，证明其系数以分拆长度为界，并利用受限标准杨表给出组合解释，同时构造双射证明该计数与所选连续数集合无关。
Diagonally symmetric alternating sign matrices
本文研究对角对称交替符号矩阵（DSASMs）的计数，得到其数量及相关统计量生成函数的Pfaffian公式，引入与DSASM一一对应的六顶点模型变体并研究相关主题。
The $q$-analogues of $γ$-positivity of Eulerian polynomials via group actions
本文通过递增二叉树模型为欧拉多项式$\gamma$-展开的$q$-模拟系数提供了组合解释，解决了Han、Jouhet和Zeng提出的问题，并证明了正性猜想。
Given a list of $n$ cells $L=[(p_1,q_1),...,(p_n, q_n)]$ where $p_i, q_i\in \textbf{Z}_{\ge 0}$, we let $Δ_L=\det |{(p_j!)^{-1}(q_j!)^{-1} x^{p_j}_iy^{q_j}_i} |$. The space of diagonally alternating polynomials is spanned by $\{Δ_L\}$ where $L$ varies among all lists with $n$ cells. For $a>0$, the operators $E_a=\sum_{i=1}^{n} y_i\partial_{x_i}^a$ act on diagonally alternating polynomials and Haiman has shown that the space $A_n$ of diagonally alternating harmonic polynomials is spanned by $\{E_λΔ_n\}$. For $t=(t_m,...,t_1)\in \textbf{Z}_{> 0}^m$ with $t_m>...>t_1>0$, we consider here the oper…
Let $Ψ$ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension $\ge 3$ over a field. Let $H$ be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_Ψ$ of the set $Ψ$. Suppose that $H$ contains the projective group and an arbitrary self-bijection of $Ψ$ transforming a triple of collinear points to a non-collinear triple. It is well-known from \cite{KantorMcDonough} that if $Ψ$ is finite then $H$ contains the alternating subgroup $\mathfrak{A}_Ψ$ of $\mathfrak{S}_Ψ$. We show in Theorem \ref{density} …
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