Shane Chern
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View article: Asymptotics for the enumeration of commuting matrices over finite fields
Asymptotics for the enumeration of commuting matrices over finite fields Open
We give asymptotic expressions for the number of commuting matrices over finite fields. For this we use product expansions for the corresponding generating functions.
View article: Signed counting of partition matrices
Signed counting of partition matrices Open
We prove that the signed counting (with respect to the parity of the ``$\operatorname{inv}$'' statistic) of partition matrices equals the cardinality of a subclass of inversion sequences. In the course of establishing this result, we intro…
View article: Domino tilings, nonintersecting lattice paths and subclasses of Koutschan-Krattenthaler-Schlosser determinants
Domino tilings, nonintersecting lattice paths and subclasses of Koutschan-Krattenthaler-Schlosser determinants Open
Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattic…
View article: Asymptotics for moments of the minimal partition excludant in congruence classes
Asymptotics for moments of the minimal partition excludant in congruence classes Open
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 frequenc…
View article: Convolutive sequences, I: Through the lens of integer partition functions
Convolutive sequences, I: Through the lens of integer partition functions Open
Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic convol…
View article: A probabilistic proof of Euler's pentagonal number theorem
A probabilistic proof of Euler's pentagonal number theorem Open
We present a probabilistic proof of Euler's pentagonal number theorem based on a shuffling model.
View article: Multiple Rogers--Ramanujan type identities for torus links
Multiple Rogers--Ramanujan type identities for torus links Open
In this paper, we establish simple $k$-fold summation expressions for the Quot and motivic Cohen--Lenstra zeta functions associated with the $(2,2k)$ torus links. Such expressions lead us to some multiple Rogers--Ramanujan type identities …
View article: q-Identities for parafermion theories
q-Identities for parafermion theories Open
In this paper, we will prove a series of q -identities suggested by the realisation of certain conformal field theories involving the so-called ‘coupled free fermions.’ We will consider q -series arising from coupled free fermions construc…
View article: Juhl type formulas for curved Ovsienko--Redou operators
Juhl type formulas for curved Ovsienko--Redou operators Open
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 conjectures of Case, Lin, and Yuan. We also offer an extension o…
View article: Hankel determinants and Jacobi continued fractions for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math>-Euler numbers
Hankel determinants and Jacobi continued fractions for -Euler numbers Open
The -analogs of Bernoulli and Euler numbers were introduced by Carlitz in 1948. Similar to recent results on the Hankel determinants for the -Bernoulli numbers established by Chapoton and Zeng, we perform a parallel analysis for the -Euler…
View article: Leading coefficient in the Hankel determinants related to binomial and $q$-binomial transforms
Leading coefficient in the Hankel determinants related to binomial and $q$-binomial transforms Open
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 $q$-binomial transforms and study the behavior of the lead…
View article: An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands}
An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands} Open
In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are built by taking unrestricted integer partitions and designating exactly one part of each size. In th…
View article: A central limit theorem for a card shuffling problem
A central limit theorem for a card shuffling problem Open
Given a positive integer $n$, consider a random permutation $τ$ of the set $\{1,2,\ldots, n\}$. In $τ$, we look for sequences of consecutive integers that appear in adjacent positions: a maximal such a sequence is called a block. Each bloc…
View article: Ramanujan's theta functions and internal congruences modulo arbitrary powers of $3$
Ramanujan's theta functions and internal congruences modulo arbitrary powers of $3$ Open
In this work, we investigate internal congruences modulo arbitrary powers of $3$ for two functions arising from Ramanujan's classical theta functions $φ(q)$ and $ψ(q)$. By letting \begin{align*} \sum_{n\ge 0} ph_3(n) q^n:=\dfrac{φ(-q^3)}{φ…
View article: Elementary Proofs of Arithmetic Properties for Schur-Type Overpartitions Modulo Small Powers of 2
Elementary Proofs of Arithmetic Properties for Schur-Type Overpartitions Modulo Small Powers of 2 Open
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 congruences satisfied by $S(n)$ modulo $2$, $4$, and $5$. In this wo…
View article: An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands
An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands Open
In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrenc…
View article: Nearly self-conjugate integer partitions
Nearly self-conjugate integer partitions Open
We investigate integer partitions $\lambda$ of $n$ that are nearly self-conjugate in the sense that there are $n - 1$ overlapping cells among the Ferrers diagram of $\lambda$ and its transpose, by establishing a correspondence, through the…
View article: Hitting a prime by rolling a die with infinitely many faces
Hitting a prime by rolling a die with infinitely many faces Open
Alon and Malinovsky recently proved that it takes on average $2.42849\ldots$ rolls of fair six-sided dice until the first time the total sum of all rolls arrives at a prime. Naturally, one may extend the scenario to dice with a different n…
View article: Hankel determinants and Jacobi continued fractions for $q$-Euler numbers
Hankel determinants and Jacobi continued fractions for $q$-Euler numbers Open
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 Chapoton and Zeng, we determine parallel evaluations for the $…
View article: $q$-Log-concavity and $q$-unimodality of Gaussian polynomials and a problem of Andrews and Newman
$q$-Log-concavity and $q$-unimodality of Gaussian polynomials and a problem of Andrews and Newman Open
We answer a nonnegativity problem of G. E. Andrews and D. Newman by the $q$-unimodality of Gaussian polynomials. Some new considerations of the $q$-log-concavity and $q$-unimodality of Gaussian polynomials from a purely partition-theoretic…
View article: Linked partition ideals and a family of quadruple summations
Linked partition ideals and a family of quadruple summations Open
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 for the two types of partitions. Such an extension is made po…
View article: On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-measure of partitions and distinct partitions
On the -measure of partitions and distinct partitions Open
The -measure of an integer partition was recently introduced by Andrews, Bhattacharjee and Dastidar. In this paper, we establish trivariate generating function identities counting both the length and the -measure for partitions and distinc…
View article: Inter‐Metal Interaction with a Threshold Effect in NiCu Dual‐Atom Catalysts for CO<sub>2</sub> Electroreduction
Inter‐Metal Interaction with a Threshold Effect in NiCu Dual‐Atom Catalysts for CO<sub>2</sub> Electroreduction Open
Dual‐atom catalysts (DACs) have become an emerging platform to provide more flexible active sites for electrocatalytic reactions with multi‐electron/proton transfer, such as the CO 2 reduction reaction (CRR). However, the introduction of a…
View article: Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions
Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions Open
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 inversion sequences can be enumerated by the OEIS entry A2182…
View article: The Ariki--Koike algebras and Rogers--Ramanujan type partitions
The Ariki--Koike algebras and Rogers--Ramanujan type partitions Open
In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike algebras $\mathcal{H}_{\mathbb{C},q;Q_1,\ldots, Q_m}\big(G(m, 1, n)\big)$ (when the parameters are roots of unity and $q\neq 1$) are labeled by the so-called Kles…
View article: General coefficient-vanishing results associated with theta series
General coefficient-vanishing results associated with theta series Open
There are a number of sporadic coefficient-vanishing results associated with theta series, which suggest certain underlying patterns. By expanding theta powers as linear combinations of products of theta functions, we present two strategie…
View article: Diagonal Hooks and a Schmidt-Type Partition Identity
Diagonal Hooks and a Schmidt-Type Partition Identity Open
In a recent paper of Andrews and Paule, several Schmidt-type partition identities are considered within the framework of MacMahon's Partition Analysis. Following their work, we derive a new Schmidt-type identity concerning diagonal hooks o…
View article: On 0012-avoiding inversion sequences and a conjecture of Lin and Ma
On 0012-avoiding inversion sequences and a conjecture of Lin and Ma Open
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 number of inversion sequences avoiding the pattern $0012$. We…