Shane Chern

Proof of the Andrews-El Bachraoui positivity conjecture

Shane Chern, Chun Wang

We prove that for $k \geq 1$ , all coefficients in the expansion of the series $n \geq 0 \sum \frac{( q ^{2 n + 2} , q ^{2 n + 2 k} ; q ^{2} ) _{\infty}}{( q ^{2 n + 1} ; q ^{2} ) _{\infty}^{2}} q^{2 n}$ are positive,…

Combinatorial proof of an identity of Andrews and Yee

Shane Chern

Recently, Andrews and Yee studied two-variable generalizations of two identities involving partition functions $p_{I ¨} (n)$ and $p_{I^} ½ (n)$ introduced by Andrews, Dixit and Yee. In this…

Congruences for partition functions related to mock theta functions

Shane Chern, Li-Jun Hao

Partitions associated with mock theta functions have received a great deal of attention in the literature. Recently, Choi and Kim derived several partition identities from the thir…

math.NT2024

Multiple Rogers--Ramanujan type identities for torus links

Shane Chern

In this paper, we establish simple $k$ -fold summation expressions for the Quot and motivic Cohen--Lenstra zeta functions associated with the $(2, 2 k)$ torus links. Such expressions…

math.NT2020

Linked partition ideals and Kanade--Russell conjectures

Shane Chern, Zhitai Li

This paper will primarily present a method of proving generating function identities for partitions from linked partition ideals. The method we introduce is built on a conjecture b…

math.NT2022

Asymmetric Rogers--Ramanujan type identities. I. The Andrews--Uncu Conjecture

Shane Chern

In this work, we start an investigation of asymmetric Rogers--Ramanujan type identities. The first object is the following unexpected relation $$\sum_{n\ge 0} \frac{(-1)^n q^{3\bin…

On the power mean of a sum analogous to the Kloosterman sum

Shane Chern

In this paper, we study the power mean of a sum analogous to the Kloosterman sum by using analytic methods for character sums.

New congruences for 2-color partitions

Shane Chern

Let $p_{k} (n)$ denote the number of $2$ -color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$ . We will prove a conjecture of Ahmed, Baruah,…

Ramanujan-type congruences for $2$ -color partition triples

Shane Chern, Chun Wang

Let $p_{3, 3} (n)$ denote the number of $2$ -color partition triples of $n$ where one of the colors appears only in parts that are multiples of $3$ . In this paper, we shall establis…

math.NT2024

Leading coefficient in the Hankel determinants related to binomial and $q$ -binomial transforms

Shane Chern, Lin Jiu, Shuhan Li +1

It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to…

Hitting a prime by rolling a die with infinitely many faces

Shane Chern

Alon and Malinovsky recently proved that it takes on average $2.42849 \dots$ rolls of fair six-sided dice until the first time the total sum of all rolls arrives at a prime. Natura…

Unlimited parity alternating partitions

Shane Chern

We introduce a new type of partitions that consists of partitions whose different parts alternate in parity (e.g., $3 + 2 + 2 + 1 + 1$ ). Various properties of this partition function are s…

New congruences for $ℓ$ -regular overpartitions

Shane Chern

Recently, Shen (2016) and Alanazi et al. (2016) studied the arithmetic properties of the $ℓ$ -regular overpartition function $A_{ℓ} (n)$ , which counts the number of o…

math.CO2022

The Ariki--Koike algebras and Rogers--Ramanujan type partitions

Shane Chern, Zhitai Li, Dennis Stanton +2

In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike algebras $H_{C, q; Q_{1}, \dots, Q_{m}} (G (m, 1, n))$ (when the parameters are root…

Note on the truncated generalizations of Gauss' square exponent theorem

Shane Chern

In this note, we investigate J.-C. Liu's work on truncated Gauss' square exponent theorem and obtain more truncations. We also discuss some possible multiple summation extensions o…

On conjectural fermionic formulas for the Macdonald index in Argyres-Douglas theories

Shane Chern, Chanh Tran, Tanay Wakhare

We prove a fermionic-bosonic duality relation for the Macdonald index in Argyres-Douglas theories of type $(A_{1}, D_{2 k + 1})$ , thereby yielding a conjectural fermionic formula due to…

Linked partition ideals and the Alladi--Schur theorem

George E. Andrews, Shane Chern, Zhitai Li

Let $S$ denote the set of integer partitions into parts that differ by at least $3$ , with the added constraint that no two consecutive multiples of $3$ occur as parts. We…

Signed counting of partition matrices

Shane Chern, Shishuo Fu

We prove that the signed counting (with respect to the parity of the `` $inv$ '' statistic) of partition matrices equals the cardinality of a subclass of inversion seq…

A further look at the truncated pentagonal number theorem

Shane Chern

In this paper, we study the asymptotic behavior of the following function $M_{k} (n) := (- 1)^{k - 1} j = 0 \sum k - 1 (p (n - j (3 j + 1) /2) - p (n - j (3 j + 5) /2 - 1)),$ which arises from Andrew…

hep-th2024

q-Identities for parafermion theories

Peter Bouwknegt, Shane Chern, Bolin Han

In this paper we will prove a series of $q$ -identities suggested by the realisation of certain conformal field theories by so-called `coupled free fermions'. We will consider $q$ -s…

New central $q$ -binomial identities

Shane Chern, Karl Dilcher, Lin Jiu

We establish several new series evaluations involving the central $q$ -binomial coefficients, with the inspiration coming from earlier work by Vignat and one of the authors on the l…

More minor summation formulae

Shane Chern, Theresia EisenkÃ¶lbl, Ilse Fischer +8

We prove determinantal-Pfaffian formulae that simultaneously generalise the Pfaffian minor summation formula of Ishikawa and Wakayama and Byun's recent minor summation formula. The…

math.NT2025

A probabilistic proof of Euler's pentagonal number theorem

Shane Chern

We present a probabilistic proof of Euler's pentagonal number theorem based on a shuffling model.

math.DG2024

Juhl type formulas for curved Ovsienko--Redou operators

Shane Chern, Zetian Yan

We prove Juhl type formulas for the curved Ovsienko--Redou operators and their linear analogues, which indicate the associated formal self-adjointness, thereby confirming two conje…

Convolutive sequences, II: Parametrizations

Shane Chern, Dennis Eichhorn, Shishuo Fu +1

In recent work, the authors defined a sequence $(a_{n})_{n \geq 0}$ to be $m$ -convolutive exactly if \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end…

Congruences and recursions for the cubic partition

Shane Chern, Manosij Ghosh Dastidar

Let $p_{2} (n)$ denote the number of cubic partitions. In this paper, we shall present two new congruences modulo $11$ for $p_{2} (n)$ . We also provide an elementary alternative proof of…

Hankel determinants and Jacobi continued fractions for $q$ -Euler numbers

Shane Chern, Lin Jiu

The $q$ -analogs of Bernoulli and Euler numbers were introduced by Carlitz. Similar to the recent results on the Hankel determinants for the $q$ -Bernoulli numbers established by Cha…

A congruence involving the quotients of Euler and its applications (III)

Hao Zhong, Shane Chern, Tianxin Cai

In this paper, we will present several new congruences involving binomial coefficients under integer moduli, which are the continuation of the previous two work by Cai \textit{et a…

Distribution of reducible polynomials with a given coefficient set

Shane Chern

For a given set of integers $S$ , let $R_{n}^{*} (S)$ denote the set of reducible polynomials $f (X) = a_{n} X^{n} + a_{n - 1} X^{n - 1} + \dots + a_{1} X + a_{0}$ over $\mathbb{Z}[…

math.NT2020

1-Shell totally symmetric plane partitions (TSPPs) modulo powers of 5

Shane Chern

Let $s (n)$ be the number of 1-shell totally symmetric plane partitions (TSPPs) of $n$ . In this paper, an infinite family of congruences modulo powers of $5$ for $s (n)$ will be dedu…

Formulas for Partition $k$ -Tuples with $t$ -Cores

Shane Chern

Let $A_{t, k} (n)$ denote the number of partition $k$ -tuples of $n$ where each partition is $t$ -core. In this paper, we establish formulas of $A_{t, k} (n)$ for some values of $t$ and…

Ramanujan's theta functions and internal congruences modulo arbitrary powers of $3$

Shane Chern, Dazhao Tang

In this work, we investigate internal congruences modulo arbitrary powers of $3$ for two functions arising from Ramanujan's classical theta functions $I ¨ (q)$ and $I ¨ (q)$ . By lettin…

An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands

Shane Chern, James A. Sellers

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{partitions with designated summands}. These are built by taking unrestricted integ…

Overpartitions with bounded part differences

Shane Chern, Ae Ja Yee

We generalize recent results of Breuer and Kronholm, and Chern on partitions and overpartitions with bounded differences between largest and smallest parts. We prove our generaliza…

Reciprocity between partitions and compositions

George Beck, Shane Chern

In this paper, we extend the work of Andrews, Beck and Hopkins by considering partitions and compositions with bounded gaps between each pair of consecutive parts. We show that bot…

math.CO2020

5-Dissections and sign patterns of Ramanujan's parameter and its companion

Shane Chern, Dazhao Tang

In 1998, Michael Hirschhorn discovered 5-dissections of the Rogers--Ramanujan continued fraction $R (q)$ and its reciprocal. In this paper, we obtain the 5-dissections for functions…

On partitions with even parts below odd parts

Shane Chern

Recently, Andrews gave a detailed study of partitions with even parts below odd parts in which only the largest even part appears an odd number of times. In this paper, we provide…

math.CO2022

Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions

Shane Chern, Shishuo Fu, Zhicong Lin

Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$ -avoiding in…

math.NT2024

General coefficient-vanishing results associated with theta series

Shane Chern, Dazhao Tang

There are a number of sporadic coefficient-vanishing results associated with theta series, which suggest certain underlying patterns. By expanding theta powers as linear combinatio…

An infinite family of congruences arising from a second order mock theta function

Shane Chern, Chun Wang

Let $I^^{2} (q) = \sum_{n \geq 0} b (n) q^{n}$ be a second order mock theta function defined by $n \geq 0 \sum \frac{q ^{n (n + 1)} ( - q ^{2} ; q ^{2} ) _{n}}{( q ; q ^{2} ) _{n + 1}^{2}} .$ In this paper, we obt…

An overpartition analogue of partitions with bounded differences between largest and smallest parts

Shane Chern

We study the generating function for overpartitions with bounded differences between largest and smallest parts, which is analogous to a result of Breuer and Kronholm on integer pa…

Note on sums involving the Euler function

Shane Chern

In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} Ï\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qq…

math.CO2023

Linked partition ideals and a family of quadruple summations

George E. Andrews, Shane Chern

Recently, $4$ -regular partitions into distinct parts are connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews fo…

math.CO2019

Linked partition ideals, directed graphs and $q$ -multi-summations

Shane Chern

Finding an Andrews--Gordon type generating function identity for a linked partition ideal is difficult in most cases. In this paper, we will handle this problem in the setting of g…

On certain weighted 7-colored partitions

Shane Chern, Dazhao Tang

Inspired by Andrews' 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities…

math.CO2019

Partitions and the maximal excludant

Shane Chern

For each nonempty integer partition $I ¨ $ , we define the maximal excludant of $I ¨ $ to be the largest nonnegative integer smaller than the largest part of $I ¨ $ that is not a part of…

q-fin.MF2017

Pricing VIX Derivatives With Free Stochastic Volatility Model

Wei Lin, Shenghong Li, Shane Chern

In this paper, we relax the power parameter of instantaneous variance and develop a new stochastic volatility plus jumps model that generalize the Heston model and 3/2 model as spe…

Finite Kleshchev bipartitions and $q$ -trinomial coefficients

Shane Chern

The Kleshchev multipartitions arise in the representation theory for the Ariki-Koike algebras. In previous work, Li, Stanton, Xue, Yee, and the author considered a refined enumerat…

math.CO2022

Linked partition ideals and Euclidean billiard partitions

Shane Chern

Euclidean billiard partitions were recently introduced by Andrews, Dragovic and Radnovic in their study of periodic trajectories of ellipsoidal billiards in the Euclidean space. Th…

math.NT2026

Tadpole Nahm sum as a Wronskian

Shane Chern, Chanh Tran, Tanay Wakhare

We show that the tadpole Nahm sum is essentially a specialization of a series studied by Bartlett and Warnaar. This overlooked connection was first discovered by the internal model…

math.CO2020

Vanishing coefficients in several $q$ -series expansions related to the Rogers--Ramanujan continued fraction

Shane Chern, Dazhao Tang

In this paper, we investigate several infinite products with vanishing Taylor coefficients in arihmetic progressions. These infinite products are closely related to the Rogers--Ram…

math.NT2026

Asymptotics for the Enumeration of Commuting Matrices over Finite Fields

Kathrin Bringmann, Shane Chern, Johann Franke +1

We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.

Parity considerations for drops in cycles on ${1, 2, \dots, n}$

Shane Chern

In 2019, A. Lazar and M. L. Wachs conjectured that the number of cycles on $[2 n]$ with only even-odd drops equals the $n$ -th Genocchi number. In this paper, we restrict our attenti…

On a conjecture of George Beck

Shane Chern

In this paper, we prove a conjecture proposed by George Beck, which involves gap-free partitions and partitions with distinct parts.

Some inequalities for Garvan's bicrank function of 2-colored partitions

Shane Chern, Dazhao Tang, Liuquan Wang

In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted v…

Some inequalities for $k$ -colored partition functions

Shane Chern, Shishuo Fu, Dazhao Tang

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$ -colored partition functions $p_{- k} (n)$ for all $k \geq 2$ . This enables us to ext…

math.NT2025

Asymptotics for moments of the minimal partition excludant in congruence classes

Shane Chern, Ernest X. W. Xia

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,…

math.AG2025

Multiple Rogers-Ramanujan type identities for inert quadratic orders

Shane Chern, Yifeng Huang

We compute the Quot and finitized Coh zeta functions of the inert quadratic orders $F_{q} [[T]] + T^{m} F_{q^{2}} [[T]]$ for every $m \geq 1$ in terms of a $2 m$ -fold mult…

math.CO2020

On 0012-avoiding inversion sequences and a Conjecture of Lin and Ma

Shane Chern

The study of pattern avoidance in inversion sequences recently attracts extensive research interests. In particular, Zhicong Lin and Jun Ma conjectured a formula that counts the nu…

A curious identity and its applications to partitions with bounded part differences

Shane Chern

In this note, we present a curious $q$ -series identity with applications to certain partitions with bounded part differences.

Elementary Proofs of Arithmetic Properties for Schur-Type Overpartitions Modulo Small Powers of 2

Shane Chern, Robson da Silva, James A. Sellers

In 2022, Broudy and Lovejoy extensively studied the function $S (n)$ which counts the number of overpartitions of \emph{Schur-type}. In particular, they proved a number of congruenc…

math.PR2023

A central limit theorem for a card shuffling problem

Shane Chern, Lin Jiu, Italo Simonelli

Given a positive integer $n$ , consider a random permutation $I ¨ $ of the set ${1, 2, \dots, n}$ . In $I ¨ $ , we look for sequences of consecutive integers that appear in adjacent posi…

math.NT2019

Asymptotics for the Taylor coefficients of certain infinite products

Shane Chern

Let $(m_{1}, \dots, m_{J})$ and $(r_{1}, \dots, r_{J})$ be two sequences of $J$ positive integers satisfying $1 \leq r_{j} < m_{j}$ for all $j = 1, \dots, J$ . Let $(I^´_{1}, \dots, I^´_{J})$ be a sequence of…

Elementary proof of congruences modulo 25 for broken $k$ -diamond partitions

Shane Chern, Dazhao Tang

Let $I^_{k} (n)$ denote the number of $k$ -broken diamond partitions of $n$ . Quite recently, the second author proved an infinite family of congruences modulo 25 for $I^_{k} (n)$ with…

Some congruences modulo 5 and 25 for overpartition

Shane Chern, Manosij Ghosh Dastidar

We present two new Ramanujan-type congruences modulo 5 for overpartition. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo…

Arithmetic properties for cubic partition pairs modulo powers of 3

Shane Chern

Let $b (n)$ denote the number of cubic partition pairs of $n$ . In this paper, we aim to provide a strategy to obtain arithmetic properties of $b (n)$ . This gives affirmative answers…

Multi-dimensional $q$ -summations and multi-colored partitions

Shane Chern, Shishuo Fu, Dazhao Tang

Motivated by Alladi's recent multi-dimensional generalization of Sylvester's classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contain…

math.CO2025

Domino tilings, nonintersecting lattice paths and subclasses of Koutschan-Krattenthaler-Schlosser determinants

Qipin Chen, Shane Chern, Atsuro Yoshida

Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determi…

On the $k$ -measure of partitions and distinct partitions

George E. Andrews, Shane Chern, Zhitai Li

The $k$ -measure of an integer partition was recently introduced by Andrews, Bhattacharjee and Dastidar. In this paper, we establish trivariate generating function identities counti…

Remarks on the distribution of the primitive roots of a prime

Shane Chern

Let $F_{p}$ be a finite field of size $p$ where $p$ is an odd prime. Let $f (x) \in F_{p} [x]$ be a polynomial of positive degree $k$ that is not a $d$ -th power in $\math…

q-fin.CP2015

Consistent Pricing of VIX and Equity Derivatives with the 4/2 Stochastic Volatility Plus Jumps Model

Wei Lin, Shenghong Li, Xingguo Luo +1

In this paper, we develop a 4/2 stochastic volatility plus jumps model, namely, a new stochastic volatility model including the Heston model and 3/2 model as special cases. Our mod…

Proof of Cigler's conjecture on $q$ -Hoggatt numbers

Shane Chern, Wenle Shi

We prove the nonnegativity and palindromicity of a family of polynomials arising from $q$ -Hoggatt numbers. The nonnegativity is derived from Stanley's $(P, I ¨ )$ -partition theory thr…

Integral right triangle and rhombus pairs with a common area and a common perimeter

Shane Chern

We prove that there are infinitely many integral right triangle and $I^¸$ -integral rhombus pairs with a common area and a common perimeter by the theory of elliptic curves.

Linked partition ideals and Russell's three-colored partitions

Shane Chern

Russell recently introduced a family of three-colored partitions in analogy with earlier work by Alladi and Gordon. He showed that their enumerations are surprisingly connected to…

math.NT2021

On a congruence involving harmonic series and Bernoulli numbers

Shane Chern

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime $p$ , $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,p)=1\Ä±+j+k=p}}\frac{1}…

math.CO2024

Multi-headed lattices and Green functions

Qipin Chen, Shane Chern, Lin Jiu

Lattice geometries and random walks on them are of great interest for their applications in different fields such as physics, chemistry, and computer science. In this work, we focu…

math.NT2015

A note on balancing binomial coefficients

Shane Chern

In 2014, T. Komatsu and L. Szalay studied the balancing binomial coefficients. In this paper, we focus on the following Diophantine equation $$\binom{1}{5}+\binom{2}{5}+...+\binom{…

Nonmodular infinite products and a conjecture of Seo and Yee

Shane Chern

We will tackle a conjecture of S. Seo and A. J. Yee, which says that the series expansion of $1/ (q, - q^{3}; q^{4})_{\infty}$ has nonnegative coefficients. Our approach relies on an approxi…

math.CO2025

Convolutive sequences, I: Through the lens of integer partition functions

Shane Chern, Dennis Eichhorn, Shishuo Fu +1

Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}…

Asymptotics for the Fourier coefficients of eta-quotients

Shane Chern

We study the asymptotics for the Fourier coefficients of a broad class of eta-quotients, $r = 1 \prod R (k \geq 1 \prod (1 - q^{m_{r} k}))^{I^´_{r}},$ where $m_1,\l…