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Mingyang Kang

Publications and source records attributed to Mingyang Kang.

5 recordsLinked to original sources
arXiv2026

Real-rootedness of the $\tau$-polynomial under graph joins

For a simple graph $G$ with $n$ vertices, write its chromatic polynomial in the rising factorial basis as $$ \chi_G(x)=\sum_{i=0}^{n}(-1)^{n-i}c_i(G)\langle x\rangle_i,$$ where $ \langle x\rangle_i=x(x+1)\cdots(x+i-1).$ The associated $\tau$-polynomial $$ \tau_G(x)=\sum_{i=0}^{n}c_i(G)x^i $$ was defined and systematically investigated by Brenti in 1992. In this paper, we prove that if the $\tau$-polynomials of two vertex-disjoint simple graphs $G$ and $H$ have only real zeros, then the $\tau$-polynomial of their join $G\vee H$ has only real zeros. This settles a conjecture posed by Brenti, Royle and Wagner since 1994.

math.CO↗
arXiv2026

A solution to Butler's positivity conjecture

Let $\lambda, \mu, \nu$ be distinct partitions such that $\lambda, \mu\subset \nu$ and $|\nu/\lambda|=|\nu/\mu|=1$. We prove Butler's positivity conjecture posed in 1994: the expansion of the Macdonald intersection polynomial \[ \frac{T_\lambda\widetilde{H}_\mu(X;q,t)-T_\mu\widetilde{H}_\lambda(X;q,t)}{T_\lambda-T_\mu} \] in terms of the Schur function basis has coefficients in $\mathbb{Z}_{\geq 0}[q,t]$.

math.CO↗
arXiv2026

A Lam--Postnikov--Pylyavskyy inequality for hybrid Grothendieck polynomials

We prove a multivariate Lam--Postnikov--Pylyavskyy type inequality for hybrid Grothendieck polynomials, unifying and refining results for stable and dual stable Grothendieck polynomials established by Chan--Chen--Pak--Soskin. We also conjecture extensions of the Lam--Postnikov--Pylyavskyy inequality and a conjecture by Thomas--Yong to the (equivariant) Schubert and Grothendieck polynomial setting.

math.CO↗
arXiv2025

Hybrid Grothendieck polynomials

For a skew shape $\lambda/\mu$, we define the hybrid Grothendieck polynomial $${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w}) =\sum_{T\in \mathrm{SVRPP}(\lambda/\mu)} \textbf{x}^{\mathrm{ircont}(T)}\textbf{t}^{\mathrm{ceq} (T)}\textbf{w}^{\mathrm{ex}(T)}$$ as a weight generating function over set-valued reverse plane partitions of shape $\lambda/\mu$. It specializes to \begin{itemize} \item[(1)] the refined stable Grothendieck polynomial introduced by Chan--Pflueger by setting all $t_i=0$; \item[(2)] the refined dual stable Grothendieck polynomial introduced by Galashin--Grinberg--Liu by setting all $w_i=0$. \end{itemize} We show that ${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w})$ is symmetric in the $\textbf{x}$ variables. By building a crystal structure on set-valued reverse plane partitions, we obtain the expansion of ${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w})$ in the basis of Schur functions, extending previous work by Monical--Pechenik--Scrimshaw and Galashin. Based on the Schur expansion, we deduce that hybrid Grothendieck polynomials of straight shapes have saturated Newton polytopes. Finally, using Fomin--Greene's theory on noncommutative Schur functions, we give a combinatorial formula for the image of ${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w})$ (in the case $t_i=\alpha$ and $w_i=\beta$) under the omega involution on symmetric functions. The formula unifies the structures of weak set-valued tableaux and valued-set tableaux introduced by Lam--Pylyavskyy. Several problems and conjectures are motivated and discussed.

math.CO↗