Bijections for hook pair identities
Christian Krattenthaler
简短，双射证明的身份多套的“钩对”（臂腿对）的细胞的某些图。这些钩对恒等式最初是由Regev发现的。
Short, bijective proofs of identities for multisets of `hook pairs' (arm-leg pairs) of the cells of certain diagrams are given. These hook pair identities were originally found by Regev.
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Valid hook configurations are combinatorial objects used to understand West's stack-sorting map. We extend existing bijections corresponding valid hook configurations to intervals in partial orders on Motzkin paths. To enumerate valid hook configurations on $312$-avoiding permutations, we build off of an existing bijection into a Motzkin poset and construct a bijection to certain well-studied closed lattice walks in the first quadrant. We use existing results about these lattice paths to show that valid hook configurations on $312$-avoiding permutations are not counted by a $D$-finite generati…
Dyck paths and a bijection for multisets of hook numbers
本文给出了Regev和Vershik关于某些斜Young图的两个钩数多集相等的一个猜想的双射证明。双射证明了一个结果，这是更强和更对称的比原来的猜想，通过建设涉及戴克路径，一种特殊类型的格路径。
Generalizations of Han's Hook Length Identities
韩最近发现了二叉树的新的钩长恒等式。本文将韩氏恒等式推广到二项式树族。此外，我们提出了一个双射证明的家庭的有序树的身份。
Based on the work of A. Vershik, we introduce two new combinatorial identities. We show how these identities can be used to prove a new hook-content identity. The main motivation for deriving this identity was a particular optimization problem in the field of quantum information processing.
We give relations between the joint distributions of multiple hook lengths and of frequencies and part sizes in partitions, extending prior work in this area. These results are discovered by investigating truncations of the Han/Nekrasov-Okounkov hooklength formula and of (k,j)-colored partitions, a unification of k-colored partitions and overpartitions. We establish the observed relations at the constant and linear terms for all n, and for j=2 in their quadratic term, with the associated hook/frequency identities. Further results of this type seem likely.
A new algorithm for inserting rim-hooks into reverse plane partitions is presented. The insertion is used to define a bijection between reverse plane partitions of a fixed shape and multi-sets of rim-hooks. In turn this yields a bijective proof of the fact that the generating function for reverse plane partitions of a fixed shape, which was first obtained by R. Stanley, factors into a product featuring the hook-lengths of this shape. Our bijection turns out to be equivalent to a map defined by I. Pak by different means, and can be related to the Hillman-Grassl correspondence and the Robinson-S…
Identities for the number of standard Young tableaux in some $(k,\ell)$ hooks
闭公式是已知的$S(k,0;n)$，数量标准的杨tableaux的大小$n$和最多$k$部分，其中$1\le k\le 5$。在这里，我们研究的模拟问题，$S(k,\ell;n)$，标准的年轻tableaux的大小$n$的数量包含在$(k,\ell)$钩。我们推导了$k+\ell\le 4$情形的一些公式。
Recently Féray, Goulden and Lascoux gave a proof of a new hook summation formula for unordered increasing trees by means of a generalization of the Prüfer code for labelled trees and posed the problem of finding a bijection between weighted increasing trees and Cayley trees. We give such a bijection, providing an answer to the problem posed by Féray, Goulden and Lascoux as well as showing a combinatorial connection to the theory of tree volumes defined by Kelmans. In addition we give two simple proofs of the hook summation formula. As an application we describe how the hook summation formula g…
We provide a bijective map from the partitions enumerated by the series side of the Rogers-Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers-Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of Gordon's theorem are also explored.
The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman--Grassl correspondence, respectively. The main new results are two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting rever…
Generalizations of wreath product identities via Garsia-Gessel bijections
本文利用 Garsia-Gessel 双射，将圈积集合上的四变量恒等式推广至任意正支配序，并简化了 Adin 和 Roichman 序下的六变量分布函数。
Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities
本文研究了由求和-乘积恒等式产生的两个划分类中t-hooks数量的不等式，通过生成函数和渐进行式证明了t=1,2时的不等式。
On Han's Hook Length Formulas for Trees
最近，Han得到了二叉树的两个钩长公式，并要求进行组合证明。其中一个公式已推广到k元树杨。萨根找到了杨延拓的概率证明。给出了杨的k元树公式和韩的二叉树公式的组合证明。我们的双射是基于k元树的结构与楼梯标号。
Recently, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. In this paper, we present a simple bijection that proves an equivalent recursive version of Naruse's result, in the same way that the celebrated hook-walk proof due to Green, Nijenhuis and Wilf gives a bijective (or probabilistic) proof of the hook-length formula for ordinary sh…
A few years ago, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes, both straight and shifted. The formula involves a sum over objects called \emph{excited diagrams}, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. Recently, the formula for skew straight shapes was proved via a simple bumping algorithm. The aim of this paper is to extend this result to skew shifted shapes. Since straight skew shapes are special cases of skew shifted shapes, this is …
Proofs of two conjectures of Berkovich and Dhar on hook lengths
本文证明了 Berkovich 和 Dhar 关于有界划分中钩长系数的两个猜想的更强形式，通过组合和解析方法建立奇数划分与不同划分中钩对数量的不等式。
This paper gives bijective proofs of some novel coinversion identities first discovered by Ayyer, Mandelshtam, and Martin (arxiv:2011.06117) as part of their proof of a new combinatorial formula for the modified Macdonald polynomials $\tilde{H}_μ$. Those authors used intricate algebraic manipulations of $q$-binomial coefficients to prove these identities, which imply the existence of certain bijections needed in their proof that their formula satisfies the axioms characterizing $\tilde{H}_μ$. They posed the open problem of constructing such bijections explicitly. We resolve that problem here.
Another involution principle-free bijective proof of Stanley's hook-content formula
另一个双射证明斯坦利的钩内容公式的生成函数的半标准tableaux一个给定的形状给出，不涉及对合原则的Garsia和米尔恩。这是一个合并的结果修改jeudetaquin的想法从作者以前的双射证明（“Stanley的hook-content公式的一个无对合原理的双射证明”，离散数学。理论。计算机科学，出现）和Novelli-Pak-Stoyanovskii双射（离散数学理论计算机科学1（1997），53-67），用于给定形状的标准杨氏表的钩公式。这个新算法也可以作为一个算法，用于随机生成一个给定的形状与有界条目的tableaux。对该算法进行适当变形，给出了在给定框内随机生成平面划分的算法。
Warnaar's bijection and colored partition identities, II
在我们的前一篇文章中，我们确定了一个统一的组合框架来看待大量的有色划分恒等式，并研究了对应于Schroeter，Russell和Ramanujan型素数阶例外模方程的五个恒等式。本文的目标是使用我们以前的工作的主双射组合显示几个新的和高度非平凡的有色划分身份。最后，我们列出了一些与证明相同类型的进一步有趣的恒等式。
The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications
本文致力于推导欧拉乘积的幂的展开公式的划分钩长，发现Nekrasov和Okounkov在他们的研究的Seiberg-Witten理论。基于t-核的一个新性质给出了一个改进，并利用Macdonald恒等式给出了一个初等证明.我们还得到了一个扩展，通过增加两个参数，这似乎是一个离散的麦克唐纳身份之间的插值和生成函数的t-核心。几个应用程序派生，包括“标记钩公式”。
A bijective proof of the hook-length formula for shifted standard tableaux
本文基于修正的jeu de taquin和Novelli，Pak和Stoyanovskii关于普通标准表的钩长公式的双射证明思想，给出了固定形状移位标准表的钩长公式的双射证明.在Novelli、Pak和Stoyanovskii的证明中，他们定义了费勒斯图的任意填充与整数$1,2,...,n$和标准tableaux和hook tabloids对之间的双射。在我们的移动版本的算法的映射从一组任意填充的移动费勒斯图到一组移动的标准tableaux是类似的建设Novelli，Pak和Stoyanovskii，然而，与他们的算法不同，我们被迫使用'rowwise'总顺序的细胞在移动费勒斯图，而不是'columnwise'总顺序的算法中的基本顺序。不幸的是，在移位的情况下，移位的钩子小报的构造更加复杂。作为一个副作用，我们得到了一个简单的随机算法生成一个给定形状的移位标准tableaux，它产生每一个这样的tableaux同样可能。
Composition sum identities related to the distribution of coordinate values in a discrete simplex
利用谱剩余的参数化，递归定义的序列，我们开发了一个通用的方法产生的身份组成和。具体的结果是通过集中在系数序列的解决方案的一阶和二阶，普通，线性微分方程。关于第一类，相应的恒等式相当于标号计数的指数公式的证明。第二类恒等式可以用来建立有界有序整数元组单形的某些几何性质。我们提出了三个定理，支持的结论，这样一个顺序单形的内部尺寸，在一定意义下，比外部尺寸更充足。此外，我们给出了一个代数证明的双射之间的两个家庭的子集的顺序单纯形，并询问建立这种双射的组合，而不是代数方法的可能性。
In a work of 1995, Alladi, Andrews, and Gordon provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the tools the authors employed, we obtain new partition identities by identifying further sets of partitions that can be explicitly put into a one-to-one correspondence by the method described in the 1995 paper. As a further result, although of a different nature, we obtain an analytical identity of Rogers-Ramanujan type, involving generating functions, for a cla…
Bijections for generalized Wilf equivalences
本研究提出从组合恒等式的容斥证明出发、用递归减法构造直接双射的方法，将其用于逆序序列连续模式的广义Wilf等价研究，给出了已有结论的直接双射证明及更强模式关系的新双射证明。
A combinatorial proof of the Rogers-Ramanujan and Schur identities
本文利用Dyson秩的一个新推广的两个对称性，给出了第一个Rogers-Ramanujan恒等式的组合证明。这些对称是由直接双射建立的。
A new proof of an Eğecioğlu--Remmel inverse Kostka matrix problem via a Garsia--Milne involution involving Sym and NSym
本文通过引入隧道钩覆盖和反向双射，给出了Eğecioğlu--Remmel逆Kostka矩阵问题的全新证明，不同于Loehr和Mendes的结果。
Bijective Proofs of Gould's and Rothe's Identities
我们首先给出了一个双射证明古尔德的身份在模型的二进制字。然后利用一个双射从Gould恒等式再次推出Rothe恒等式，这也是$q$ -Chu-Vandermonde公式的一个双和推广.
Warnaar's bijection and colored partition identities, I
我们提供了一个一般的和统一的组合框架，一些有色划分的身份，其中包括五个，最近证明的分析B。Berndt，它们对应于H.施罗特河Russell和S.拉马努金我们的方法推广了S。Kim给出了这五个恒等式中两个的双射证明，即模7恒等式（也称为Farkas-Kra恒等式）和模3恒等式。作为我们的方法的结果，我们确定双射证明也为两个高度非平凡的身份模5和11，从而留下开放的组合只有一个模23。
Combinatorial approach to Andrews-Gordon and Bressoud type identities
本研究受Warnaar工作启发，通过构造整数分拆相关集合间显式保权重保长度双射，给出Andrews-Gordon与Bressoud恒等式和式侧的组合解释，还发现并证明了奇偶条件相反的Kurşungöz恒等式的q级数版本，同时证明了一批该类已知及新恒等式。
Combinatorial proof of identities involving partitions with distinct even parts and 4-regular partitions
针对Andrews与Bachraoui用基本超几何级数证明并寻求组合解释的互异偶数部分分拆与4-正则分拆恒等式，本文通过构造双射给出组合证明。
Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams
本文研究双重Grothendieck多项式的Cauchy恒等式，提出pipe dream矫正算法构建pipe dreams间的权重保持双射，并基于新流算子揭示其对称性，最终恢复对偶RSK对应变体。
Weighted 2-Motzkin Paths
本文的动机是最近提出的两个问题Coker的组合身份有关的Narayana多项式和Catalan数。我们发现，陈，Deutsch和Elizalde的双射可以用来提供组合解释的身份的Coker时，它适用于加权平面树。为了介绍我们的组合对应关系，我们提供了一个描述的双射陈，Deutsch和Elizalde在一个稍微不同的方式在一个直接建设的形式，从平面树的2-Motzkin路径没有中间步骤涉及的戴克路径。
We provide new Schmidt-type results through an investigation of two bijections, which are results involving partitions with parts counted only at given indices. Mork's bijection, the first of these, was originally given as a proof of Schmidt's theorem. We show that a version of Sylvester's bijection is equivalent to Mork's bijection applied to 2-modular diagrams, which implies refinements of existing results and new generating function identities. We then develop a bijection based on an idea appearing in a recent paper of Andrews and Keith, that places partitions counted at the indices $r$, $t…
Path counting and random matrix theory
我们建立了三个身份涉及戴克路径和交替Motzkin路径，其证明是基于同一双射的变种。我们用半直线上的闭随机游动来解释这些恒等式。我们解释如何产生这些身份的组合解释的某些属性的$β$ -厄米和$β$ -拉盖尔合奏的随机矩阵理论。最后，我们提出了其他两个身份以同样的方式获得，其中找到组合证明是一个开放的问题。
Combinatorial proofs of several partition identities of Andrews and Merca
本文基于Fu和Tang的双射，为Andrews和Merca的若干分拆恒等式提供了组合证明，并引入两种分拆权重扩展了其中一个恒等式。
Combinatorial Proofs for Two-Color Partition Identities Involving Overpartitions and Minimal Excludants
研究涉及超分割和最小排除数的双色分拆恒等式，通过构造显式保权双射和奇偶反转对合给出组合证明，还给出相关超分割精细化的新组合证明。
We present combinatorial bijections and identities between certain skew Young tableaux, Dyck paths, triangulations, and dissections.
A Unified Combinatorial Approach to Several Poincare Series Identities
Mendes最近证明了一个恒等式，它将等变多项式映射空间的Poincaré级数从$\mathbb{R}^{n}$简化为$Sym^{2}(\mathbb{R}^{n})$的一个子表示.我们将展示如何证明这个身份使用一个相当简单的整数分区双射。首先，我们从表示论给出了一个相似的著名恒等式的双射证明。然后，我们表明，这种双射可以推广到证明其他庞加莱系列的身份，包括一个版本的身份由门德斯以及它的改进。
A short bijective proof of dimension identities of Erickson and Hunziker
本文针对 Erickson 和 Hunziker 关于 $(-m)$-不对称分拆的维度恒等式，提供了一个简洁的双射证明，替代了原有的钩子含量公式运算方法。
Growth diagram proofs for the Littlewood identities
本文利用 Fomin 增长图与新提出的投影恒等式，将算子交换关系双射化，构造出推广经典情形的 Littlewood 恒等式无穷双射族。
Generalized Cauchy identities, trees and multidimensional Brownian motions. Part I: bijective proof of generalized Cauchy identities
在本系列文章中，我们研究柯西等式的多维推广的组合学与多维布朗运动和布朗桥等连续对象之间的联系。在第一部分的系列，我们提出了一个双射证明多维推广的柯西身份。我们的双射使用定向平面树配备了一些线性订单。
Let $\mathcal O_r(n)$ be the set of $r$-regular partitions of $n$, $\mathcal D_r(n)$ the set of partitions of $n$ with parts repeated at most $r-1$ times, $\mathcal O_{1,r}(n)$ the set of partitions with exactly one part (possibly repeated) divisible by $r$, and let $\mathcal D_{1,r}(n)$ be the set of partitions in which exactly one part appears at least $r$ times. If $E_{r, t}(n)$ is the excess in the number of parts congruent to $t \pmod r$ in all partitions in $\mathcal O_r(n)$ over the number of different parts appearing at least $t$ times in all partitions in $\mathcal D_r(n)$, then $E_{r…
Affine partitions and affine Grassmannians
给出了仿射外尔群模其外尔群的商中元素与某些色划分之间的一个双射.根据博特公式，这些有色分区产生了一些分区恒等式。在某些类型中，这些身份以前曾出现在Bousquet-Melou-Eriksson，Eriksson-Eriksson和Reiner的工作中。在其他类型的身份似乎是新的。对于类型$A_{n}$，仿射着色划分形成具有$n+1$ -核心划分和$n$ -有界划分的双射中的另一族组合对象。我们的主要应用是仿射Grassmannian的仿射分区和推广的Young格，细化弱序，是一个子集的Bruhat秩序的合理光滑的舒伯特品种的特点。有几个证明是计算机辅助的。
Some beautiful identities involving hook contents of Young diagrams have been found in the field of quantum information processing, along with a combinatorial proof. We here give a representation theoretic proof of these identities and a number of generalizations. Our proof is based on trace identities for elements belonging to a class of permutation centralizer algebras. These algebras have been found to underlie the combinatorics of composite gauge invariant operators in quantum field theory, with applications in the AdS/CFT correspondence. Based on these algebras, we discuss some analogies …
Transition Matrices between Plethystic Bases of Polysymmetric Functions via Bijective Methods
本文通过双射和符号反转对合，为多对称函数代数中十二对Plethystic基之间的转移矩阵元素提供组合解释，并给出六个相关OEIS序列的新解释。
How to decompose a permutation into a pair of labeled Dyck paths by playing a game
我们给出了长度为2n的置换和某些Dyck路径对与标签的下降步骤之间的双射。双射来自一个游戏，在这个游戏中，两个玩家从一组2n个项目中交替选择：排列编码了玩家对项目的偏爱序，而戴克路径编码了在最佳游戏下选择项目的顺序。我们列举排列的某些统计数据，AA反转和BB反转，这在游戏方面有自然的解释。给出了经典恒等式如\sum_p \prod_{i=1}^n q^{h_i-1} [h_i]_q = [1]_q [3]_q. [2n-1]_q，其中和是在所有长度为2n的Dyck路p上，h_i是p的下台阶的高度。
In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra $A_{n-1}^{(1)}.$ In this first paper, we focus on combinatorial aspects. We give a $n^2$-coloured generalisation of Primc's identity by constructing a $n^2 \times n^2$ matrix of difference conditions, Primc's original identities corresponding to $n=2$ and $n=3$. While most coloured partition identities in the literature connect partitions with difference conditions to partitio…
We prove combinatorially some identities related to Euler's partition identity (the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts). They were conjectured by Beck and proved by Andrews via generating functions. Let $a(n)$ be the number of partitions of $n$ such that the set of even parts has exactly one element, $b(n)$ be the difference between the number of parts in all odd partitions of $n$ and the number of parts in all distinct partitions of $n$, and $c(n)$ be the number of partitions of $n$ in which exactly one part is repeated. Then,…
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