arXiv subjects

Jianxi Mao

Publications and source records attributed to Jianxi Mao.

12 recordsLinked to original sources

arXiv2026

Engel's Interval Packing Problem in the Boolean Lattice

Let $\mathcal{B}_n$ be the Boolean lattice of all subsets of $[n]$ and let $\mathcal{P}_{n;\ell,u}$ be the subposet of $\mathcal{B}_n$ induced by the consecutive levels $\ell,\ell+1,\ldots,u$. We determine $ν_{n;\ell,u}$, the maximum size of a family of pairwise disjoint maximal intervals in $\mathcal P_{n;\ell,u}$, whenever $u\le ({n+\ell^2})/({\ell+1})$. This completely settles Engel's problem~[Combin. Probab. Comput., 1996]. The proof is constructive. We also record consequences for weakly cross-intersecting set-pair systems and discuss the three-level case.

math.CO↗

arXiv2026

Proof of Almkvist's conjecture on the unimodality of partition polynomials

For integers $r\ge2$ and $n\ge1$, let $$ F_{r,n}(q)=\prod_{k=1}^{n}\frac{1-q^{rk}}{1-q^k}. $$ The coefficient of $q^j$ in $F_{r,n}(q)$ counts partitions of $j$ into parts at most $n$, each occurring at most $r-1$ times. Hughes proved that $F_{2,n}(q)=\prod_{k=1}^{n}(1+q^k)$ is unimodal for every $n\ge 1$. This result was reproved by Stanley using an algebraic approach and Odlyzko and Richmond using an analytic approach. Almkvist conjectured that $F_{r,n}(q)$ is unimodal in the following two cases: every even $r$ and every $n\ge 1$; every odd $r$ and every $n\ge 11$. He proved that this conjecture is true for $3\le r \le 20$ and $r=100,101$. In this paper, we completely settle the conjecture.

math.CO↗

arXiv2026

Preservation of log-concavity under Hadamard products

For a nonzero real polynomial $p$, let $\W(p)$ denote the numerator of its ordinary generating function. We prove that if the coefficients of both $\W(p)$ and $\W(q)$ are nonnegative and log-concave with no internal zeros, then so are the coefficients of $\W(pq)$. This provides an affirmative answer to a question of Brändén, Ferroni, and Jochemko. As applications, we derive corresponding results for finite products and for Cartesian products of lattice polytopes, answering a question of Ferroni and Higashitani.

math.CO↗

arXiv2026

Eulerian triangle is totally positive of order 3

Brenti conjectured that the Eulerian triangle is totally positive. In this paper, we prove that the Eulerian triangle is totally positive of order 3, thereby providing a partial affirmative answer to Brenti's conjecture. In addition, we establish total positivity of order 3 for several families of generalized Eulerian triangles.

math.CO↗

arXiv2026

Pfaffian--Toeplitz identities, Schur positivity, and the $q$-log-convexity of Baxter polynomials

We use the Pfaffian minor summation formula together with Jacobi--Trudi Toeplitz matrices to derive Pfaffian expansions in skew Schur functions. This yields explicit Schur expansions for several generating functions involving products of skew Schur functions. In particular, using sparse skew-symmetric matrices, we provide a Pfaffian proof of a Schur-positive identity arising in the study of the $q$-log-convexity of the Narayana polynomials. As the main application, we prove that the Baxter polynomials form a $q$-log-convex sequence. We further show that the Baxter transformation defined by the refined Baxter numbers preserves log-convexity. Finally, by realizing the $q$-refined Baxter numbers as principal specializations of rectangular Schur functions, we prove that the array of $q$-refined Baxter numbers is $q$-log-concave both along each row and along each column. Both $q$-log-concavity results extend naturally to the $q$-analogues of the $d$-Hoggatt numbers.

math.CO↗

arXiv2026

The Narayana transformation

For $m\in\mathbb{Z}_{\geq 0}$, let \[ N_{n,m}(x)={}_2F_1(-n,-n-m;m+1;x), \] which specializes to the Narayana polynomials of types $B$ and $A$ for $m=0$ and $m=1$, respectively. We prove that the associated basis transformation \[ T_{N_m}\left(\sum_{k=0}^n a_kx^k\right)=\sum_{k=0}^n a_kN_{k,m}(x) \] maps every real-rooted polynomial with nonnegative coefficients to a real-rooted polynomial. The proof is based on the rectangular additive convolution of polynomials. We then apply this result to products of lower triangular matrices and obtain a general criterion ensuring that their row generating functions remain real-rooted. As consequences, we recover this property for powers and products of several classical triangular matrices, including Pascal's triangle, the Stirling triangles, and the Narayana triangles of types $A$ and $B$. We conclude with conjectures concerning the squares of the Eulerian and Delannoy triangles.

math.CO↗

arXiv2026

Total positivity of transformation matrices for uniform subdivisions

The transformation of the $h$-vector of a finite simplicial complex under an $\mathcal{F}$-uniform subdivision is encoded by a transformation matrix. Mu and Welker conjectured that the transformation matrix of the barycentric subdivision is totally positive. In this paper, we give a new combinatorial proof of this conjecture. We also prove the total positivity of the transformation matrix of the interval subdivision. In addition, we establish a sufficient condition for the transformation matrix of a uniform subdivision to be totally positive of order $2$ (TP$_2$), thereby partially answering a question of Mu and Welker. As an application, we show that the transformation matrix of the $r$-colored barycentric subdivision is TP$_2$.

math.CO↗

arXiv2026

The analytic properties of Hoggatt triangles

The $d$-Hoggatt triangle is a lower triangular matrix whose entries are given by specific minors of Pascal's triangle formed by consecutive $d$ rows and $d$ columns. The cases $d=1,2,3$ correspond to Pascal's triangle, the Narayana triangle, and the Baxter triangle, respectively. In this paper, we present the infinite log-concavity of the row and column sequences, the log-concavity of the sequences along transversals, and the eventual log-convexity of the sequences along rays of the $d$-Hoggatt triangle. In addition, we prove the asymptotic normality of the row sequences and total positivity of the $d$-Hoggatt triangle.

math.CO↗

arXiv2021

Equidistributions around special kinds of descents and excedances

We consider a sequence of four variable polynomials by refining Stieltjes' continued fraction for Eulerian polynomials. Using combinatorial theory of Jacobi-type continued fractions and bijections we derive various combinatorial interpretations in terms of permutation statistics for these polynomials, which include special kinds of descents and excedances in a recent paper of Baril and Kirgizov. As a by-product, we derive several equidistribution results for permutation statistics, which enables us to confirm and strengthen a recent conjecture of Vajnovszki and also to obtain several compagnion permutation statistics for two bistatistics in a conjecture of Baril and Kirgizov.

math.CO↗

arXiv2021

Yet another criterion for the total positivity of Riordan arrays

Let $R=\mathcal{R}(d(t),h(t))$ be a Riordan array, where $d(t)=\sum_{n\ge 0}d_nt^n$ and $h(t)=\sum_{n\ge 0}h_nt^n$. We show that if the matrix \begin{equation*} \left[\begin{array}{ccccc} d_0 & h_0 & 0 & 0 &\cdots\\ d_1 & h_1 & h_0 & 0 &\\ d_2 & h_2 & h_1 & h_0 &\\ \vdots&\vdots&&&\ddots \end{array}\right] \end{equation*} is totally positive, then so is the Riordan array $R$.

math.CO↗

arXiv2020

Eulerian polynomials and excedance statistics

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials. Our formulae are comparable with Zhuang's generalizations [Adv. in Appl. Math. 90 (2017) 86-144] using descent statistics of permutations. Our methods include permutation enumeration techniques involving variations of classical bijections from permutations to Laguerre histories, explicit continued fraction expansions of combinatorial generating functions in Shin and Zeng [European J. Combin. 33 (2012), no. 2, 111--127] and cycle version of modified Foata-Strehl action. We also prove similar formulae for restricted permutations such as derangements and permutations avoiding certain patterns. Moreover, we provide new combinatorial interpretations for the $γ$-coefficients of the inversion polynomials restricted on $321$-avoiding permutations.

math.CO↗

arXiv2019

Further equidistribution of set-valued statistics on permutations

We construct bijections to show that two pairs of sextuple set-valued statistics of permutations are equidistributed on symmetric groups. This extends a recent result of Sokal and the second author valid for integer-valued statistics as well as a previous result of Foata and Han for bivariable set-valued statistics.

math.CO↗