SearcharxivSearch

arXiv subjects

Byungchan Kim

Publications and source records attributed to Byungchan Kim.

At least 19 recordsLinked to original sources

Moments of partition statistics, Bell polynomials and Eisenstein-type series

We develop a systematic method to express generating functions for moments of combinatorial statistics in terms of partition traces. We employ an algebraic approach based on the complete Bell polynomials and their inversion formula, alongside an analytic approach via Fa\`a di Bruno's formula. Our approach can be applied to a wide class of combinatorial statistics, such as the largest part of an integer partition, the partition crank and rank, and the unimodal sequence rank.

math.CO

Multi-Persona Debate System for Automated Scientific Hypothesis Generation

Modern scientific discovery is bottlenecked not by data scarcity, but by the inability to synthesize fragmented knowledge into actionable hypotheses. This challenge is especially acute in battery materials research, where electrochemical performance, interfacial behavior, and manufacturing feasibility must be optimized simultaneously. Here, we present the Multi-Persona Debate System (MPDS), a literature-grounded framework for automated scientific hypothesis generation that combines literature retrieval, long-context large language model reasoning, corpus-driven persona induction, and structured multi-agent debate. MPDS constructs literature snapshots of up to 500 papers, grounds agents in role-specific evidence pools, and conducts a three-round citation-aware debate followed by moderator synthesis, enabling negotiation between personas while preserving evidence traceability. We evaluate MPDS using a temporally controlled protocol excluding direct access to target papers, including two held-out battery-materials case studies and a blinded comparison across 30 matched cases. In sodium-ion anode and all-solid-state battery cathode design tasks, MPDS recovered design logics aligned with experimentally validated solution spaces and generated more mechanistically explicit, process-aware proposals than simpler baselines. To assess the impact of personas and debate, we introduce Integrative Hypothesis Quality scoring. In ablation studies, MPDS achieved the highest mean score among five conditions, with its largest advantage in cross-perspective integration. A laboratory follow-up suggests utility as a diagnostic aid for identifying practical bottlenecks in workflows. These results indicate that structured debate over literature snapshots improves hypothesis formation under coupled engineering constraints and provides a reusable workflow for text-intensive scientific discovery.

cs.CL

Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities

Motivated by recent study on the number of $t$-hooks in partitions arising from Euler's partition identity, we investigate the number of $t$-hooks in the sets from the first Rogers-Ramanujan identity and the first little Göllitz identity. In particular, for $t=1,2$, we obtain the generating functions for the number of $t$-hooks and prove $t$-hook inequalities by deriving asymptotic formulas.

math.NT

Modularity of moments of reciprocal sums for partitions into distinct parts

In this paper, we determine modularity properties of the generating function of $s_k(n)$ which sums $k$-th power of reciprocals of parts throughout all of the partitions of $n$ into distinct parts. In particular, we show that the generating function for $s_k (n)$ is related to Maass Eisenstein series and sesquiharmonic Maass forms.

math.NT

On the distribution of $t$-hooks of doubled distinct partitions

Recently, Griffin, Ono, and Tsai examined the distribution of the number of $t$-hooks in partitions of $n$, which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of $t$-hooks in self-conjugate partitions of $n$. Motivated by these studies, in this paper, we further investigate the number of $t$-hooks in some subsets of partitions. More specifically, we obtain the generating functions for the number of $t$-hooks in doubled distinct partitions and the number of $t$-shifted hooks in strict partitions. Based on these generating functions, we prove that the number of $t$-hooks in doubled distinct partitions and the number of $t$-shifted hooks in strict partitions are both asymptotically normally distributed.

math.CO

Improved asymptotics for moments of reciprocal sums for partitions into distinct parts

In this paper we strongly improve asymptotics for $s_1(n)$ (respectively $s_2(n)$) which sums reciprocals (respectively squares of reciprocals) of parts throughout all the partitions of $n$ into distinct parts. The methods required are much more involved than in the case of usual partitions since the generating functions are not modular and also do not posses product expansions.

math.NT

Alder-type partition inequality at the general level

A Known Alder-type partition inequality of level $a$, which involves the second Rogers-Ramanujan identity when the level $a$ is 2, states that the number of partitions of $n$ into parts differing by at least $d$ with the smallest part being at least $a$ is greater than or equal to that of partitions of $n$ into parts congruent to $\pm a \pmod{d+3}$, excluding the part $d+3-a$. In this paper, we prove that for all values of $d$ with a finite number of exceptions, an arbitrary level $a$ Alder-type partition inequality holds without requiring the exclusion of the part $d+3-a$ in the latter partition.

math.CO

Asymptotics and sign patterns for coefficients in expansions of Habiro elements

We prove asymptotics and study sign patterns for coefficients in expansions of elements in the Habiro ring which satisfy a strange identity. As an application, we prove asymptotics and discuss positivity for the generalized Fishburn numbers which arise from the Kontsevich-Zagier series associated to the colored Jones polynomial for a family of torus knots. This extends Zagier's result on asymptotics for the Fishburn numbers.

math.NT

Dissections of strange $q$-series

In a study of congruences for the Fishburn numbers, Andrews and Sellers observed empirically that certain polynomials appearing in the dissections of the partial sums of the Kontsevich-Zagier series are divisible by a certain $q$-factorial. This was proved by the first two authors. In this paper we extend this strong divisibility property to two generic families of $q$-hypergeometric series which, like the Kontsevich-Zagier series, agree asymptotically with partial theta functions.

math.NT

An overpartition analogue of $q$-binomial coefficients, II: combinatorial proofs and $(q,t)$-log concavity

In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization $\overline{m+n \brack n}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of Delannoy numbers. First we obtain finite versions of classical $q$-series identities such as the $q$-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gaussian polynomials. Then, by constructing involutions, we obtain an identity involving a finite theta function and prove the $(q,t)$-log concavity of $\overline{m+n \brack n}_{q,t}$. We particularly emphasize the role of combinatorial proofs and the consequences of our results on Delannoy numbers. We conclude with some conjectures about the unimodality of $\overline{m+n \brack n}_{q,t}$.

math.CO

Congruences for a mock modular form on $\operatorname{SL}_2(\mathbb{Z})$ and the smallest parts function

Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight $3/2$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo primes $\ell\geq 5$. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo $\ell$. As another application, we show that congruences modulo $\ell$ for the smallest parts function are rare in a precise sense.

math.NT

On the number of even and odd strings along the over partitions of n

Recently, Andrews, Chan, Kim and Osburn introduced the even strings and the odd strings in the overpartitions. We show that their conjecture $A_k (n) \geq B_k (n)$ holds for large enough positive integers n, where A_k(n) (resp. B_k(n)) is the number of odd (resp. even) strings along the overpartitions of n. We introduce m-strings and show that how this new combinatorial object is related with another positivity conjecture of Andrews, Chan, Kim, and Osburn. Finally, we confirm that the positivity conjecture is also true for large enough integers.

math.NT

On spt-crank type functions

In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function $\operatorname{spt}_ω(n)$, which is closely related to Ramanujan's third order mock theta function $ω(q)$. Garvan and Jennings-Shaffer introduce a crank function which explains congruences for $\operatorname{spt}_ω(n)$. In this note, we study asymptotic behavior of this crank function and confirm a positivity conjecture of the crank asymptotically. We also study a sign pattern of the crank and congruences for $\operatorname{spt}_ω(n)$.

math.NT

An overpartition analogue of the $q$-binomial coefficients

We define an overpartition analogue of Gaussian polynomials (also known as $q$-binomial coefficients) as a generating function for the number of overpartitions fitting inside the $M \times N$ rectangle. We call these new polynomials over Gaussian polynomials or over $q$-binomial coefficients. We investigate basic properties and applications of over $q$-binomial coefficients. In particular, via the recurrences and combinatorial interpretations of over q-binomial coefficients, we prove a Rogers-Ramaujan type partition theorem.

math.CO

On the asymptotic behavior of unimodal rank generating functions

In a recent paper, J. Lovejoy and the second author conjectured that ranks for four types of unimodal like sequences satisfy certain inequalities. In this paper, we prove these conjectures asymptotically. For this, we extend Wright's Circle Method and analyze the asymptotic behavior of certain general partial theta functions.

math.NT

The first positive rank and crank moments for overpartitions

In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.

math.NT

Dissections of a "strange" function

The "strange" function of Kontsevich and Zagier is defined by \[F(q):=\sum_{n=0}^\infty(1-q)(1-q^2)\dots(1-q^n).\] This series is defined only when $q$ is a root of unity, and provides an example of what Zagier has called a "quantum modular form." In their recent work on congruences for the Fishburn numbers $ξ(n)$ (whose generating function is $F(1-q)$), Andrews and Sellers recorded a speculation about the polynomials which appear in the dissections of the partial sums of $F(q)$. We prove that a more general form of their speculation is true. The congruences of Andrews-Sellers were generalized by Garvan in the case of prime modulus, and by Straub in the case of prime power modulus. As a corollary of our theorem, we reprove the known congruences for $ξ(n)$ modulo prime powers.

math.NT

Eichler integrals and harmonic weak Maass forms

Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more general groups, namely, $H$-groups by employing the theory of supplementary functions introduced and developed by M. I. Knopp and S. Y. Husseini. In particular, we show that the set of Eichler integrals, which have polynomial period functions, is the same as the set of holomorphic parts of harmonic weak Maass forms of which the non-holomorphic parts are certain period integrals of cusp forms. From this we deduce relations among period functions for harmonic weak Maass forms.

math.NT