Anna Weigandt
Motivated by a recent conjecture of R. P. Stanley we offer a lower bound for the sum of the coefficients of a Schubert polynomial in terms of $132$-pattern containment.
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We study the action of a differential operator on Schubert polynomials. Using this action, we first give a short new proof of an identity of I. Macdonald (1991). We then prove a determinant conjecture of R. Stanley (2017). This conjecture implies the (strong) Sperner property for the weak order on the symmetric group, a property recently established by C. Gaetz and Y. Gao (2018).
We prove, combinatorially, that the product of a Schubert polynomial by a Stanley symmetric polynomial is a truncated Schubert polynomial. Using Monk's rule, we derive a nonnegative combinatorial formula for the Schubert polynomial expansion of a truncated Schubert polynomial. Combining these results, we give a nonnegative combinatorial rule for the product of a Schubert and a Schur polynomial in the Schubert basis.
We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions. We also give combinatorial rules for the positive structure constants of these bases that generalize …
Denote by $u(n)$ the largest principal specialization of the Schubert polynomial: $ u(n) := \max_{w \in S_n} \mathfrak{S}_w(1,\ldots,1) $ Stanley conjectured in [arXiv:1704.00851] that there is a limit $\lim_{n\to \infty} \, \frac{1}{n^2} \log u(n), $ and asked for a limiting description of permutations achieving the maximum $u(n)$. Merzon and Smirnov conjectured in [arXiv:1410.6857] that this maximum is achieved on layered permutations. We resolve both Stanley's problems restricted to layered permutations.
Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman--Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial …
We prove that the well-known condition of being a balanced labeling can be characterized in terms of the sliding algorithm on tower diagrams. The characterization involves a generalization of authors' Rothification algorithm. Using the characterization, we obtain descriptions of Schubert polynomials and Stanley symmetric functions.
Dual Schubert polynomials via a Cauchy identity
本文通过组合证明建立了Postnikov-Stanley对偶舒伯特多项式公式与经典柯西恒等式的等价性，从而为Huang和Pylyavskyy的插入算法提供了自然解释。
In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula is general enough to give new expressions for all known types of Schubert polynomials. In the present paper we discuss the relationship between this formula and Stanley symmetric functions. We show that the coefficients obtained when a Stanley symmetric function is expressed in the basis of Schur functions are special cases of a class of generalized Littlewood-Richardson coefficients appearing in the formula for quiver varieties. We also prove that Fulton's universa…
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 …
We study the back stable $K$-theory Schubert calculus of the infinite flag variety. We define back stable (double) Grothendieck polynomials and double $K$-Stanley functions and establish coproduct expansion formulae. Applying work of Weigandt, we extend our previous results on bumpless pipedreams from cohomology to $K$-theory. We study finiteness and positivity properties of the ring of back stable Grothendieck polynomials, and divided difference operators in $K$-homology.
In recent work, the authors used an order lowering operator $\nabla$, introduced by Stanley, to prove the strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt interpreted $\nabla$ as a differential operator on Schubert polynomials and used this to prove a new identity for Schubert polynomials and a determinant conjecture of Stanley. In this paper we study a raising operator $Δ$ for the \emph{strong} Bruhat order, which is in many ways dual to $\nabla$. We prove a Schubert identity dual to that of Hamaker et al. and derive formulas fo…
We study the intersections of general Schubert varieties X_w with permuted big cells, and give an inductive degeneration of each such "Schubert patch" to a Stanley-Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of [Knutson 2007] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments. The underlying simplicial complex of the Stanley-Reisner scheme is a subword complex, as introduced for slightly different purposes in [Knutson-Miller 2004], and is homeomorphi…
Schur polynomials are special cases of Schubert polynomials, which in turn are special cases of dual characters of flagged Weyl modules. The principal specialization of Schur and Schubert polynomials has a long history, with Macdonald famously expressing the principal specialization of any Schubert polynomial in terms of reduced words. We study the principal specialization of dual characters of flagged Weyl~modules. Our result yields an alternative proof of a conjecture of Stanley about the principal specialization of Schubert polynomials, originally proved by Weigandt.
Postnikov--Stanley polynomials are Lorentzian
该研究证明了 Postnikov--Stanley 多项式是 Lorentzian 的，通过将其表示为 Richardson 簇的次数多项式，推广了已有结果并解决了 M-凸支撑猜想。
This paper deals with decreasing operators on back stable Schubert polynomials. We study two operators $ξ$ and $\nabla$ of degree $-1$, which satisfy the Leibniz rule. Furthermore, we show that all other such operators are linear combinations of $ξ$ and $\nabla$. For the case of Schur functions, these two operators fully determine the product of Schur functions, i.e., it is possible to define the Littlewood-Richardson coefficients only from $ξ$ and $\nabla$. This new point of view on Schur functions gives us an elementary proof of the Giambelli identity and of Jacobi-Trudi identities. For the …
In this paper we discuss analogs of Füredi-Hajnal and Stanley-Wilf conjectures for $t$-dimensional matrices with $t>2$.
We prove Stanley's plethysm conjecture for the $2 \times n$ case, which composed with the work of Black and List provides another proof of Foulkes conjecture for the $2 \times n$ case. We also show that the way Stanley formulated his conjecture, it is false in general, and suggest an alternative formulation.
We give two simple proofs of a conjecture of Richard Stanley concerning the equidistribution of derangements and alternating permutations with the maximal number of fixed points.
The Neggers-Stanley conjecture (also known as the Poset conjecture) asserts that the polynomial counting the linear extensions of a partially ordered set on $\{1,2,...,p\}$ by their number of descents has real zeros only. We provide counterexamples to this conjecture.
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.
The purpose of this note is to explain that the combinatorial local log-concavity conjecture introduced by Gross, Mansour, Tucker and Wang (Eur. J. Comb. 52, 207-222, 2016) in fact follows from a result of Stanley (Eur. J. Comb. 32 (6), 937-943, 2011).
Stanley, building on work of Stern, defined an array of numbers by the recurrence $s(n, 2k) = s(n-1, k)$, $s(n, 2k+1) = s(n-1, k) + s(n-1, k+1)$. Stanley showed that, for each positive integer $r$, the sequence $s_n^r:= \sum_k s(n,k)^r$ obeys a homogeneous linear recurrence in $n$ of length $r/2+O(1)$. Numerical evidence, however, suggested that $s_n^r$ obeys shorter recurrences, of length $r/3+O(1)$. We prove Stanley's conjecture.
We establish strict growth for the rank function of an r-differential poset. We do so by exploiting the representation theoretic techniques developed by Reiner and the author for studying related Smith forms.
In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank $1$ boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.
We consider the loci of invertible linear maps $f : \mathbb{C}^n \to {(\mathbb{C}^n)}^*$ together with pairs of flags $(E_\bullet, F_\bullet)$ in $\mathbb{C}^n$ such that the various restrictions $f : F_j \to E_i^*$ have specified ranks. Identifying an invertible linear map with its graph viewed as a point in a Grassmannian, we show that the closures of these loci have cohomology classes represented by the back-stable Schubert polynomials of Lam, Lee, and Shimozono. As a special case, we recover the result of Knutson, Lam, and Speyer that Stanley symmetric functions represent the classes of gr…
This paper contributes to a programme initiated by the first author: `How much information about a graph is revealed in its Potts partition function?'. We show that the $W$-polynomial distinguishes non-isomorphic weighted trees of a \emph{good} family. The framework developed to do so also allows us to show that the $W$-polynomial distinguishes non-isomorphic caterpillars. This establishes Stanley's isomorphism conjecture for caterpillars, an extensively studied problem.
R. Stanley has found a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape. Then, he has given a conjectural generalisation for any shape. Here, we will prove this formula using shifted Schur functions and Jucys-Murphy elements.
We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur's P and Q-functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type C Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one.
It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n by n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem. The unimodality of this sequance, which follows, was conjectured by Stanley (1983). Several generalizations are given.
A characteristic polynomial is an important invariant in the field of hyperplane arrangement. For the Linial arrangement of any irreducible root system, Postnikov and Stanley conjectured that all roots of the characteristic polynomial have the same real part. In relation to this conjecture, Yoshinaga obtained an explicit relationship between the characteristic quasi-polynomial and the Ehrhart quasi-polynomial for the fundamental alcove. In this paper, we calculate Yoshinaga's explicit formula through the decomposition of the Ehrhart quasi-polynomial into several quasi-polynomials and a modifie…
Towards the characterization of minimum weight codewords of Schubert codes
本文验证了Schubert码最小重量码字猜想，证明了Schubert分解可码字在有限多个例外情况下成立。
The concept of pattern avoidance respectively containment in permutations can be extended to permutations on multisets in a straightforward way. In this note we present a direct proof of the already known fact that the well-known Stanley-Wilf Conjecture, stating that the number of permutations avoiding a given pattern does not grow faster than exponentially, also holds for permutations on multisets.
A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system $A_{n-1}$. The proof is based on an explicit formula for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The con…
We construct a sequence of finite automata that accept subclasses of the class of 4231-avoiding permutations. We thereby show that the Wilf-Stanley limit for the class of 4231-avoiding permutations is bounded below by 9.35. This bound shows that this class has the largest such limit among all classes of permutations avoiding a single permutation of length 4 and refutes the conjecture that the Wilf-Stanley limit of a class of permutations avoiding a single permutation of length k cannot exceed (k-1)^2.
The Stern poset $\mathcal{S}$ is a graded infinite poset naturally associated to Stern's triangle, which was defined by Stanley analogously to Pascal's triangle. Let $P_n$ denote the interval of $\mathcal{S}$ from the unique element of row $0$ of Stern's triangle to the $n$-th element of row $r$ for sufficiently large $r$. For $n\geq 1$ let \begin{align*} L_n(q)&=2\cdot\left(\sum_{k=1}^{2^n-1}A_{P_k}(q)\right)+A_{P_{2^n}}(q), \end{align*} where $A_{P}(q)$ represents the corresponding $P$-Eulerian polynomial. For any $n\geq 1$ Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(…
Stanley's conjecture on the Schur positivity of distributive lattices
本文解决了Stanley于1998年提出的关于分配格的开放问题，通过构造两族反例分配格，分别证明了存在不好且非Schur正性的分配格，以及好但非Schur正性的分配格。
Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of "admissible sequences" by Richardson and Springer. They are maximal chains in Richardson and Springer's weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain …
Counterexamples to Stanley's conjecture on dimer coverings
通过构造三角代数单位间的乘法恒等式，证明了Stanley关于二聚体覆盖生成函数分母多项式仅有单根的猜想在k=13时首次失效。
In our previous work arXiv:1305.3543, we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schutzenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of arXiv:0908.3628, which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elu…
Inversions Tableaux
介绍用于舒伯特多项式和斯坦利对称函数的逆序表格这一组合模型，它是平衡阶梯表格的修改，明确描述相关表格及作用，为后续证明鲁贝的滑道移动猜想建立背景。
The Stanley Conjecture Revisited
本文基于 Stanley 猜想，提出 Jack Littlewood-Richardson 系数的窗口化性质，并将 Schur L-R 的分解性质推广到 Jack 情形，从而建立大量关系并实现一类情形的直接计算。
We show that the Stanley's Conjecture holds for an intersection of three monomial primary ideals of a polynomial algebra S over a field.
Eschers and Stanley's chromatic e-positivity conjecture in length-2
本研究针对长度为2的分拆情形，给出了Stanley色e正性猜想的简短证明，推进了该组合数学猜想的验证工作。
Schubert polynomials $\mathfrak{S}_w$ are polynomial representatives for cohomology classes of Schubert varieties in a complete flag variety, while Grothendieck polynomials $\mathfrak{G}_w$ are analogous representatives for the $K$-theory classes of the structure sheaves of Schubert varieties. In the special case that $\mathfrak{S}_w$ is a multiplicity-free sum of monomials, K. Mészáros, L. Setiabrata, and A. St. Dizier conjectured that $\mathfrak{G}_w$ can be easily computed from $\mathfrak{S}_w$ via Möbius inversion on a certain poset. We prove this conjecture.
In 1977 Stanley conjectured that the $h$-vector of a matroid independence complex is a pure $O$-sequence. In this paper we use lexicographic shellability for matroids to motivate a combinatorial strengthening of Stanley's conjecture. This suggests that a pure $O$-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank $d$ matroids on at most $2d$ elements, then it holds for all matro…
We prove that the number of partitions of an integer into at most b distinct parts of size at most n forms a unimodal sequence for n sufficiently large with respect to b. This resolves a recent conjecture of Stanley and Zanello.
We prove that an inclusion-exclusion inspired expression of Schubert polynomials of permutations that avoid the patterns 1432 and 1423 is nonnegative. Our theorem implies a partial affirmative answer to a recent conjecture of Yibo Gao about principal specializations of Schubert polynomials. We propose a general framework for finding inclusion-exclusion inspired expression of Schubert polynomials of all permutations.
In this paper, we give a formula for the number of lattice points in the dilations of Schubert matroid polytopes. As applications, we obtain the Ehrhart polynomials of uniform and minimal matroids as special cases, and give a recursive formula for the Ehrhart polynomials of $(a,b)$-Catalan matroids. Ferroni showed that uniform and minimal matroids are Ehrhart positive. We show that all sparse paving Schubert matroids are Ehrhart positive and their Ehrhart polynomials are coefficient-wisely bounded by those of minimal and uniform matroids. This confirms a conjecture of Ferroni for the case of s…
A proof of the Stanley--Stembridge conjecture
本文通过概率解释证明了单位间隔图的色准对称函数展开式系数，并由此推导出斯坦利-斯蒂姆布里奇猜想的成立。
Two infinite families of counterexamples to the Stanley--Gasharov conjecture
该研究通过结合无爪图的阶数普查数据，证明了Stanley–Gasharov猜想的最小反例，并构造出无穷族连通线图作为该猜想的无穷反例。
Non-vanishing of Single, Double, and Triple Schubert Structure Constants
本文研究双、三Schubert系数的非零性问题，证明三Schubert系数非零性由单Schubert系数决定，得到三Littlewood--Richardson系数的饱和性质，提出双Schubert系数非零性相关猜想并证明部分情形成立。
We prove that the Stanley-Wilf limit of any layered permutation pattern of length $\ell$ is at most $4\ell^2$, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern. If the conjecture is true that the maximum Stanley-Wilf limit for patterns of length $\ell$ is attained by a layered pattern then this implies an upper bound of $4\ell^2$ for the Stanley-Wilf limit of any pattern of length $\ell$. We also …
Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric reasons. This results in a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. Given a permutation w in S_n, we consider a determinantal ideal I_w whose generators are certain minors in the generic n x n matrix (filled with independent variables). Using `multidegrees' as simple algebraic substitutes for torus-equivar…
Stanley introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. Stanley later gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree.
A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible $h$-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid $h$-vectors are pure ${O}$-sequences. Finally, we completely characterize the Stanley-Reisner ideals of 1-dimensional matroid complexes.
The Stanley's Conjecture on Cohen-Macaulay multigraded modules is studied especially in dimension 2. In codimension 2 similar results were obtained by Herzog, Soleyman-Jahan and Yassemi. As a consequence of our results Stanley's Conjecture holds in 5 variables.
D. Grigoriev-G. Koshevoy recently proved that tropical Schur polynomials have (at worst) polynomial tropical semiring complexity. They also conjectured tropical skew Schur polynomials have at least exponential complexity; we establish a polynomial complexity upper bound. Our proof uses results about (stable) Schubert polynomials, due to R. P. Stanley and S. Billey-W. Jockusch-R. P. Stanley, together with a sufficient condition for polynomial complexity that is connected to the saturated Newton polytope property.
We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Toeplitz matrices. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for $E$-polynomials of series-parallel posets and column-strict labelled Ferrers posets.
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.
Let $S$ be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of $I$ determines the Stanley projective dimension of $S/I$ or $I$. Our main result is that this conjecture implies the Stanley conjecture for $I$, and it also implies that \[ \operatorname{sdepth} S/I \geq \operatorname{depth} S/I - 1.\] Recently, Duval et al. found a counterexample to the Stanley conjecture, and their counterexample satisfies $\operatorname{sdepth} S/I = \operatorname{depth} S/I - 1$. So if our conjecture is true, then the conclusion is…
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