本文针对 Stanley 特征多项式最高次项系数对应的带装饰平面树，构造了其与轮换最小因子分解之间的明确双射。
摘要：
Stanley 和 Féry 给出了一个与多矩形 Young 图相关的对称群不可约特征的公式。该公式表明，该特征是多矩形坐标中的一个多项式，并通过计数某些带装饰的映射（即在曲面上绘制的图）对其系数给出了明确的组合解释。在本文中，我们集中研究 Stanley 特征多项式中最高次单项式的系数，这对应于计数某些带装饰的平面树。我们给出了此类树与一个轮换的最小因子分解之间的明确双射。
英文摘要：
Stanley and Féray gave a formula for the irreducible character of the symmetric group related to a multi-rectangular Young diagram. This formula shows that the character is a polynomial in the multi-rectangular coordinates and gives an explicit combinatorial interpretation for its coefficients in terms of counting certain decorated maps (i.e., graphs drawn on surfaces). In the current paper we concentrate on the coefficients of the top-degree monomials in the Stanley character polynomial, which corresponds to counting certain decorated plane trees. We give an explicit bijection between such trees and minimal factorizations of a cycle.
We prove a recent conjecture by René Marczinzik involving certain statistics on Dyck paths that originate in the representation theory of Nakayama algebras of a linearly oriented quiver. We do so by analysing the effect of the Billey-Jockusch-Stanley bijection between Dyck paths and 321-avoiding permutations on these statistics, which was suggested by the result of a query issued to the online database FindStat.
Lam, Lee and Shimozono introduced the structure of bumpless pipedreams in their study of back stable Schubert calculus. They found that a specific family of bumpless pipedreams, called EG-pipedreams, can be used to interpret the Edelman-Greene coefficients appearing in the expansion of a Stanley symmetric function in the basis of Schur functions. It is well known that the Edelman-Greene coefficients can also be interpreted in terms of reduced word tableaux for permutations. Lam, Lee and Shimozono proposed the problem of finding a shape preserving bijection between reduced word tableaux for a p…
We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a simplex. We apply this result to verify the Stanley conjecture for quotients of monomial ideals with up to six generators. For seven generators we obtain a partial result.
In a delightful article, Richard Stanley derived, algebraically, the surprisingly simple formula, 3 times 7 to the power n-1, for the sum of the cubes of the n-th row of Stern's diatomic array. In this note, we find an elegant bijective proof of this surprising fact, that explains it and gives insight. The novelty is that this gorgeous bijection was discovered by a computer (SBE), with minimal guidance by a human (DZ). This debunks the conventional wisdom, held by some human supremacists, that computers can only compute, but they can't give insight
The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynom…
In this paper we answer a question posed by R. Stanley in his collection of Bijection Proof Problems (Problem 240). We present a bijective proof for the enumeration of walks of length $k$ a chess rook can move along on an $m\times n$ board starting and ending on the same square.
In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in $\mathbb{R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection be…
We develop a notion of a dual of a graph, generalizing the definition of Goulden and Yong (which only applied to trees), and reproving their main result using our new notion. We in fact give three definitions of the dual: a graph-theoretic one, an algebraic one, and a combinatorial "mind-body" dual, showing that they are in fact the same, and are also the same (on trees) as the topological dual developed by Goulden and Yong. Goulden and Yong use their dual to define a bijection between the vertex labeled trees and the factorizations of the permutation $(n, \ldots, 1)$ into $n-1$ transpositions…
We consider a variant of the game of Brussels Sprouts that, like Conway's original version, ends in a predetermined number of moves. We show that the endstates of the game are in natural bijection with noncrossing trees and that the game histories are in natural bijection with both parking functions and factorizations of a cycle of $S_n$.
We are interested in random uniform minimal factorizations of the $n$-cycle which are factorizations of $(1~2\dots n)$ into a product of $n-1$ transpositions. Our main result is an explicit formula for the joint probability that 1 and 2 appear a given number of times in a uniform minimal factorization. For this purpose, we combine bijections with Cayley trees together with explicit computations of multivariate generating functions.
We introduce a new method for showing that the roots of the characteristic polynomial of certain finite lattices are all nonnegative integers. This method is based on the notion of a quotient of a poset which will be developed to explain this factorization. Our main theorem will give two simple conditions under which the characteristic polynomial factors with nonnegative integer roots. We will see that Stanley's Supersolvability Theorem is a corollary of this result. Additionally, we will prove a theorem which gives three conditions equivalent to factorization. To our knowledge, all other theo…
We develop direct bijections between the set $F_n^k$ of minimal factorizations of the long cycle $(0\,1\,\cdots\, kn)$ into $(k+1)$-cycle factors and the set $R_n^k$ of rooted labelled forests on vertices $\{1,\ldots,n\}$ with edges coloured with $\{0,1,\ldots,k-1\}$ that map natural statistics on the former to generalized Mahonian statistics on the latter. In particular, we examine the generalized major index on forests $R_n^k$ and show that it has a simple and natural interpretation in the context of factorizations. Our results extend those by the present authors (2021), which treated the ca…
Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by …
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…
Kohnert proposed the first monomial positive formula for Schubert polynomials as the generating polynomial for certain unit cell diagrams obtained from the Rothe diagram of a permutation. Billey, Jockusch and Stanley gave the first proven formula for Schubert polynomials as the generating polynomial for compatible sequences of reduced words of a permutation. In this paper, we give an explicit bijection between these two models, thereby definitively proving Kohnert's rule for Schubert polynomials.
Generalized Stirling permutations, families of increasing trees and urn models
Bona [2007+]研究了Gessel和Stanley [1978]引入的Stirling排列类中上升、平稳和下降的分布。最近，Janson [2008+]证明了Stirling置换和平面递归树之间的联系，并证明了Bona考虑的参数的联合正规律。这里我们将考虑广义Stirling置换，推广了Bona和Janson的早期结果，并将它们与某些广义平面递归树族以及$(k+1)$ -元增树联系起来。我们还给出了两个不同的双射之间的某些家庭的增长树，这两个作为一个特殊情况的双射之间的三元增长树和平面递归树。为了描述感兴趣的参数的（渐近）行为，我们研究了三个（广义）Polya瓮模型使用各种方法。
In a recent paper, Ayyer and Behrend present for a wide class of partitions factorizations of Schur polynomials with an even number of variables where half of the variables are the reciprocals of the others into symplectic and/or orthogonal group characters, thereby generalizing results of Ciucu and Krattenthaler for rectangular shapes. Their proofs proceed by manipulations of determinants underlying the characters. The purpose of the current paper is to provide bijective proofs of such factorizations. The quantities involved have known combinatorial interpretations in terms of Gelfand-Tsetlin…
We show that J_n, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n-3)-dimensional simplicial associahedron A_n. This resolution is not minimal for n > 5; in this case the Betti numbers of J_n are strictly smaller than the f-vector of A_n. We show that in fact the Betti numbers of J_n are in bijection with the number of standard Young tableaux of shape (d+1, 2, 1^{n-d-3}). This complements the fact that the number of (d-1)-dimensional faces of A_n are given by the number of standard Young tableaux of (super)shape (d+1, d+1, 1^{n-d-3}); a bijective proof of thi…
We investigate random minimal factorizations of the $n$-cycle, that is, factorizations of the permutation $(1 \, 2 \cdots n)$ into a product of cycles $τ_1, \ldots, τ_k$ whose lengths $\ell(τ_1), \ldots, \ell(τ_k)$ verify the minimality condition $\sum_{i=1}^k(\ell(τ_i)-1)=n-1$. By associating to a cycle of the factorization a black polygon inscribed in the unit disk, and reading the cycles one after an other, we code a minimal factorization by a process of colored laminations of the disk, which are compact subsets made of red noncrossing chords delimiting faces that are either black or white.…
Recent work by Huang, Lewis, Morales, Reiner, and Stanton suggests that the regular elliptic elements of $\mathrm{GL}_n \mathbb{F}_q$ are somehow analogous to the $n$-cycles of the symmetric group. In 1981, Stanley enumerated the factorizations of permutations into products of $n$-cycles. We study the analogous problem in $\mathrm{GL}_n \mathbb{F}_q$ of enumerating factorizations into products of regular elliptic elements. More precisely, we define a notion of cycle type for $\mathrm{GL}_n \mathbb{F}_q$ and seek to enumerate the tuples of a fixed number of regular elliptic elements whose produ…
Characters of symmetric groups in terms of free cumulants and Frobenius coordinates
自由累积量是Young图形状的良好而有用的泛函，特别是它们给出了对称群S（n）在极限n \to \infty中的规范特征标的渐近性.给出了对称群的正规特征标用自由累积量表示的一个显式组合公式。我们也用Frobenius坐标来表示字符。我们的公式涉及到计算给定置换的某些因子分解。主要工具是斯坦利多项式，它给出了多矩形杨图上的字符值。简历自由累积量是函数的可积函数，特别是在\to \infty中对称群S（n）的特征正规化的渐近性。我们给出了一个明确的组合公式，用于自由累积量基础上对称组的标准化特征。我们将以弗罗贝纽斯的坐标为基础来检验这些特征。Nos formules font intervenir le nombre de certaines factorisations d'une permutation donnée.主要是斯坦利的多项式族，它具有杨的多矩形曲线上的特征值。
Fix natural numbers $n \geq 1$, $t \geq 2$ and a primitive $t^{\text{th}}$ root of unity $ω$. In previous work with A. Ayyer (J. Alg., 2022), we studied the factorization of specialized irreducible characters of $\text{GL}_{tn}$, $\text{SO}_{2tn+1},$ $\text{Sp}_{2tn}$ and $\text{O}_{2tn}$ evaluated at elements to $ω^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$. In this work, we extend the results to the groups $\text{GL}_{tn+m}$ $(0 \leq m \leq t-1)$, $\text{SO}_{2tn+3}$, $\text{Sp}_{2tn+2}$ and $\text{O}_{2tn+2}$ evaluated at similar specializations: (1) for the $\text{GL}_{tn+m}(\mat…
Recently, Lazar and Wachs (arXiv:1910.07651) showed that the (median) Genocchi numbers play a fundamental role in the study of the homogenized Linial arrangement and obtained two new permutation models (called D-permutations and E-permutations) for (median) Genocchi numbers. They further conjecture that the distributions of cycle numbers over the two models are equal. In a follow-up, Eu et al. (arXiv:2103.09130) further proved the gamma-positivity of the descent polynomials of even-odd descent permutations, which are in bijection with E-permutations by Foata's fundamental transformation. This …
