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 the diagonal action of the automorphism group $\mathcal{G}_{\underlineλ}= Aut(\mathcal{A}_{\underlineλ})$, is a polynomial in $q$ with integer coefficients. Positivity conjecture states that these coefficients are in fact non-negative. In this article, we prove this conjecture.
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.
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".
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.
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.
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+…
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…
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 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 $B$.
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.
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.
We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map $F_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1})$ are given by noncommutative Laurent polynomials with nonnegative integer coefficients.
We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map $F_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1})$ are given by noncommutative Laurent polynomials with nonnegative integer coefficients.
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…
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.
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 introduced expressions for the normalized characters of the symmetric group and stated some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanley's positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results by Goulden and the present author. We shall see that the generating series $C(t)$ introduced by them is critical to our discussion.
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 prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur functions and resolve the Schur positivity and $k$-branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.
We give a combinatorial interpretation of a certain positivity conjecture of Chern-Schwartz-MacPherson classes, as stated by P. Aluffi and the author in a previous paper. It translates into a positivity property for a sum of p by p determinants consisting of binomial coefficients, generalizing the classical Theorem of Lindstrom-Gessel-Viennot et al. which computes these determinants in terms of non-intersecting lattice paths. We prove this conjecture for p=2,3.
One deals with r-regular bipartite graphs with 2n vertices. In a previous paper Butera, Pernici, and the author have introduced a quantity d(i), a function of the number of i-matchings, and conjectured that as n goes to infinity the fraction of graphs that satisfy Delta^k d(i) for all k and i, approaches 1. Here Delta is the finite difference operator. This conjecture we called the 'graph positivity conjecture'. In this paper it is formally shown that for each i and k the probability that Delta^k d(i) goes to 1 with n going to infinity. We call this weaker result the 'weak graph positivity con…
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 deal with $r$-regular bipartite graphs with $2n$ vertices. In a previous paper, Butera, Pernici and the author have introduced a quantity $u(i)$, $u(i) = -\ln(i!m\_i)$, a function of the number of $i$-matchings, $m\_i$, and conjectured that the fraction of graphs that violate $Δ^k u(i) > 0$ for $k > 1$ vanishes as $n$ goes to infinity. Here $Δ$ is the finite difference operator. We now more particularly define the "Virial Positivity Conjecture" as the conjecture that the fraction of graphs that satisfy $Δ^k u(i)$ go to 0 for all $k > 1$ and $i$, approaches 1 as $n$ goes to infinity. The "We…
Let $Λ$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_λ[X/(1-t)]\}_{λ_1\leq k}$. We introduce a new family of symmetric polynomials, $\{A_λ^{(k)}[X;t]\}_{λ_1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A_λ^{(k)}[X;t]$ form a basis for $V_k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but wo…
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.
We study the isospectral Hilbert scheme X_n, defined as the reduced fiber product of C^2n with the Hilbert scheme H_n of points in the plane, over the symmetric power S^n C^2. We prove that X_n is normal, Cohen-Macaulay, and Gorenstein, and hence flat over H_n. We derive two important consequences. (1) We prove the strong form of the "n! conjecture" of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients K_{lambda,mu}(q,t). This establishes the Macdonald positivity conjecture, that K_{lambda,mu}(q,t) is always a polynomial with non-negati…
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…
We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace, and a rather non-trivial expression for their evaluation. We study the limiting cases q=0 and q=\infty, which lead to two families of Hall-Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q=t=0 or q=t=\infty seem to generalize naturally the…
We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with non-negative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of X are uniquely determined by Witten's presentation of QH(X) and the fact that they are non-negative. We conjecture that the same is true for any flag variety X = G/P of simply laced Lie type. For the variety GL(n)/B of complete flags, this conjecture is equ…
One deals with r-regular bipartite graphs with 2n vertices. In a previous paper Butera, Pernici, and the author have introduced a quantity d(i), a function of the number of i-matchings, and conjectured that as n goes to infinity the fraction of graphs that satisfy Delta^k( d(i)) is non-negative, for all k and i, approaches 1. Here Delta is the finite difference operator. This conjecture we called the "graph positivity conjecture". "Weak graph positivity" is the conjecture that for each i and k the probability that Delta^k (d(i) is non-negative goes to 1 as n goes to infinity. Here we prove thi…
Solving a conjecture of M. H. Freedman, L. Lovász and A. Schrijver we prove that a graph parameter is edge reflection positive and multiplicative if and only if it can be represented by an edge coloring model.
The Brown-Erdős-Sós conjecture, one of the central conjectures in extremal combinatorics, states that for any integer $m\geq 6,$ if a 3-uniform hypergraph on $n$ vertices contains no $m$ vertices spanning at least $m-3$ edges, then the number of edges is $o(n^2).$ We prove the conjecture for triple systems coming from finite abelian groups.
Several new transformations for q-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new q-binomial transformations are also applied to obtain multisum Rogers--Ramanujan identities, to find new representations for the Rogers--Szego polynomials, and to make some progress on Bressoud's generali…
F. Bergeron recently asked the intriguing question whether $\binom{b+c}{b}_q -\binom{a+d}{d}_q$ has nonnegative coefficients as a polynomial in $q$, whenever $a,b,c,d$ are positive integers, $a$ is the smallest, and $ad=bc$. We conjecture that, in fact, this polynomial is also always unimodal, and combinatorially show our conjecture for $a\le 3$ and any $b,c\ge 4$. The main ingredient will be a novel (and rather technical) application of Zeilberger's KOH theorem.
The problem to decide whether a given multivariate (quasi-)rational function has only positive coefficients in its power series expansion has a long history. It dates back to Szego in 1933 who showed certain quasi-rational function to be positive, in the sense that all the series coefficients are positive, using an involved theory of special functions. In contrast to the simplicity of the statement, the method was surprisingly difficult. This dependency motivated further research for positivity of (quasi-)rational functions. More and more (quasi-)rational functions have been proven to be posit…
An explicit formula for Koornwinder moments and Rains' positivity conjecture
Sungkyunkwan University（成均馆大学）；Ewha Womans University（梨花女子大学）；Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学与交叉科学研究中心）；Yonsei University（延世大学）
We show that the Frobenius character of the equivariant Borel-Moore homology of a certain positive $GL_n$-version of the unramified affine Springer fiber $Z_k$ studied by Goreski, Kottwitz and MacPherson is computed by the matrix coefficients of the $\nabla^k$-operator, which acts diagonally in the modified Macdonald basis. We do this by relating the combinatorial formula for the $\nabla^k$-operator we obtained in an earlier paper to the GKM paving of $Z_k$, and we give an algebraic presentation of the above homology as an explicit submodule of the Kostant-Kumar nil Hecke algebra. We then stud…
