Arthur Yang

• Nankai University

As introduced by Gutman and Harary, the independence polynomial of a graph serves as the generating polynomial of its independent sets. In 1987, Alavi, Malde, Schwenk and Erd\H{o}s conjectured that the independence polynomials of all trees are unimodal. In this paper we come up with a new way for proving log-concavity of independence polynomials of...

Motivated by the notion of nice graphs, we introduce the concept of strongly nice property, which can be used to study the Schur positivity of symmetric functions. We show that a graph and all its induced subgraphs are strongly nice if and only if it is claw-free, which strengthens a result of Stanley and provides further evidence for the well-know...

Ferroni and Larson gave a combinatorial interpretation of the braid Kazhdan-Lusztig polynomials in terms of series-parallel matroids. As a consequence, they confirmed an explicit formula for the leading Kazhdan-Lusztig coefficients of braid matroids with odd rank, as conjectured by Elias, Proudfoot, and Wakefield. Based on Ferroni and Larson’s work...

In this paper we solve an open problem on distributive lattices, which was proposed by Stanley in 1998. This problem was motivated by a conjecture due to Griggs, which equivalently states that the incomparability graph of the boolean algebra $B_n$ is nice. Stanley introduced the idea of studying the nice property of a graph by investigating the Sch...

Inspired by the notion of equivariant log-concavity, we introduce the concept of induced log-concavity for a sequence of representations of a finite group. For an equivariant matroid equipped with a symmetric group action or a finite general linear group action, we transform the problem of proving the induced log-concavity of matroid invariants to...

We give new proof of the (generalized) Jacobi-Trudi identity, which expresses the (skew) Schur polynomial as a determinant of the Jacobi-Trudi matrix whose entries are the complete homogeneous symmetric polynomials. The proof is based on interpreting Kostka numbers as tensor product multiplicities in the BGG category $\mathcal{O}$ of $\mathfrak{s l...

The Hankel matrix of type B Narayana polynomials was proved to be totally positive by Wang and Zhu, and independently by Sokal. Pan and Zeng raised the problem of giving a planar network proof of this result. In this paper, we present such a proof by constructing a planar network allowing negative weights, applying the Lindström-Gessel-Viennot lemm...

Brenti conjectured that, for any finite Coxeter group, the descent generating polynomial has only real zeros, and he left the type D case open. Dilks, Petersen, and Stembridge proposed a companion conjecture, which states that, for any irreducible finite Weyl group, the affine descent generating polynomial has only real zeros, and they left the typ...

We give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to produce inequa...

The Hankel matrix of type B Narayana polynomials was proved to be totally positive by Wang and Zhu, and independently by Sokal. Pan and Zeng raised the problem of giving a planar network proof of this result. In this paper, we present such a proof by constructing a planar network allowing negative weights, applying the Lindstr\"om-Gessel-Viennot le...

By using Chen, Hou and Mu’s extended Zeilberger algorithm, the authors obtain two recurrence relations for Callan’s generalization of Narayana polynomials. Based on these recurrence relations, the authors further prove the real-rootedness and asymptotic normality of Callan’s Narayana polynomials.

In this paper we give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to...

Motivated by Stanley and Stembridge's conjecture about the $e$-positivity of claw-free incomparability graphs, Hamel and her collaborators studied the $e$-positivity of $(claw, H)$-free graphs, where $H$ is a four-vertex graph. In this paper we establish the $e$-positivity of generalized pyramid graphs and $2K_2$-free unit interval graphs, which ar...

Motivated by the concepts of the inverse Kazhdan-Lusztig polynomial and the equivariant Kazhdan-Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan-Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse Kazhdan-Lusztig polynomial of a matroid is very useful for determining its equivariant Kazhdan-Lus...

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias et al. (2016) [4]. Let Um,d denote the uniform matroid of rank d on a set of m+d elements. Gedeon et al. (2017) [7] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of Um,d using equivariant Kazhdan-Lusztig polynomials. In this paper we give a...

Motivated by Stanley's $\mathbf{(3+1)}$-free conjecture on chromatic symmetric functions, Foley, Ho\`{a}ng and Merkel introduced the concept of strong $e$-positivity and conjectured that a graph is strongly $e$-positive if and only if it is (claw, net)-free. In order to study strongly $e$-positive graphs, they further introduced the twinning operat...

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{...

Recursive matrices are ubiquitous in combinatorics, which have been extensively studied. We focus on the study of the sums of 2 × 2 minors of certain recursive matrices, the alternating sums of their 2 × 2 minors, and the sums of their 2 × 2 permanents. We obtain some combinatorial identities related to these sums, which generalized the work of Sun...

Motivated by Stanley's conjecture about the $e$-positivity of claw-free incomparability graphs, Hamel and her collaborators studied the $e$-positivity of $(claw, H)$-free graphs, where $H$ is a four-vertex graph. In this paper we establish the $e$-positivity of generalized pyramid graphs and $2K_2$-free unit interval graphs, which are two important...

Recursive matrices are ubiquitous in combinatorics, which have been extensively studied. We focus on the study of the sums of $2\times 2$ minors of certain recursive matrices, the alternating sums of their $2\times 2$ minors, and the sums of their $2\times 2$ permanents. We obtain some combinatorial identities related to these sums, which generaliz...

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot, and Wakefield [{\it Adv. Math. 2016}]. Let $U_{m,d}$ denote the uniform matroid of rank $d$ on a set of $m+d$ elements. Gedeon, Proudfoot, and Young [{\it J. Combin. Theory Ser. A, 2017}] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polyno...

For any positive integers $r$, $s$, $m$, $n$, an $(r,s)$-order $(n,m)$-dimensional rectangular tensor ${\cal A}=(a_{i_1\cdots i_r}^{j_1\cdots j_s}) \in ({\mathbb R}^n)^r\times ({\mathbb R}^m)^s$ is called partially symmetric if it is invariant under any permutation on the lower $r$ indexes and any permutation on the upper $s$ indexes. Such partiall...

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield, whose properties need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was recently conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomi...

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield, whose properties need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was recently conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomi...

n this paper, we prove the real-rootedness of a family of generalized Narayana polynomials which arose in the study of the infinite log-concavity of the Boros-Moll polynomials. We establish certain recurrence relations for these Narayana polynomials, from which we derive the real-rootedness. In order to prove the real-rootedness, we use a sufficien...

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, B\'{o}na, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subset...

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, B\'{o}na, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subset...

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 $B$.

We prove that, for any positive q, the q-Eulerian polynomial of type D has only real zeros. This settles an open problem of Brenti in 1994. For q = 1, our result reduces to the real-rootedness of the Eulerian polynomials of type D, which was originally conjectured by Brenti and recently proved by Savage and Visontai.

The Chinese A-share Stock Market has been suffering from massive volatility since the popping of a bubble on June 15, 2015. About a third of the values of A-shares in Shanghai Stock Exchange was lost within one month of the event. Although the Chinese government enacted many measures to halt the fall, the turbulence of the Chinese Stock Market cont...

In this paper, we prove the real-rootedness of two classes of generalized Narayana polynomials: one arising as the $h$-polynomials of the generalized associahedron associated to the finite Weyl groups, the other arising in the study of the infinite log-concavity of the Boros-Moll polynomials. For the former, Br\"{a}nd\'{e}n has already proved that...

In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov's unimodality conjecture with the help of Newton's inequality. By using an equidistribution property between desce...

In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov's unimodality conjecture with the help of Newton's inequality. By using an equidistribution property between desce...

Let $f(m,c)=\sum_{k=0}^{\infty} (km+1)^{k-1} c^k e^{-c(km+1)/m} / (m^kk!)$. For any positive integer $m$ and positive real $c$, the identity $f(m,c)=f(1,c)^{1/m}$ arises in the random graph theory. In this paper, we present two elementary proofs of this identity: a pure combinatorial proof and a power-serial proof. We also proved that this identity...

In this paper, we present a direct bijective proof of the hook-length formula for standard immaculate tableaux, which arose in the study of non-commutative symmetric functions. Our proof is along the spirit of Novelli, Pak and Stoyanovskii's combinatorial proof of the hook-length formula for standard Young tableaux.

We prove the log-concavity of the Fennessey-Larcombe-French sequence based on its three-term recurrence relation, which was recently conjectured by Zhao. The key ingredient of our approach is a sufficient condition for log-concavity of a sequence subject to certain three-term recurrence.

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at $1$, and will be an integer polynomial after multiplying it by a product of the first consecut...

Based on the Hermite--Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $\mathbf{s}$-Eulerian polynomials. We also confirm Hyatt's conjectures on the interlacing p...

We use the method of mutual interlacing to prove two conjectures on the real-rootedness of Eulerian-like polynomials: Brenti's conjecture on $q$-Eulerian polynomials for Weyl groups of type $D$, and Dilks, Petersen, and Stembridge's conjecture on affine Eulerian polynomials for irreducible finite Weyl groups. For the former, we obtain a refinement...

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 $\Lambda_{\mathbb{R}}$ denote the algebra of symmetric functions with real coefficients. If $\varphi$ is the homomorphism from $\Lambda_{\mathbb{R}}$ to $\mathbb{...

We prove two conjectures of Br\"{a}nd\'{e}n on the real-rootedness of polynomials $Q_n(x)$ and $R_n(x)$ which are related to the Boros-Moll polynomials $P_n(x)$. In fact, we show that both $Q_n(x)$ and $R_n(x)$ form Sturm sequences. The first conjecture implies the 2-log-concavity of $P_n(x)$, and the second conjecture implies the 3-log-concavity o...

We propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M,s;A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distributi...

Wang and Yeh proved that if $P(x)$ is a polynomial with nonnegative and nondecreasing coefficients, then $P(x+d)$ is unimodal for any $d>0$. A mode of a unimodal polynomial $f(x)=a_0+a_1x+\cdots + a_mx^m$ is an index $k$ such that $a_k$ is the maximum coefficient. Suppose that $M_*(P,d)$ is the smallest mode of $P(x+d)$, and $M^*(P,d)$ the greatest...

The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let P(x) be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of P(x + 1), which leads to the log-concavity of P(x + c) for any c ≥ 1 due to Llamas and Martínez-Bernal. As a consequence, we obtain the ratio monotonicity...

We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri's rule and the Jacobi–Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. By the principal specialization, this le...

We show that the shifted rank, or srank, of any partition $\lambda$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_{\lambda}$ function in terms of power sum symmetric functions. This gives an affirmative answer to a conjecture of Clifford. As pointed out by Clifford, the notion of the srank can be...

Using the decomposition theory of modular and integral flow polynomials, we answer a problem of Beck and Zaslavsky, by providing a general situation in which the integral flow polynomial is a multiple of the modular flow polynomial.

We propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M, s; A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribu...

We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence $\{\sum_{k=0}^n{n\choose k}^2q^k\}_{n\geq 0}$. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then...

Gioan showed that the number of cycle reversing classes of totally cyclic orientations of a given graph can be calculated as an evaluation of the corresponding Tutte polynomial. We note that the concept of cycle reversing classes of orientations coincides with that of Eulerian-equivalence classes considered by Chen and Stanley, and Kochol. Based on...

We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.

We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity. We begin with a formula of Brändén expressing the q-Narayana numbers as a specialization of Schur functions and, by deriving several symmetric function identiti...

We present an affirmative answer to Stanley's zrank conjecture, namely, the zrank and the rank are equal for any skew partition. We show that certain classes of restricted Cauchy matrices are nonsingular and further-more, the signs are determined by the number of zero entries. We also give a characterization of the rank in terms of the Giambelli-ty...

We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.

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 Br\"and$\mathrm{\acute{e}}$n which expres...

Following the treatment of Blass and Sagan, we present an algorithmic bijection between the Eulerian equivalence classes of totally cyclic orientations and the spanning trees without internal activity edges for a given graph.

We introduce a statistic pmaj(P) for partitions of [n], and show that it is equidistributed with cr2, the number of 2-crossings, over all partitions of [n] with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inv and maj.

We introduce a statistic $\pmaj$ on partitions of $[n]=\{1,2,..., n\}$, and show that it is equidistributed with the number of 2-crossings over partitions of $[n]$ with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inversion number and major...

By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli-type matrices for a given skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the transpose of the dual Jacobi-Trudi matrix are stably equivalent over symmetr...

We present an affirmative answer to Stanley's zrank problem, namely, the zrank and rank are equal for any skew partition. We show that certain classes of restricted Cauchy matrices are nonsingular and furthermore, the signs depend on the number of zero entries. Similar to notion of the jrank of a skew partition, we give a characterization of the ra...

By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli type matrices of a skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the dual Jacobi-Trudi matrix are stably equivalent over symmetric functions. This gives...

The rank of a skew partition λ/μ, denoted rank(λ/μ), is the smallest number r such that λ/μ is a disjoint union of r border strips. Let sλ/μ(1t) denote the skew Schur function sλ/μ evaluated at x1 = ⋯ = xt = 1, xi = 0 for i > t. The zrank of λ/μ, denoted zrank(λ/μ), is the exponent of the largest power of t dividing sλ/μ(1t). Stanley conjectured th...

Given two polynomials f(x) and g(x), we extend the formula expressing the remainder in terms of the roots of these two polynomials to the case where f(x) is a Laurent polynomial. This allows us to give new expressions of a Schur function, which generalize the Giambelli identity.

The rank of a skew partition $\lambda/\mu$, denoted $rank(\lambda/\mu)$, is the smallest number $r$ such that $\lambda/\mu$ is a disjoint union of $r$ border strips. Let $s_{\lambda/\mu}(1^t)$ denote the skew Schur function $s_{\lambda/\mu}$ evaluated at $x_1=...=x_t=1, x_i=0$ for $i>t$. The zrank of $\lambda/\mu$, denoted $zrank(\lambda/\mu)$, is...

We consider a ∞agged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing difie...

We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice...

We introduce the notion of the cutting strip of an outside decomposition of a skew shape, and show that cutting strips are in one-to-one correspondence with outside decompositions for a given skew shape. Outside decompositions are introduced by Hamel and Goulden and are used to give an identity for the skew Schur function that unifies the determina...

We obtain a flagged form of the Cauchy determinant and establish a correspondence between this determinant and nonintersecting lattice paths, from which it follows that Cauchy identity on Schur functions. By choosing di#erent origins and destinations for the lattice paths, we are led to an identity of Gessel on the Cauchy sum of Schur functions in...