Michel Lassalle
We present a positivity conjecture for the coefficients of the development of Jack polynomials in terms of power sums. This extends Stanley's ex-conjecture about normalized characters of the symmetric group. We prove this conjecture for partitions having a rectangular shape.
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A positivity conjecture on the structure constants of shifted Jack functions
本文提出关于移位Jack函数结构常数的正性猜想，扩展了Stanley的原始猜想。
Monotonicity for generalized binomial coefficients and Jack positivity
本文证明了Schur多项式与Jack多项式相关广义二项式系数的单调性，导出了对称函数不等式，并得出Schur正性与Jack正性结果。
Positive formula for Jack polynomials, Jack characters and proof of Lassalle's conjecture
本文给出了Jack多项式幂和展开的显式公式，通过二分图映射解释Jack特征，证明了Lassalle猜想，并利用创生算子提供了新公式。
Majorization via positivity of Jack and Macdonald polynomial differences
本文通过Jack和Macdonald多项式差的正性，扩展了主化不等式的研究，并证明了主化和弱主化的特征刻画。
Introduced by Goulden and Jackson in their 1996 paper, the matchings-Jack conjecture and the hypermap-Jack conjecture (also known as the $b$-conjecture) are two major open questions relating Jack symmetric functions, the representation theory of the symmetric groups and combinatorial maps. They show that the coefficients in the power sum expansion of some Cauchy sum for Jack symmetric functions and in the logarithm of the same sum interpolate respectively between the structure constants of the class algebra and the double coset algebra of the symmetric group and between the numbers of orientab…
We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian numbers, Stirling numbers, quasi-Yamanouchi tableaux, and rook boards. These results also lead to further conjectures about the fundamental quasisymmetric expansions of these bases, which we prove for special cases.
Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformation $τ_b$ of the generating series of bipartite maps, which generalizes the partition function of $β$-ensembles of random matrices. The Matching-Jack conjecture suggests that the coefficients $c^λ_{μ,ν}$ of the function $τ_b$ in the power-sum basis are non-negative integer polynomials in the deformation parameter $b$. Dołęga and Féray have proved in 2016 the "polynomiality" part in the Matching-Jack conjecture, namely that coefficients $c^λ_{μ,ν}$ are in $\mathbb{Q}[b]$. In this paper, we prove the "integrality…
Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $ψ(x, y, z; t, 1+β)$ that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They made the following conjecture: the coefficients of $ψ(x, y, z; t, 1+β)$ in the power-sum basis are polynomials in $β$ with nonnegative integer coefficients (by construction, these coefficients are rational functions in $β$). We prove partially this conjecture, nowadays called $b$-conjecture, by showing that coefficients of $ψ(x, y, z; t, 1+ β)$ are polynomials in $…
In 1996 Goulden and Jackson introduced a family of coefficients $( c_{π, σ}^λ ) $ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $( J^{(α)}_π)$. The coefficients $ c_{π, σ}^λ $ can be viewed as an interpolation between the structure constants of the class algebra and the double coset algebra. Goulden and Jackson suggested that the coefficients $ c_{π, σ}^λ $ are polynomials in the variable $β:= α-1$ with non-negative integer coefficients and that there is a combinatorics of matching hidden behind them. This \emph{Matching…
We prove a positivity result for interpolation polynomials that was conjectured by Knop and Sahi. These polynomials were first introduced by Sahi in the context of the Capelli eigenvalue problem for Jordan algebras, and were later shown to be related to Jack polynomials by Knop-Sahi and Okounkov-Olshanski. The positivity result proved here is an inhomogeneous generalization of Macdonald's positivity conjecture for Jack polynomials. We also formulate and prove the non-symmetric version of the Knop-Sahi conjecture, and in fact we deduce everything from an even stronger positivity result. This la…
Some conjectures on the Schur expansion of Jack polynomials
本文提出Jack多项式Schur展开的正性猜想，并探讨了相关基底中的组合学内容及特殊情形的证明。
Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations on non-oriented surfaces that have recently been introduced by Chapuy and Dołęga. A key step in the proof is an encoding of constellations with tuples of matchings. We consider a one parameter deformation of the generating series of constellations using Jack polynomials and we introduce the coefficients $c^λ_{μ^0,...,μ^k}(b)$ obtained by the expansion of thes…
We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter $b$ with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate $b$-weighting that "measures" in some sense their non-orientability. Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so fa…
We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about them. We conjecture existence of a weight on non-oriented maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any given Jack character as a weighted sum of some simple functions indexed by maps. We provide a candidate for this weight which gives a positive answer to our conjecture in some, but unfortunately not all, cases. In particular…
We show that no tree on twenty vertices with maximum degree ten has Schur positive chromatic symmetric function, thereby providing a counterexample to a conjecture from the paper "Schur and e-positivity of trees and cut vertices".
Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley conjectured that if the Littlewood-Richardson coefficient for a triple of Schur polynomials is 1, then the corresponding coefficient for Jack polynomials can be expressed as a product of weighted hooks of the Young diagrams associated to the partitions indexing the coefficient. We prove a special case of this conjecture in which the partitions indexing the Littlewood-Richardson coefficient have at most 3 par…
We study pfaffian analogues of immanants, which we call pfaffinants. Our main object is the TL-pfaffinants which are analogues of Rhoades and Skandera's TL-immanants. We show that TL-pfaffinants are positive when applied to planar networks and explain how to decompose products of complementary pfaffians in terms of TL-pfaffinants. We conjecture in addition that TL-pfaffinants have positivity properties related to Schur Q-functions.
The $e$-positivity conjecture and the $e$-unimodality conjecture of chromatic quasisymmetric functions are proved for some classes of natural unit interval orders. Recently, J. Shareshian and M. Wachs introduced chromatic quasisymmetric functions as a refinement of Stanley's chromatic symmetric functions and conjectured the $e$-positivity and the $e$-unimodality of these functions. The $e$-positivity of chromatic quasisymmetric functions implies the $e$-positivity of corresponding chromatic symmetric functions, and our work resolves Stanley's conjecture on chromatic symmetric functions of $(3+…
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。
This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack symmetric functions. First introduced by Goulden and Jackson (1996) these numbers indexed by three partitions of a given integer $n$ and the Jack parameter $α$ are defined as the coefficients in the power sum expansion of the Cauchy sum for Jack symmetric functions. While very little is known about them, examples of computations for small values of $n$ tend…
A note on the map expansion of Jack polynomials
本文猜想 Jack 多项式幂和展开中映射边装饰为单射的变体形式，并针对两列划分情形结合组合方法与微分算子计算给出证明。
We prove Okounkov's conjecture, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibon's conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. An alternative proof of this result is provided. We also give an intriguing log-concavity property of Schur functions.
We prove a strong factorization property of interpolation Macdonald polynomials when $q$ tends to $1$. As a consequence, we show that Macdonald polynomials have a strong factorization property when $q$ tends to $1$, which was posed as an open question in our previous paper with Féray. Furthermore, we introduce multivariate $q,t$-Kostka numbers and we show that they are polynomials in $q,t$ with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate $q,t$-Kostka numbers are in fact polynomials in $q,t$ with nonnegative integer c…
Macdonald characters from a new formula for Macdonald polynomials
本文引入了一个新的算子Γ，用于获得Macdonald多项式生成公式，连接了Macdonald算子理论与移位Macdonald多项式，并提出关于广义特征的正性猜想。
In a 2016 ArXiv posting F. Bergeron listed a variety of symmetric functions $G[X;q]$ with the property that $G[X;1+q]$ is $e$-positive. A large subvariety of his examples could be explained by the conjecture that the Dyck path LLT polynomials exhibit the same phenomenon. In this paper we list the results of computer explorations which suggest that other examples exhibit the same phenomenon. We prove two of the resulting conjectures and propose algorithms that would prove several of our conjectures. In writing this paper we have learned that similar findings have been independently discovered b…
In the study of Zeilberger's conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let $(t)_n$ denote the rising factorial, and let $Λ_{\mathbb{R}}$ denote the algebra of symmetric functions with real coefficients. If $φ$ is the homomorphism from $Λ_{\mathbb{R}}$ to $\mathbb{R}$ defined by $φ(h_n)={1}/{((t)_nn!)}$ for some $t>0$, then for any Schur function $s_λ$, the value $φ(s_λ)$ is positive. In this paper, we provide an affirmative answer to Lassalle's conjecture by using the Laguerre-Pólya-Schur theory of multiplier sequences.
We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of admissible tableaux of the non-symmetric Jack polynomials. In the final step of the proof appears an identity on weighted sums of partitions that we demonstrate using the methods of Gessel-Viennot.
Ehrhart positivity for paving positroids
本文证明任意铺砌正拟阵的基多面体具有正埃哈特系数，为费罗尼等人提出的正拟阵均为埃哈特正的核心猜想提供了新的正例与重要证据。
This article is devoted to the computation of Jack connection coefficients, a generalization of the connection coefficients of two classical commutative subalgebras of the group algebra of the symmetric group: the class algebra and the double coset algebra. The connection coefficients of these two algebraic structures are of significant interest in the study of Schur and zonal polynomials as well as the irreducible characters of the symmetric group and the zonal spherical functions. Furthermore they play an important role in combinatorics as they give the number of factorizations of a permutat…
We present Pieri rules for the Jack polynomials in superspace. The coefficients in the Pieri rules are, except for an extra determinant, products of quotients of linear factors in $α$ (expressed, as in the usual Jack polynomial case, in terms of certain hook-lengths in a Ferrers' diagram). We show that, surprisingly, the extra determinant is related to the partition function of the 6-vertex model. We give, as a conjecture, the Pieri rules for the Macdonald polynomials in superspace.
A solution to Butler's positivity conjecture
该研究针对不同分拆满足λ、μ⊂ν且|ν/λ|=|ν/μ|=1的情况，证明了1994年提出的巴特勒正性猜想，即麦克唐纳相交多项式在舒尔函数基下的展开系数为非负整数系数的二元多项式。
Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis. This conjecture has been proved recently by Lin. We conjecture that an analogous statement holds for simple permutations, and use the substitution decomposition tree of a permutation (by repeated inflation) to show that this would imply the Gessel-Lin result. We provide supporting evidence for this stronger conjecture.
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。
Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chen's log-concavity conjecture, Bóna, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subsets. They obtained the log-concavity of the corresponding generating functions for these subsets by using the hook-length formula. In this paper, we generalize and prove their results by establishing the Schur positivity of certain symmetric functions. Thi…
LLT polynomials are $q$-analogues of product of Schur functions that are known to be Schur-positive by Grojnowski and Haiman. However, there is no known combinatorial formula for the coefficients in the Schur expansion. Finding such a formula also provides Schur positivity of Macdonald polynomials. On the other hand, Haiman and Hugland conjectured that LLT polynomials for skew partitions lying on $k$ adjacent diagonals are $k$-Schur positive, which is much stronger than Schur positivity. In this paper, we prove the conjecture for $k=2$ by analyzing unicellular LLT polynomials. We first present…
Interpolation Polynomials, Binomial Coefficients, and Symmetric Function Inequalities
本文研究插值多项式及其关联系数，证明了二项系数的正性和单调性，以及Littlewood-Richardson系数的部分正性，并提出一个新的对称函数不等式，建立Jack展开正性与分拆包含顺序之间的对偶关系。
We investigate chromatic symmetric functions in the relation to the algebra $Γ$ of symmetric functions generated by Schur $Q$-functions. We construct natural bases of $Γ$ in terms of chromatic symmetric functions. We also consider the $p$-positivity of skew Schur $Q$-functions and find a class of $p$-positive ribbon Schur $Q$-functions, making a conjecture that they are \emph{all}. We include many concrete computational results that support our conjecture.
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。
Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the free cumulants of the associated Young diagram. We present two positivity conjectures for their coefficients. The latter are stronger than the positivity conjecture of Kerov-Biane, recently proved by Feray.
Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition λ, we denote by λ^c its complement in a square partition (m^m). We conjecture a Schur-positivity criterion for symmetric functions of the form s_{μ'}s_{μ^c}-s_{λ'}s_{λ^c}, where λis a partition of weight |μ|-1 contained in μand the complement of μis taken in the same square partition as the complement of λ. We prove the conjecture in many cases.
We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $α$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $α$…
We say that a permutation $π=π_1π_2\cdots π_n \in \mathfrak{S}_n$ has a peak at index $i$ if $π_{i-1} < π_i > π_{i+1}$. Let $\mathcal{P}(π)$ denote the set of indices where $π$ has a peak. Given a set $S$ of positive integers, we define $\mathcal{P}_S(n)=\{π\in\mathfrak{S}_n:\mathcal{P}(π)=S\}$. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers $S$ and sufficiently large $n$, $| \mathcal{P}_S(n)|=p_S(n)2^{n-|S|-1}$ where $p_S(x)$ is a polynomial depending on $S$. They gave a recursive formula for $p_S(x)$ involving an alternating sum, and they conjectured that the …
We prove the s-positivity of the chromatic functions of interval hypergraphs and a conjecture of Taylor~\cite{Taylor15} as a consequence. We also give a new proof of Gasharov's theorem on the s-positivity of the chromatic symmetric functions of $(3+1)$-free posets.
We extend the Lee-Schiffler Dyck path model to give a proof of the Kontsevich non-commutative cluster positivity conjecture with unequal parameters.
We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of series coefficients, and apply the inverse of this operator to the rational functions. We obtain new rational functions which seem to have only positive coefficients, whose positivity would imply positivity of the original series, and which, in a certain sense, cannot be improved any further.
A counterexample to the claw-free Schur-positivity conjecture
该研究针对Stanley记载的无爪图Schur正性猜想，构造出12顶点的反例，经计算验证其色对称函数系数为负，且12顶点是最小反例阶数，还发现另一个同构类反例。
Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the $q$-log-convexity of the Narayana polynomials, and a proof of the second conjecture that the Narayana transformation preserves the log-convexity. Based on a formula of Bränd$\mathrm{\acute{e}}$n which expresses the $q$-Narayana numbers as the specializations of Schur functions, we derive several symmetric function identities using the Littlewood-Richardson rule for the product of Schur functions, and obtain the strong $q$-log-convexity of the Narayana pol…
Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $π$ angle of the skew diagram. It follows that, in some special cases, the coefficients of the skew Jack symmetric functions with respect to the basis of the monomial symmetric functions are polynomials with nonnegative integer coefficients.
In this article we prove the $e$-positivity of $G_{\mathbfν}[X;q+1]$ when $G_{\mathbfν}[X;q]$ is a vertical strip LLT polynomial. This property has been conjectured by Alexandersson and Panova, and by Garsia, Haglund, Qiu and Romero, and it implies several $e$-positivities conjectured by them and also by Bergeron. We make use of a result of Carlsson and Mellit that shows that a vertical strip LLT polynomial can be obtained by applying certain compositions of operators of the Dyck path algebra to the constant $1$. Our proof gives in fact an algorithm to expand these symmetric functions in the e…
Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler
该研究针对Guo和Krattenthaler的正性猜想，通过分拆注入法、推广的Andrews引理结合鞍点分析，关联图二项式系数，证明了该猜想对大n成立并给出轨道计数解释。
We prove the total positivity of the Narayana triangles of type $A$ and type $B$, and thus affirmatively confirm a conjecture of Chen, Liang and Wang and a conjecture of Pan and Zeng. We also prove the strict total positivity of the Narayana squares of type $A$ and type $B$.
Double orthodontia formulas and Lascoux positivity
基于Magyar的图表正畸算法提出双Grothendieck多项式新公式，推导双Schubert多项式类似公式，证明旗置换下的Lascoux正性结果并提出对所有置换成立的猜想。
$m$-Symmetric functions, non-symmetric Macdonald polynomials and positivity conjectures
本文定义并研究 $m$-对称函数空间及其 Macdonald、Schur 基，提出 Macdonald 按 Schur 正展开的猜想，并建立其与 $(q,t)$-Koska 系数及非对称 Hall–Littlewood 展开的联系。
Proofs of Two Positivity Conjectures of Guo
本文利用Warnaar-Zudilin正性定理和q-Chu-Vandermonde求和，证明了关于高斯系数交错和与阶乘比的两个正性猜想。
Hidden Structure of Jack Littlewood-Richardson Coefficients
研究揭示Jack Littlewood-Richardson系数是新型多项式的特例，通过特定分区三元组证明其在Johnson图的自守群下不变，并推测多项式在超平面上的因子化性质。
Haglund's conjecture states that $\dfrac{\langle J_λ(q,q^k),s_μ\rangle}{(1-q)^{|λ|}} \in \mathbb{Z}_{\geq 0}[q]$ for all partitions $λ,μ$ and all non-negative integers $k$, where $J_λ$ is the integral form Macdonald symmetric function and $s_μ$ is the Schur function. This paper proves Haglund's conjecture in the cases when the pair $(λ,μ)$ satisfies $K_{λ,μ}=1$ or $K_{μ',λ'}=1$ where $K$ denotes the Kostka number. We also obtain some general results about the transition matrix between Macdonald symmetric functions and Schur functions.
Proof of the Andrews-El Bachraoui positivity conjecture
通过q-超几何方法证明了安德鲁斯-埃尔巴哈里oui正性猜想，证实了该猜想的正确性。
The problem of deciding the positivity of Kronecker coefficients and Saxl conjecture
本文研究了确定Kronecker系数正性的问题，通过半群性质和广义块方法探讨Saxl猜想中不可约表示的分解问题。
In this note we prove positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally when studying log-concavity of rising factorial series. We propose several conjectures concerning zeros and coefficients of a generalized form of those polynomials. We also consider polynomials whose generating functions are higher order Toeplitz determinants formed by rising factorial series. We make three conjectures about these polynomials. All proposed conjectures are supported by numerical evidence.
For a partition $\underlineλ = (λ_{1}^{ρ_1}>λ_{2}^{ρ_2}>λ_{3}^{ρ_3}>\ldots>λ_{k}^{ρ_k})$ and its associated finite $\mathcal{R}$-module $\mathcal{A}_{\underlineλ}=\underset{i=1}{\overset{k}{\oplus}} (\mathcal{R}/π^{λ_i}\mathcal{R})^{ρ_i}$, where $\mathcal{R}$ is a discrete valuation ring, with maximal ideal generated by a uniformizing element $π$, having finite residue field ${\bf k}=\mathcal{R}/π\mathcal{R}\cong \mathbb{F}_q$, the number of orbits of pairs $n_{\underlineλ}(q)= \mid \mathcal{G}_{\underlineλ}\backslash \big(\mathcal{A}_{\underlineλ}\times \mathcal{A}_{\underlineλ}\big)\mid$ for…
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