arXiv subjects

JiSun Huh

Publications and source records attributed to JiSun Huh.

At least 19 recordsLinked to original sources

arXiv2026

New lower bounds on domination--packing ratios in connected subcubic and cubic graphs

For a graph $G$, let $\gamma(G)$ and $\rho(G)$ denote its domination number and packing number, respectively. Let $c_{\mathrm{sub}}$ and $c_{\mathrm{cub}}$ denote the respective limsups of $\gamma(G)/\rho(G)$ over connected subcubic and connected cubic graphs as $\rho(G)\to\infty$. We prove \[ c_{\mathrm{sub}}\geq\frac{13}{6}, \qquad c_{\mathrm{cub}}\geq\frac{17}{8}, \] by constructing two explicit binary branching families. The connected noncubic subcubic graphs $\widehat B_t^\star$ satisfy \[ |V(\widehat B_t^\star)|=76\cdot2^t-12,\qquad \gamma(\widehat B_t^\star)=26\cdot2^t-4,\qquad \rho(\widehat B_t^\star)=12\cdot2^t-2, \] whereas the connected cubic graphs $\widehat B_t^\bullet$ satisfy \[ |V(\widehat B_t^\bullet)|=108\cdot2^t-14,\qquad \gamma(\widehat B_t^\bullet)=34\cdot2^t-4,\qquad \rho(\widehat B_t^\bullet)=16\cdot2^t-2. \] The constructions use the same binary connector composition and closing lemma, with different connectors and initial assemblies. As a consequence, both families give unbounded additive violations of $\gamma(G)\leq2\rho(G)+1$, disproving the proposed inequality even for connected cubic graphs.

math.CO↗

arXiv2026

A Comparison of cluster algebra structures arising from $i$-boxes and Demazure weaves

We compare two cluster algebras related to a positive element $\mathtt{b}$ in the braid group of finite $ADE$ type. One is the localized bosonic extension ${\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})$ equipped with an initial seed arising from an admissible chain $\mathfrak{C}$ of $i$-boxes, which is deeply connected to monoidal categorification. The other is the coordinate ring $\mathbb{C}[X({\underline{\Delta}} {\boldsymbol{i}})]$ of the braid variety $X({\underline{\Delta}} {\boldsymbol{i}})$ equipped with an initial seed arising from a Demazure weave $\mathfrak{W}$, where ${\boldsymbol{i}}$ and ${\underline{\Delta}}$ are expression sequences of $\mathtt{b}$ and the half twist $\Delta$, respectively. We explicitly construct a Demazure weave $\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})$ for each admissible chain $\mathfrak{C}$ associated with ${\boldsymbol{i}}$, and prove that there exists an algebra isomorphism $\varphi_{{\boldsymbol{i}}}\colon {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})\to\mathfrak{C}[X({\underline{\Delta}} {\boldsymbol{i}})]$ which is compatible with the two seeds arising from $\mathfrak{C}$ and $\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})$. Moreover, the isomorphism $\varphi_{{\boldsymbol{i}}}$ sends the PBW vectors ${\overline{\mathsf{p}}}_{{\boldsymbol{i}},k} \in {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})$ to the coordinates $z_k \in \mathfrak{C}[X({\underline{\Delta}} {\boldsymbol{i}})]$ indexed by the letters of ${\boldsymbol{i}}$. As applications, we investigate a connection between Demazure weaves and signed words via the $i$-boxes and interpret the isomorphism $\varphi_{{\boldsymbol{i}}}$ from the viewpoint of monoidal categorification using Hernandez--Leclerc categories.

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arXiv2026

Thrall's problem for two rows

In this paper, we study Thrall's problem for the higher Lie modules $L_\lambda$. Our main result provides a tableau-theoretic description of the Schur expansion of the character of $L_\lambda$ when $\lambda$ has two rows, thereby solving Thrall's problem in this case. This formula is expressed in terms of standard Young tableaux with major index congruence conditions and a spin-parity condition defined through bijections with Yamanouchi domino tableaux. We also obtain tableau formulas for hook shapes and partitions with distinct parts, and these results extend to all partitions in which each part greater than $2$ occurs at most twice.

math.CO↗

arXiv2025

Bounded Littlewood identities with fixed number of odd rows or odd columns

A Littlewood identity is an identity equating a sum of Schur functions with an infinite product. A bounded Littlewood identity is one where the sum is taken over the partitions with a bounded number of rows or columns. The price to pay is that the infinite product has to be replaced by a determinant. The focus of this article is on refinements of such bounded Littlewood identities where one also prescribes the number of odd-length rows or columns of the partitions. Goulden [{\it Discrete Math.} {\bf99} (1992), 69--77] had given such a refinement in which the number of columns is bounded and the number of odd-length rows is prescribed. We provide refinements where the number of columns is bounded and the number of odd-length columns is prescribed. Furthermore, we present new formulations of such bounded Littlewood identities involving skewing operators. As corollaries we obtain non-standard formulas for numbers of standard Young tableaux with restricted shapes as above. In the last part of the article we discuss combinatorial interpretations of such identities in terms of up-down tableaux. As corollaries, we obtain identities between numbers of standard Young tableaux and numbers of (marked) vacillating tableaux.

math.CO↗

arXiv2025

On 102-avoiding inversion sequences

In this article, we provide a bijection between the set of inversion sequences avoiding the pattern 102 and the set of 2-Schr\"{o}der paths having neither peaks nor valleys and ending with a diagonal step. To achieve this, we introduce two intermediate objects, called UVD paths and labeled $F$-paths, and establish bijections among all four families. For each of these combinatorial objects, we define a natural statistic and enumerate the corresponding structures with respect to this statistic. In addition, we study inversion sequences avoiding 102 and another pattern of length 3, providing refined enumerations according to the same statistic.

math.CO↗

arXiv2025

Refinement of Hikita's $e$-positivity theorem via Abreu--Nigro's $g$-functions and restricted modular law

We study the symmetric functions $ g_{\mm,k}(x;q) $, introduced by Abreu and Nigro for a Hessenberg function $ \mm $ and a positive integer $ k $, which refine the chromatic symmetric function. Building on Hikita's recent breakthrough on the Stanley--Stembridge conjecture, we prove the $ e $-positivity of $ g_{\mm,k}(x;1) $, refining Hikita's result. We also provide a Schur expansion of the sum $ \sum_{k=1}^n e_k(x) g_{\mm,n-k}(x;q) $ in terms of $ P $-tableaux with 1 in the upper-left corner. We introduce a restricted version of the modular law as our main tool. Then, we show that any function satisfying the restricted modular law is determined by its values on disjoint unions of path graphs.

math.CO↗

arXiv2024

Bijections on pattern avoiding inversion sequences and related objects

The number of inversion sequences avoiding two patterns $101$ and $102$ is known to be the same as the number of permutations avoiding three patterns $2341$, $2431$, and $3241$. This sequence also counts the number of Schr\"{o}der paths without triple descents, restricted bicolored Dyck paths, $(101,021)$-avoiding inversion sequences, and weighted ordered trees. We provide bijections to integrate them together by introducing $F$-paths. Moreover, we define three kinds of statistics for each of the objects and count the number of each object with respect to these statistics. We also discuss direct sums of each object.

math.CO↗

arXiv2023

Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties

A \emph{Hessenberg Schubert variety} is an irreducible component of the intersection of a Schubert variety and a Hessenberg variety, defined as the closure of a Schubert cell intersected with the Hessenberg variety. We consider the smoothness of Hessenberg Schubert varieties of regular semisimple Hessenberg varieties of type $A$ in this paper. We consider the smoothness of the intersection of a Schubert variety and a Hessenberg variety to ensure the smoothness of the corresponding Hessenberg Schubert variety. Specifically, we analyze the structure of the GKM graphs of the intersection of a Schubert variety and a Hessenberg variety. Our results show that the regularity of these GKM graphs is completely characterized in terms of pattern avoidance, which is a necessary and sufficient condition for the intersection to be smooth. This shows that our pattern avoidance provides a sufficient condition for the smoothness of a Hessenberg Schubert variety.

math.CO↗

arXiv2023

Bounded Littlewood identities for cylindric Schur functions

The identities which are in the literature often called ``bounded Littlewood identities" are determinantal formulas for the sum of Schur functions indexed by partitions with bounded height. They have interesting combinatorial consequences such as connections between standard Young tableaux of bounded height, lattice walks in a Weyl chamber, and noncrossing matchings. In this paper we prove affine analogs of the bounded Littlewood identities. These are determinantal formulas for sums of cylindric Schur functions. We also study combinatorial aspects of these identities. As a consequence we obtain an unexpected connection between cylindric standard Young tableaux and $ r $-noncrossing and $ s $-nonnesting matchings.

math.CO↗

arXiv2022

Combinatorics on bounded free Motzkin paths and its applications

In this paper, we construct a bijection from a set of bounded free Motzkin paths to a set of bounded Motzkin prefixes that induces a bijection from a set of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give bijections between a set of bounded cornerless Motzkin paths and a set of $t$-core partitions, and a set of bounded cornerless symmetric Motzkin paths and a set of self-conjugate $t$-core partitions. As an application, we get explicit formulas for the number of ordinary and self-conjugate $t$-core partitions with a fixed number of corners.

math.CO↗

arXiv2022

Results on bar-core partitions, core shifted Young diagrams, and doubled distinct cores

Simultaneous bar-cores, core shifted Young diagrams (or CSYDs), and doubled distinct cores have been studied since Morris and Yaseen introduced the concept of bar-cores. In this paper, our goal is to give a formula for the number of these core partitions on $(s,t)$-cores and $(s,s+d,s+2d)$-cores for the remaining cases that are not covered yet. In order to achieve this goal, we observe a characterization of $\bar{s}$-core partitions to obtain characterizations of doubled distinct $s$-core partitions and $s$-CSYDs. By using them, we construct $NE$ lattice path interpretations of these core partitions on $(s,t)$-cores. Also, we give free Motzkin path interpretations of these core partitions on $(s,s+d,s+2d)$-cores.

math.CO↗

arXiv2022

Toric varieties of Schr\"{o}der type

A dissection of a polygon is obtained by drawing diagonals such that no two diagonals intersect in their interiors. In this paper, we define a toric variety of Schr\"{o}der type as a smooth toric variety associated with a polygon dissection. Toric varieties of Schr\"{o}der type are Fano generalized Bott manifolds, and they are isomorphic if and only if the associated Schr\"{o}der trees are the same as unordered rooted trees. We describe the cohomology ring of a toric variety of Schr\"{o}der type using the associated Schr\"{o}der tree and discuss the cohomological rigidity problem.

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arXiv2020

Self-conjugate $(s,s+d,\dots,s+pd)$-core partitions and free rational Motzkin paths

A partition is called an $(s_1,s_2,\dots,s_p)$-core partition if it is simultaneously an $s_i$-core for all $i=1,2,\dots,p$. Simultaneous core partitions have been actively studied in various directions. In particular, researchers concerned with properties of such partitions when the sequence of $s_i$ is an arithmetic progression. In this paper, for $p\geq 2$ and relatively prime positive integers $s$ and $d$, we propose the $(s+d,d;a)$-abacus of a self-conjugate partition and establish a bijection between the set of self-conjugate $(s,s+d,\dots,s+pd)$-core partitions and the set of free rational Motzkin paths with appropriate conditions. For $p=2,3$, we give formulae for the number of self-conjugate $(s,s+d,\dots,s+pd)$-core partitions and the number of self-conjugate $(s,s+1,\dots,s+p)$-core partitions with $m$ corners.

math.CO↗

arXiv2020

On toric ideals arising from signed graphs

A signed graph is a pair $(G,τ)$ of a graph $G$ and its sign $τ$, where a \textit{sign} $τ$ is a function from $\{ (e,v)\mid e\in E(G),v\in V(G), v\in e\}$ to $\{1,-1\}$. Note that graphs or digraphs are special cases of signed graphs. In this paper, we study the toric ideal $I_{(G,τ)}$ associated with a signed graph $(G,τ)$, and the results of the paper give a unified idea to explain some known results on the toric ideals of a graph or a digraph. We characterize all primitive binomials of $I_{(G,τ)}$, and then focus on the complete intersection property. More precisely, we find a complete list of graphs $G$ such that $I_{(G,τ)}$ is a complete intersection for every sign $τ$.

math.CO↗

arXiv2020

The $(s,s+d,\dots,s+pd)$-core partitions and the rational Motzkin paths

In this paper, we propose an $(s+d,d)$-abacus for $(s,s+d,\dots,s+pd)$-core partitions and establish a bijection between the $(s,s+d,\dots,s+pd)$-core partitions and the rational Motzkin paths of type $(s+d,-d)$. This result not only gives a lattice path interpretation of the $(s,s+d,\dots,s+pd)$-core partitions but also counts them with a closed formula. Also we enumerate $(s,s+1,\dots,s+p)$-core partitions with $k$ corners and self-conjugate $(s,s+1,\dots,s+p)$-core partitions.

math.CO↗

arXiv2019

An analogue of chromatic bases and $p$-positivity of skew Schur $Q$-functions

We investigate chromatic symmetric functions in the relation to the algebra $Γ$ of symmetric functions generated by Schur $Q$-functions. We construct natural bases of $Γ$ in terms of chromatic symmetric functions. We also consider the $p$-positivity of skew Schur $Q$-functions and find a class of $p$-positive ribbon Schur $Q$-functions, making a conjecture that they are \emph{all}. We include many concrete computational results that support our conjecture.

math.CO↗

arXiv2019

Counting self-conjugate (s,s+1,s+2)-core partitions

We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions. A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays above the $x$-axis and consists of the up $U=(1,1)$, down $D=(1,-1)$, and flat $F=(1,0)$ steps. We say that a Motzkin path of length $n$ is symmetric if its reflection about the line $x=n/2$ is itself. In this paper, we show that the number of self-conjugate $(s,s+1,s+2)$-cores is equal to the number of symmetric Motzkin paths of length $s$, and give a closed formula for this number.

math.CO↗

arXiv2018

Melting lollipop chromatic quasisymmetric functions and Schur expansion of unicellular LLT polynomials

In this work, we generalize and utilize the linear relations of LLT polynomials introduced by Lee \cite{Lee}. By using the fact that the chromatic quasisymmetric functions and the unicellular LLT polynomials are related via plethystic substitution and thus they satisfy the same linear relations, we can apply the linear relations to both sets of functions. As a result, in the chromatic quasisymmetric function side, we find a class of $e$-positive graphs, called \emph{melting lollipop graphs}, and explicitly prove the $e$-unimodality. In the unicellular LLT side, we obtain Schur expansion formulas for LLT polynomials corresponding to certain set of graphs, namely, complete graphs, path graphs, lollipop graphs and melting lollipop graphs.

math.CO↗