Dominique Foata, Doron Zeilberger
A generalization of the classical statistics ``maj'' and ``inv'' (the major index and number of inversions) on words is introduced, parameterized by arbitrary graphs on the underlying alphabet. The question of characterizing those graphs that lead to equi-distributed "inv" and "maj" is posed and answered.
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Foata and Zeilberger defined the graphical major index, $\mathrm{maj}'_U$, and the graphical inversion index, $\mathrm{inv}'_U$, for words. These statistics are a generalization of the classical permutation statistics $\mathrm{maj}$ and $\mathrm{inv}$ indexed by directed graphs $U$. They showed that $\mathrm{maj}'_U$ and $\mathrm{inv}'_U$ are equidistributed over all rearrangement classes if and only if $U$ is bipartitional. In this paper we strengthen their result by showing that if $\mathrm{maj}'_U$ and $\mathrm{inv}'_U$ are equidistributed on a single rearrangement class then $U$ is essenti…
An operation on species corresponding to the inner plethysm of their associated cycle index series is constructed. This operation, the inner plethysm of species, is generalized to n-sorted species. Polynomial maps on species are studied and used to extend inner plethysm and other operations to virtual species. Finally, inner plethysm and other operations on species are applied to various problems in graph theory. In particular, regular graphs, and digraphs in which every vertex has outdegree k, are enumerated.
We present a short proof of MacMahon's classic result that the number of permutations with $k$ inversions equals the number whose major index (sum of positions at which descents occur) is $k$
Inspired by the $k$-inversion statistic for LLT polynomials, we define a $k$-inversion number and $k$-descent set for words. Using these, we define a new statistic on words, called the $k$-major index, that interpolates between the major index and inversion number. We give a bijective proof that the $k$-major index is equidistributed with the major index, generalizing a classical result of Foata and rediscovering a result of Kadell. Inspired by recent work of Haglund and Stevens, we give a partial extension of these definitions and constructions to standard Young tableaux. Finally, we give an …
We introduce the Major MacMahon map and show how this map interacts with the pyramid and bipyramid operators. When the Major MacMahon map is applied to the ab-index of a simplicial poset, it yields the q-analogue of n! times the h-polynomial of the poset. Applying the map to the Boolean algebra gives the distribution of the major index on the symmetric group, a seminal result due to MacMahon. Similarly, when applied to the cross-polytope we obtain the distribution of one of the major indexes on signed permutations due to Reiner.
Bonin, Shapiro and Simion (1993) gave two formulas on the distribution of major index for Schröder paths, and proved their result for the case $E<D<N$. In this short note, we correct an error in their proof, and give a complete proof for all cases.
In [S. Effler, F. Ruskey, A CAT algorithm for listing permutations with a given number of inversions, {\it I.P.L.}, 86/2 (2003)] the authors give an algorithm, which appears to be CAT, for generating permutations with a given major index. In the present paper we give a new algorithm for generating a Gray code for subexcedant sequences. We show that this algorithm is CAT and derive it into a CAT generating algorithm for permutations with a given major index.
Major Index on Catalan Combinatorics
本文研究了 $r$-Dyck 路径等 Catalan 组合对象上的逆序数和 major index 统计量，并证明了它们与 Dyck 路径上的对应统计量具有等分布性。
We extend the family of statistics maj_d, introduced for permutations by Kadell (1985), to standard Young tableaux. At one extreme, we have the traditional Major index statistic maj_1 for tableaux. At the other end, maj_n = inv, the inversion statistic introduced by Haglund and Stevens (2006). This is answers a question of Assaf (2008), who defined maj_2 and maj_3 for tableaux.
Super Major Index Cyclic Sieving
本文研究了矩形形状的带符号标准 Young 表的循环筛分现象，结合 Schützenberger 提升和符号循环移位，证明其由超主指数生成函数描述，扩展了 Rhoades 的经典结果，并推广到任意非矩形形状。
In a recent paper, Baxter and Zeilberger show that the two most important Mahonian statistics, the inversion number and the major index, are asymptotically independently normally distributed on permutations. In another recent paper, Canfield, Janson and Zeilberger prove the result, already known to statisticians, that the Mahonian distribution is asymptotically normal on words. This leaves one question unanswered: What, asymptotically, is the joint distribution of the inversion number and the major index on words? We answer this question by establishing convergence to a bivariate normal distri…
We introduce a statistic $\pmaj$ on partitions of $[n]=\{1,2,..., n\}$, and show that it is equidistributed with the number of 2-crossings over partitions of $[n]$ with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inversion number and major index.
For a permutation $π$ the major index of $π$ is the sum of all indices $i$ such that $π_i > π_{i+1}$. It is well known that the major index is equidistributed with the number of inversions over all permutations of length $n$. In this paper, we study the distribution of the major index over pattern-avoiding permutations of length $n$. We focus on the number $M_n^m(Π)$ of permutations of length $n$ with major index $m$ and avoiding the set of patterns $Π$. First we are able to show that for a singleton set $Π= \{σ\}$ other than some trivial cases, the values $M_n^m(Π)$ are monotonic in the sense…
We propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M, s; A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north-east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classical equidistribution results for the permutation statistics inv a…
Major index on involutions
本文确定了对称群中对合共轭类上主指标的范围，并证明除无固定点情形外所有值均可取到，无固定点时仅缺失两端各一个值。
In this paper we provide a bijective proof of a theorem of Garsia and Gessel describing the generating function of the major index over the set of all permutations of [n]={1,...,n} which are shuffles of given disjoint ordered sequences whose union is [n]. Two special cases are singled out: If the single element j is inserted into any permutation P of the remaining elements of [n], then the theorem states that inserting j into P increases the major index of P by some element of {0,1,...,n-1}, the increase determined uniquely by the index of insertion. We provide a direct proof of this fact usin…
We present refined enumeration formulas for lattice paths in $\mathbb{Z}^2$ with two kinds of steps, by keeping track of the number of descents (i.e., turns in a given direction), the major index (i.e., the sum of the positions of the descents), and the number of crossings. One formula considers crossings between a path and a fixed line; the other considers crossings between two paths. Building on the first paper of the series, which used lattice path bijections to give the enumeration with respect to major index and crossings, we obtain a refinement that keeps track of the number of descents.…
We consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We use cumulants to classify all possible limit laws for any sequence of such shapes in terms of a simple auxiliary statistic, aft, generalizing earlier results of Canfield--Janson--Zeilberger, Chen--Wang--Wang, and others. These results can be interpreted as giving a very precise description of the distribution of irreducible representations in different degrees of coinvariant algebras of certain complex reflection groups. We conclude with some conjectures concerning unimo…
We solve two problems regarding the enumeration of lattice paths in $\mathbb{Z}^2$ with steps $(1,1)$ and $(1,-1)$ with respect to the major index, defined as the sum of the positions of the valleys, and to the number of certain crossings. The first problem considers crossings of a single path with a fixed horizontal line. The second one counts pairs of paths with respect to the number of times they cross each other. Our proofs introduce lattice path bijections with convenient visual descriptions, and the answers are given by remarkably simple formulas involving $q$-binomial coefficients.
For a fixed positive integer n, let S_n denote the symmetric group of n! permutations on n symbols, and let maj(sigma) denote the major index of a permutation sigma. For positive integers k<m not greater than n and non-negative integers i and j, we give enumerative formulas for the cardinality of the set of permutations sigma in S_n with maj(sigma) congruent to i mod k and maj(sigma^(-1)) congruent to j mod m. When m divides n-1 and k divides n, we show that for all i,j, this cardinality equals (n!)/(km).
The study of permutation and partition statistics is a classical topic in enumerative combinatorics. The major index statistic on permutations was introduced a century ago by Percy MacMahon in his seminal works. In this extended abstract, we study the well-known generalization of the major index to standard Young tableaux. We present several new results. In one direction, we introduce and study two partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. The existence of such ranked poset structures allows us t…
We generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index. We also define similar variants of this map, that regards alternative models for the modified Macdonald polynomials at $t=0$, thus partially answer a question by J. Haglund. These maps imply certain uniqueness property regarding inversion-- and coinversion-free fillings, which allows us to generalize the notion of charge to a non-symmetric setting, thus answering a question by A. Lascoux. The analogous question in the symmetric setting proves a conjecture by K. Ne…
Crystal skeleton polynomials with major index, charge and depth
本文提出晶体骨架多项式，用于改进标准Young表的计数及与Schur函数展开的关联，同时深化对主指数、电荷等参数的理解。
We use recurrences (alias difference equations) to prove the longstanding conjecture that the two most important permutation statistics, namely the number of inversions and the major index, are asymptotically joint-independently-normal. We even derive more-precise-than needed asymptotic formulas for the (normalized) mixed moments. This is the fully revised second edition, incorportating the many insightful comments of nine conscientious NON-anonymous referees listed under the authors' names. This article is exclusively published in the on-line journal "Personal Journal of Shalosh B. Ekhad and …
Closed forms for $f_{λ,i} (q) := \sum_{τ\in SYT(λ) : des(τ) = i} q^{maj(τ)}$, the distribution of the major index over standard Young tableaux of given shapes and specified number of descents, are established for a large collection of $λ$ and $i$. Of particular interest is the family that gives a positive answer to a question of Sagan and collaborators. All formulas established in the paper are unimodal, most by a result of Kirillov and Reshetikhin. Many can be identified as specializations of Schur functions via the Jacobi-Trudi identities. If the number of arguments is sufficiently large, it…
An open conjecture in pattern avoidance theory is that the distribution of the major index among 321-avoiding permutations is distributed unimodally. We construct a formula for this distribution, and in the case of 2 descents prove unimodality, with unimodality for 3 through 5 descents likely being little more complicated. The formula refines the $q$-analogue of the Frame-Robinson-Thrall hooklength formula for two-rowed partitions, and in the latter part of the paper we discuss another theorem of the same type, and further exploration toward this question. We also give observations on the anal…
In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been com…
A signed labeled forest is defined as a (plane) forest labeled by {1,2,..., n} along with minus signs associated to some vertices. Signed labeled forests can be viewed as an extension of signed permutations. We define the inversion number, the flag major index and the R-major index on signed labeled forests. They can be considered as type B analogues of the indices for labeled forests introduced by Bjorner and Wachs. The flag major index for signed labeled forests is based on the flag major index on signed permutations introduced by Adin and Roichman, whereas the R-major index for signed label…
Descents and flag major index on conjugacy classes of colored permutation groups without short cycles
本文研究着色置换群共轭类上的下降与旗标主指标，证明无短循环时矩的一致性并建立渐近正态性，推广了 Fulman 关于对称群的结果。
We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index $r$ mod $n$, for all $r$. Our result generalizes the $r=1$ case due essentially to (1974) and proves a recent conjecture due to Sundaram (2016) for the $r=0$ case. A byproduct of the proof is an asymptotic equidistribution result for "almost all" shapes. The proof uses a representation-theoretic formula involving Ramanujan sums and normalized symmetric group character estimates. Further estimates involving "opposite" hook lengths are given which are well-adapted …
Super major index and Thrall's problem
本文推广Thrall问题至自由Lie超代数，证明Brandt公式、Klyachko对偶识别及Kraskiewicz-Weyman分解公式，并引入超表主指标新版本，对应Macdonald的q,t-钩子公式。
The generating functions of the major index and of the flag-major index, with each of the one-dimensional characters over the symmetric and hyperoctahedral group, respectively, have simple product formulas. In this paper, we give a factorial-type formula for the generating function of the D-major index with sign over the Weyl groups of type D. This completes a picture which is now known for all the classical Weyl groups.
We construct a bijection between 231-avoiding permutations and Dyck paths that sends the sum of the major index and the inverse major index of a 231-avoiding permutation to the major index of the corresponding Dyck path. Furthermore, we relate this bijection to others and exhibit a bistatistic on 231-avoiding permutations which is related to the q,t-Catalan numbers.
A specialisation of a transformation formula for multi-dimensional elliptic hypergeometric series is used to provide compact, non-determinantal formulae for the generating function with respect to the major index of standard Young tableaux of skew shapes of the form "staircase minus rectangle".
A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type $B$, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.
Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and a polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content. There is a natural notion of refinement for many CSP's. We formulate and prove a refinement, with respect to the major index statistic, of this CSP on words of fixed content by also fixing the cyclic descent type. The argu…
We give a new description of the flag major index, introduced by Adin and Roichman, by using a major index defined by Reiner. This allows us to establish a connection between an identity of Reiner and some more recent results due to Chow and Gessel. Furthermore we generalize the main identity of Chow and Gessel by computing the four-variate generating series of descents, major index, length, and number of negative entries over Coxeter groups of type $B$ and $D$.
Five results on maximizing topological indices in graphs
本文证明了关于图形指数的五个结果，包括确定极值图、推广已有工作、提供替代证明，并解决两个猜想，涉及维纳指数、外部维纳指数、加权塞格德指数等。研究问题聚焦于给定匹配数或独立数等条件下的指数最大化，核心方法是推导与证明，主要贡献在于推广和解决相关猜想。
Given a function defined over a parabolic subgroup of a Coxeter group, equidistributed with the length, we give a procedure to construct a function over the entire group, equidistributed with the length. Such a procedure permits to define functions equidistributed with the length in all the finite Coxeter groups. We can establish our results in the general setting of graded posets which satisfy some properties. These results apply to some known functions arising in Coxeter groups as the major index, the negative major index and the D-negative major index defined in type $A$, $B$ and $D$ respec…
In this paper, we construct bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations, which send the area statistic on Dyck paths to the inversion number on noncrossing partitions and on 231-avoiding permutations. This bijection has the additional property that it simultaneously sends the major index on Dyck paths to the sum of the major index and the inverse major index on noncrossing partitions and on 231-avoiding permutations, respectively. Moreover, we provide generalizations of these constructions to the group of signed permutations.
In this paper, we firstly extend a result of Bonin, Shapiro and Simion by giving the distribution of the major index over generalized Schröder paths. Then by providing a bijection between generalized Schröder paths and row-increasing tableaux of skew shapes with two rows, we obtain the distribution of the major index and the amajor index over these tableaux, which extends a result of Du, Fan and Zhao. We also generalize a result of Pechenik and give the distribution of the major index over increasing tableaux of skew shapes with two rows. Especially, a bijection from row-increasing tableaux wi…
Natural q analogues of classical statistics on the symmetric groups $S_n$ are introduced; parameters like: the q-length, the q-inversion number, the q-descent number and the q-major index. MacMahon's theorem about the equi-distribution of the inversion number and the reverse major index is generalized to all positive integers q. It is also shown that the q-inversion number and the q-reverse major index are equi-distributed over subsets of permutations avoiding certain patterns. Natural q analogues of the Bell and the Stirling numbers are related to these q statistics -- through the counting of…
The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincare polynomial of a finite Coxeter group.
We introduce and study three new statistics on the hyperoctahedral group $B_{n}$, and show that they give two generalizations of Carlitz's identity for the descent number and major index over $S_{n}$. This answers a question posed by Foata.
Major Index Distribution
本文针对对称群上的 Major Index 分布（与 q-Plancherel 测度相关），提出一种简洁透明的采样算法，无需表示论即可证明已知结果，并由此获得该分布的极限形状、模式正态性与循环结构等性质。
Consider S_n, the symmetric group on n letters, and let maj pi denote the major index of a permutation pi in S_n. Given positive integers k,l and nonnegative integers i,j, define m_n^{k,l}(i,j) := number of pi in S_n such that maj pi = i (mod k) and maj pi^{-1} = j (mod l). We prove bijectively that if k,l are relatively prime and at most n then m_n^{k,l}(i,j) = n!/(kl) which, surprisingly, does not depend on i and j. Equivalently, if m_n^{k,l}(i,j) is interpreted as the (i,j)-entry of a matrix m_n^{k,l}, then this is a constant matrix under the stated conditions. This bijection is extended to…
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