Weigen Yan, Yeong-Nan Yeh
In this paper we prove that two quantities relating to the length of permutations defined on trees are independent of the structures of trees. We also find that these results are closely related to the results obtained by Graham and Pollak (Bell System Tech. J. 50(1971) 2495--2519) and by Bapat, Kirkland, and Neumann (Linear Alg. Appl. 401(2005) 193--209).
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We obtain an upper and lower bound for the number of reduced words for a permutation in terms of the number of braid classes and the number of commutation classes of the permutation. We classify the permutations that achieve each of these bounds, and enumerate both cases.
Relating tournaments and permutations with xrays
本文引入k-锦标赛的传递锦标赛分解，通过排列构造1-和2-锦标赛，使其得分序列与排列的X射线一致，并证明两个猜想是更一般猜想的特例。
We give a simple proof of a major index determinant formula in the symmetric group discovered by Krattenthaler and first proved by Thibon using noncommutative symmetric functions. We do so by proving a factorization of an element in the group ring of the symmetric group. By applying similar methods to the groups of signed permutations and colored permutations, we prove determinant formulas in these groups as conjectured by Krattenthaler.
Chen et al. recently established bijections for $(d+1)$-noncrossing/ nonnesting matchings, oscillating tableaux of bounded height $d$, and oscillating lattice walks in the $d$-dimensional Weyl chamber. Stanley asked what is the total number of such tableaux of length $n$ and of any shape. We find a determinant formula for the exponential generating function. The same idea applies to prove Gessel's remarkable determinant formula for permutations with bounded length of increasing subsequences. We also give short algebraic derivations for some results of the reflection principle.
Determinantal representations of alternating run polynomials
本文基于函数商高阶导数的行列式公式，推导了Eulerian多项式、Andre多项式及多种交错游程多项式的行列式表示，并揭示了其递推关系及与对偶Stirling排列的联系。
As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a $q$-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian…
In this note we generalize an identity of John Riordan and Robert Donaghey relating the enumerator for Stirling permutations to the Eulerian polynomials.
We consider Hankel determinants of the sequence of Catalan numbers modulo 2 (interpreted as integers 0 and 1) and more generally Hankel determinants where the sum over all permutations reduces to a single signed permutation.
We show that certain weighted average of the alpha-determinant of a $kn$ by $kn$ matrix of the form $A\otimes1_{1,k}$, the Kronecker product of a $kn$ by $n$ matrix $A$ and $1$ by $k$ all one matrix $1_{1,k}$, over permutations of $kn$ letters is reduced to the $k$-wreath determinant of $A$ up to constant. The constant is exactly given by the modified content polynomial for the Young diagram $(k^n)$. As a corollary, we give a `determinantal' formula for certain functions on the symmetric groups which are invariant under the left and right translation by a Young subgroup, especially the values …
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.
On the set of permutations of a finite set, we construct a bijection which maps the 3-vector of statistics $(maj-exc,des,exc)$ to a 3-vector $(maj\_2,\widetilde{des\_2},inv\_2)$ associated with the $q$-Eulerian polynomials introduced by Shareshian and Wachs in \textit{Chromatic quasisymmetric functions, arXiv:1405.4269(2014).}
The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relati…
We study the determinant of the pxp circulant matrix whose first row is (1,-x,0,...,0,-y,0,...,0), the -y being in position q+1. The coefficients of this polynomial are integers that count certain classes of permutations. We show that all of the permutations that contribute to a fixed monomial x^ry^s have the same sign, and we determine that sign. We prove that a monomial x^ry^s appears if and only if p divides r+sq. Finally, we show that the size of the largest coefficient of the monomials that appear grows exponentially with p. We do this by proving that the permanent of the circulant whose …
We prove an identity relating the permanent of a rank $2$ matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the possibility that these are actually special cases of some more general identity (or class of identities) connecting permanents and determinants. The proof combines some basic facts from the theory of symmetric functions with an application of a famous theorem of Binet and Cauchy in linear algebra.
Analysing the Moments of the Determinant of a Random Matrix Via Analytic Combinatorics of Permutation Tables
本文利用置换表上的解析组合学方法，显著简化了此前对任意分布随机矩阵行列式四阶矩和零均值分布六阶矩的复杂递推分析。
We prove a formal power series identity, relating the arithmetic sum-of-divisors function to commuting triples of permutations. This establishes a conjecture of Franklin T. Adams-Watters.
Boson-fermion complementarity in a linear interferometer: An identity relating the determinant and permanent of a matrix
本文研究了玻色子和费米子在任意线性干涉仪中的多粒子干涉，推导出一个关联行列式和 permanents 的数学恒等式，揭示了玻色-费米互补性与经典粒子概率的一致性。
Determinants of modular Collatz graphs and variants
本文研究了模Collatz图和其变体的决定因素，并揭示了其中的数论特性。
We present a simple bijection between permutation matrices and descending plane partitions without special parts. This bijection is already mentioned in work of P. Lalonde (without giving the details); it involves the inversion words of permutations and the (well-known) representation of descending plane partitions as families of non--intersecting lattice paths. (Taking a short detour, we will also exhibit how the (well--known) enumeration of descending plane partitions follows easily from the evaluation of Andrew's determinant.)
We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions $m \times 1$ for some $m$, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape. We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated…
In this paper, we prove determinant formulas for the $K$-theory classes of the structure sheaves of degeneracy loci classes associated to vexillary permutations in type $A$. As a consequence we obtain determinant formulas for Lascoux-Schützenberger's double Grothendieck polynomials associated to vexillary permutations. Furthermore, we generalize the determinant formula to algebraic cobordism.
We give a characterization of the effect of sequences of pivot operations on a graph by relating it to determinants of adjacency matrices. This allows us to deduce that two sequences of pivot operations are equivalent iff they contain the same set S of vertices (modulo two). Moreover, given a set of vertices S, we characterize whether or not such a sequence using precisely the vertices of S exists. We also relate pivots to perfect matchings to obtain a graph-theoretical characterization. Finally, we consider graphs with self-loops to carry over the results to sequences containing both pivots a…
About the determinant of complete non-ambiguous trees
本文针对完全非歧义树（CNATs）中关于行列式统计量分布的未决猜想，通过构造双射的方法完成了证明，解决了Chen和Ohlig提出、Selig和Zhu遗留的开放问题。
Combinatorial Reduction of Set Functions and Matroid Permutations through Minor Product Assignment
本文提出基于矩阵行列式展开的代数模型，通过形变与Grassmann-Plücker关系刻画集合函数组合约简，证明其与可逆次因子乘积的等价性，并用于拟阵置换的检验与恢复。
Under binary matrices we mean matrices whose entries take one of two values. In this paper, explicit formulae for calculating the determinant of some type of binary Toeplitz matrices are obtained. Examples of the application of the determinant of binary Toeplitz matrices for the enumeration of even and odd permutations of different types are given.
We use the Robinson-Schensted correspondence, followed by symbol-crunching, in order to derive explicit expressions for the quantities mentioned in the title. We follow it by number crunching, in order to compute the first terms of these sequences. As an encore, we cleverly implement Ira Gessel's celebrated determinant formula for the generating functions of these sequences, to crank out many terms. This modest tribute is dedicated to one of the greatest enumerators alive today (and definitely the most modest one!), Ira Martin Gessel, who is turning 64 years-old today
The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in ${\bf N}$, and standard Young tableaux by semistandard ones. For $r>0$, the Robinson-Schensted correspondence can be trivially extended, using the $r$-quotient map, to one between coloured permutations and pairs of standard $r$-ribbon tableaux built on a fixed $r$-core (the Stanton-White correspondence). This correspondence can also be generalised to arbitrary matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r…
There are several combinatorial methods that can be used to produce type A Demazure characters (key polynomials). The alcove path model of Lenart and Postnikov provides a procedure that inputs a semistandard tableau $T$ and outputs a saturated chain in the Bruhat order. The final permutation in this chain determines a family of Demazure characters for which $T$ contributes its weight. Separately, the right key of $T$ introduced by Lascoux and Schützenberger also determines a family of Demazure characters for which $T$ contributes its weight. In this paper we show that the final permutation in …
An identity for second Eulerian numbers via lattice-point counting
利用 Ehrhart 理论，给出 Bernoulli 数与第二类欧拉数之间两个恒等式的简单概念性证明。
The Sixth Moment of Random Determinants for Arbitrarily Distributed Random Entries
本文推广随机行列式六阶矩公式至任意分布元素的随机矩阵，通过标记置换表分解方法得到其指数生成函数的闭式形式，并经数值验证结果的正确性。
These lectures present a survey of recent developments in the area of random matrices (finite and infinite) and random permutations. These probabilistic problems suggest matrix integrals (or Fredholm determinants), which arise very naturally as integrals over the tangent space to symmetric spaces, as integrals over groups and finally as integrals over symmetric spaces. An important part of these lectures is devoted to showing that these matrix integrals, upon apropriately adding time-parameters, are natural tau-functions for integrable lattices, like the Toda, Pfaff and Toeplitz lattices, but …
We describe immanants as trivial modules of the symmetric group and show that any homogeneous polynomial of degree n on the space of n by n matrices preserved up to scalar by left and right action by diagonal matrices and conjugation by permutation matrices is a linear combination of immanants. we prove that the identity component of the stabilizer of any immanant (except determinant, permanent) is the expected one. We also prove that for n>5 the stabilizer of the immanant of any non-symmetric partition (except determinant and permanent) is again the expected one.
It is known from the work of Baik, Deift, and Johansson [1999] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the Ewens distribution and more generally for a class of random permutation with distribution invariant under conjugation. Moreover, we obtain the convergence of the first components of the associated Young tableaux to the Airy Ensemble as well as the global convergence to the Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that the limiting…
Counting permutations with no long monotone subsequence via generating trees and the kernel method
本文通过生成树和核方法，重新证明并推广了Gessel关于无长单调子序列排列的生成函数行列式公式，并扩展到对合情形，同时跟踪固定点数。
We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, $q$-analogs (which we name $q$TNN and $q$SFL properties). %In our discussion, essential sets, introduced by Fult…
We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation $w\in \Sn$ is at most the number of elements below $w$ in the Bruhat order, and (B) that equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinator…
On longest increasing subsequences in words in which all multiplicities are equal
本文针对所有重数相等的多重集排列推广了Gessel的Bessel行列式公式，提出一个表示为Gessel型Toeplitz行列式与对称群轮换指标多项式乘积多重积分的生成函数，用于统计不含特定长度递增子序列的词。
The authors consider the length, $l_N$, of the length of the longest increasing subsequence of a random permutation of $N$ numbers. The main result in this paper is a proof that the distribution function for $l_N$, suitably centered and scaled, converges to the Tracy-Widom distribution [TW1] of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest decent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 [DZ1] in the context of integrable systems. The applicability of the Riemann-Hilbert technique…
Torus Actions on Matrix Schubert and Kazhdan-Lusztig Varieties, and their Links to Statistical Models
本文研究由置换产生的矩阵舒伯特簇与Kazhdan-Lusztig簇在环面作用下的结构、复杂度及其与条件独立和拟独立统计模型的联系。
Considered are the large $N$, or large intensity, forms of the distribution of the length of the longest increasing subsequences for various models. Earlier work has established that after centring and scaling, the limit laws for these distributions relate to certain distribution functions at the hard edge known from random matrix theory. By analysing the hard to soft edge transition, we supplement and extend results of Baik and Jenkins for the Hammersley model and symmetrisations, which give that the leading correction is proportional to $z^{-2/3}$, where $z^2$ is the intensity of the Poisson…
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