Searcharxiv⌕ Search

arXiv subjects

Ernest X. W. Xia

Publications and source records attributed to Ernest X. W. Xia.

18 recordsLinked to original sources
arXiv2026

On Chen's conjecture for the log-concavity of the longest increasing subsequence distribution

Let $P_{n,k}$ count permutations of $[n]=\{1,\ldots,n\}$ whose longest increasing subsequence has length $k$. Chen conjectured that $(P_{n,k})_{k=1}^n$ is log-concave for every positive integer $n$. In this paper, we prove that Chen's conjecture is true when $n\ge 10^{10^{65}}$. The proof combines Gessel's determinantal formula with quantitative coefficient extraction, Tracy--Widom density estimates, and exact inequalities for the extreme lengths.

math.CO↗
arXiv2026

Proofs of eighteen conjectural identities related to the modulo nine Kanade--Russell identities

Uncu and Zudilin studied reflections of finite forms of the modulo nine Kanade--Russell identities under $q\mapsto q^{-1}$, leading to ten conjectural sum--product evaluations, including two proposed by Warnaar. Hickerson conjectured four further identities in his study of contiguous relations for the associated parameterized double sums. By constructing finite forms of Hickerson's sums and studying their reflections, Konenkov proposed two additional product identities, whose conjugates give two more. In this paper, we prove all eighteen conjectural identities by combining Nahm-sum dual evaluations established by the author with parameterized divided-difference identities. For the symmetric reflections, Schur-type decompositions and finite rational identities reduce the required evaluations to nonsingular two-by-two linear systems. The asymmetric and cyclotomic cases are treated through $q$-Airy--theta connection formulas, whereas the Hickerson identities are derived from low-order divided differences.

math.NT↗
arXiv2026

The modulo 9 Kanade--Russell identities and their Nahm-sum duals

Kanade and Russell initiated a family of conjectural Rogers--Ramanujan type identities of moduli $9$ and $12$, which ultimately comprised five modulo $9$ identities and eleven modulo $12$ identities. The eleven modulo $12$ conjectures were subsequently settled through the work of Bringmann, Jennings-Shaffer, and Mahlburg and of Rosengren. In this paper, we prove all five modulo $9$ Kanade--Russell sum-product identities, four individual generalized Nahm-sum dual identities, and a product formula for the natural dual companion of the fifth Kanade--Russell identity, which is expressed as a linear combination of two negative-mixed-term generalized Nahm sums. The first three individual dual identities settle Conjecture~3.6 of Wang and Wang, while the fourth proves the corresponding conjecture of Li and Wang. Our results also connect directly with the recent Dynkin-diagram framework of Sun and Wang for generalized Nahm sums. They identified the rank-two pairs $(T_1,G_2)$ and $(G_2,T_1)$ as unresolved cases whose modularity would follow, respectively, from the first modulo $9$ Kanade--Russell identity and its Wang-Wang dual. The present results prove precisely these two required identities and hence establish the corresponding modularity statements unconditionally.

math.NT↗
arXiv2026

A proof of Andrews-El Bachraoui's conjecture on the parity of coefficients of a $q$-series

Recently, Andrews and El Bachraoui studied a partition function $s_1(n)$, which counts the number of two-color partitions into distinct parts of $n$ whose smallest part occurs in one prescribed color only, while every larger part may occur in either color or in both colors. They obtained a complete description modulo 4 for $s_1(n)$. They also considered a $q$-series $T_{o}(q)$ which is the odd companion series of the generating function for $s_1(n)$. At the end of their paper, they presented a conjecture on the parity of the coefficients of $T_o(q)$. In this paper, we confirm this conjecture. Moreover, we establish an infinite family of congruences modulo 8 for the coefficients of $S_1(q)$ and prove that the set of integers satisfying $s_1(n)\equiv 0\pmod 8$ has natural density one.

math.NT↗
arXiv2025

Combinatorial proof of an inequality on some partitions separated by parity

In 2019, Andrews investigated integer partitions in which all parts of a given parity are smaller than those of the opposite parity and introduced eight partition functions based on the parity of the smaller parts and parts of a given parity appearing at most once or an unlimited number of times. Recently, Bringmann, Craig and Nazaroglu studied the asymptotic behavior of the eight partition functions proved several inequalities for sufficiently large $n$. At the end of their paper, they asked for combinatorial proofs of those inequalities. In this paper, we prove that an inequality on partitions separated by parity holds for $n\geq 373$ by a combinatorial method. This answers a question posed by Bringmann, Craig and Nazaroglu.

math.CO↗
arXiv2025

Asymptotics for moments of the minimal partition excludant in congruence classes

The minimal excludant statistic, which denotes the smallest positive integer that is not a part of an integer partition, has received great interest in recent years. In this paper, we move on to the smallest positive integer whose frequency is less than a given number. We establish an asymptotic formula for the moments of such generalized minimal excludants that fall in a specific congruence class. In particular, our estimation reveals that the moments associated with a fixed modulus are asymptotically ``equal''.

math.NT↗
arXiv2023

Some identities on Lin-Peng-Toh's partition statistic of $k$-colored partitions

Recently, Andrews proved two conjectures on a partition statistic introduced by Beck. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and To introduced a partition statistic of $k$-colored partitions $NB_k(r,m,n)$ which counts the total number of parts of $π^{(1)}$ in each $k$-colored partition $π$ of $n$ with ${\rm crank}_k(π)$ congruent to $r$ modulo $m$ and proved a number of congruences for $NB_k(r,m,n)$. In this paper, we prove some identities on $NB_k(r,m,n)$ which are analogous to Ramanujan's ``most beautiful identity". Moreover, those identities imply some congruences proved by Lin, Peng and Toh.

math.NT↗
arXiv2023

A proof of a conjecture of Mao on Beck's partition statistics modulo 8

Beck introduced two partition statistics $NT(r,m,n)$ and $M_ω(r,m,n)$,which denote the total number of parts in the partition of $n$ with rank congruent to $r$ modulo $m$ and the total number of ones in the partition of $n$ with crank congruent to $r$ modulo $m$, respectively. In recent years, a number of congruences and identities on $NT(r,m,n)$ and $M_ω(r,m,n)$ for some small $m $ have been established.In this paper, we prove an identity on $NT(r,8,n)$ and $M_ω(r,4,n)$ which confirm a conjecture given by Mao.

math.NT↗
arXiv2022

On the log-concavity of $n$-th root of a sequence

In recent years, the log-concavity of $\{\sqrt[n]{S_n}\}_{n\geq 1}$ have been received a lot of attention. Very recently, Sun posed the following conjecture in his new book: the sequences $\{\sqrt[n]{a_n}\}_{n\geq 2}$ and $\{ \sqrt[n]{b_n}\}_{n\geq 1}$ are log-concave, where \[ a_n:= \frac{1}{n}\sum_{k=0}^{n-1} \frac{{n-1\choose k}^2{n+k\choose k}^2 }{4k^2-1} \] and \[ b_n:= \frac{1}{n^3}\sum_{k=0}^{n-1} (3k^2+3k+1){n-1\choose k}^2 {n+k\choose k}^2. \] In this paper, two methods, semi-automatic and analytic methods, are used to confirm Sun's conjecture. The semi-automatic method relies on a criterion on the log-concavity of $\{\sqrt[n]{S_n}\}_{n\geq 1}$ given by us and a mathematica package due to Hou and Zhang, while the analytic method relies on a result due to Xia.

math.CO↗
arXiv2022

Proofs of some conjectures of Chan-Mao-Osburn on Beck's partition statistics

Recently, George Beck introduced two partition statistics $NT(m,j,n)$ and $M_ω(m,j,n)$, which denote the total number of parts in the partition of $n$ with rank congruent to $m$ modulo $j$ and the total number of ones in the partition of $n$ with crank congruent to $m$ modulo $j$, respectively. Andrews proved a congruence on $NT(m,5,n)$ which was conjectured by Beck. Very recently, Chan, Mao and Osburn established a number of Andrews-Beck type congruences and posed several conjectures involving $NT(m,j,n)$ and $M_ω(m,j,n)$. Some of those conjectures were proved by Chern and Mao. In this paper, we confirm the remainder three conjectures of Chan-Mao-Osburn and two conjectures due to Mao. We also present two new conjectures on $M_ω(m,j,n)$ and $NT(m,j,n)$.

math.CO↗
arXiv2013

Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions

Let $\bar{p}(n)$ denote the number of overpartitions of $n$. It was conjectured by Hirschhorn and Sellers that $\bar{p}(40n+35)\equiv 0\ ({\rm mod\} 40)$ for $n\geq 0$. Employing 2-dissection formulas of quotients of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for $\bar{p}(40n+35)$ modulo 5. Using the $(p, k)$-parametrization of theta functions given by Alaca, Alaca and Williams, we give a proof of the congruence $\bar{p}(40n+35)\equiv 0\ ({\rm mod\} 5)$. Combining this congruence and the congruence $\bar{p}(4n+3)\equiv 0\ ({\rm mod\} 8)$ obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we give a proof of the conjecture of Hirschhorn and Sellers.

math.CO↗
arXiv2010

2-Log-concavity of the Boros-Moll Polynomials

The Boros-Moll polynomials $P_m(a)$ arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that $P_m(a)$ is 2-log-concave for any $m\geq 2$. Let $d_i(m)$ be the coefficient of $a^i$ in $P_m(a)$. We also show that the sequence $\{i (i+1)(d_i^{\,2}(m)-d_{i-1}(m)d_{i+1}(m))\}_{1\leq i \leq m}$ is log-concave. This leads another proof of Moll's minimum conjecture.

math.CO↗
arXiv2010

The 2-log-convexity of the Apery Numbers

We present an approach to proving the 2-log-convexity of sequences satisfying three-term recurrence relations. We show that the Apery numbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4 and the large Schroder numbers are all 2-log-convex. Numerical evidence suggests that all these sequences are k-log-convex for any $k\geq 1$ possibly except for a constant number of terms at the beginning.

math.CO↗
arXiv2010

Interlacing Log-concavity of the Boros-Moll Polynomials

We introduce the notion of interlacing log-concavity of a polynomial sequence $\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of $P_m(x)$ interlace the ratios of consecutive coefficients of $P_{m+1}(x)$ for any $m\geq 0$. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which implies that some classical combinatorial polynomials are interlacing log-concave.

math.CO↗
arXiv2009

A Proof of Moll's Minimum Conjecture

Let $d_i(m)$ denote the coefficients of the Boros-Moll polynomials. Moll's minimum conjecture states that the sequence $\{i(i+1)(d_i^2(m)-d_{i-1}(m)d_{i+1}(m))\}_{1\leq i \leq m}$ attains its minimum with $i=m$. This conjecture is a stronger than the log-concavity conjecture proved by Kausers and Paule. We give a proof of Moll's conjecture by utilizing the spiral property of the sequence $\{d_i(m)\}_{0\leq i \leq m}$, and the log-concavity of the sequence $\{i!d_i(m)\}_{0\leq i \leq m}$.

math.CO↗
arXiv2009

The Ratio Monotonicity of the Boros-Moll Polynomials

In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this paper, we show that the Boros-Moll polynomials possess the ratio monotone property which implies the log-concavity and the spiral property. We conclude with a conjecture which is stronger than Moll's conjecture on the $\infty$-log-concavity.

math.CO↗
arXiv2008

The q-WZ Method for Infinite Series

Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove nonterminating basic hypergeometric series identities. This observation enables us to extend the q-WZ method to identities on infinite series. As examples, we will give the q-WZ pairs for some classical identities such as the q-Gauss sum, the $_6ϕ_5$ sum, Ramanujan's $_1ψ_1$ sum and Bailey's $_6ψ_6$ sum.

math.CO↗