arXiv subjects

Xiaowei Lin

Publications and source records attributed to Xiaowei Lin.

4 recordsLinked to original sources

arXiv2026

Combinatorial formulas for Macdonald polynomials by superizations

In this paper, we derive new combinatorial formulas for symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ and non-symmetric Macdonald polynomials $E_{\gamma}(X;q,t)$, in terms of several new statistics and the major index, for a partition $\lambda$ and a weak composition $\gamma$. Compared to previous formulas, these new formulas contain the fewest terms and lead to explicit $(q,t)$-formulas for the coefficients in the monomial expansion of $P_{\lambda}(X;q,t)$. In particular, the combinatorial formula for $E_{\gamma}(X;q,t)$ extends the one for $E_{\lambda}(X;q,t)$ indexed by a partition $\lambda$, due to Corteel, Mandelshtam and Williams (2022). Three existing formulas for $P_{\lambda}(X;q,t)$ established by Corteel, Mandelshtam and Williams (2022), by Corteel, Haglund, Mandelshtam, Mason and Williams (2022), and by Mandelshtam (2025) are recovered. Our proof relies on two new statistics on super fillings, employing the superization formulas of Haglund--Haiman--Loehr (2005) and Ayyer--Mandelshtam--Martin (2023), together with our recent approach to modified Macdonald polynomials.

math.CO↗

arXiv2025

The monomial expansions of modified Macdonald polynomials

We discover a family $A$ of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma\in T(\lambda)}x^{\sigma}q^{maj(\sigma)}t^{\eta(\sigma)}$$ for each statistic $\eta\in A$. Building upon this new formula, we establish four compact formulas for the modified Macdonald polynomials, namely, $$\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma}d_{\varepsilon}(\sigma)x^{\sigma}q^{maj(\sigma)}t^{\eta(\sigma)}$$ which is summed over all canonical or dual canonical fillings of a Young diagram and $d_{\varepsilon}(\sigma)$ is a product of $t$-multinomials. Finally, the compact formulas enable us to derive four explicit expressions for the monomial expansion of modified Macdonald polynomials, one of which coincides with the formula given by Garbali and Wheeler (2020).

math.CO↗

arXiv2024

Modified Macdonald polynomials and mu-Mahonian statistics

The Haglund--Haiman--Loehr theorem provides the following combinatorial formula for the modified Macdonald polynomials: $$\tilde{H}_{\mu}(X;q,t)=\sum_{\sigma: \mu\rightarrow \mathbb{P}}x^{\sigma}t^{maj(\sigma)}q^{inv(\sigma)}.$$ Inspired by Martin's multiline-queue formula for the stationary distribution of multitype asymmetric simple exclusion processes, Corteel, Haglund, Mandelshtam, Mason and Williams recently introduced the queue inversion statistic $quinv$ and conjectured that the tableaux formula for $\tilde{H}_{\mu}(X;q,t)$ is invariant if the inversion statistic $inv$ is replaced by $quinv$. This was subsequently resolved by Ayyer, Mandelshtam and Martin, who proposed a stronger conjecture on the equivalence of the two refined formulas for $\tilde{H}_{\mu}(X;q,t)$. Our main result confirms this Ayyer--Mandelshtam--Martin conjecture. We establish an equidistribution between the pairs $(inv,maj)$ and $(quinv,maj)$ of $\mu$-Mahonian statistics on any row-equivalency class $[\tau]$, where $\tau$ is a filling of the Young diagram of $\mu$. As a byproduct of our approach, we show that if $\tau$ is a rectangular filling, the triples $(inv,quinv,maj)$ and $(quinv,inv,maj)$ have the same distribution over $[\tau]$.

math.CO↗

arXiv2014

Hybrid Ant Colony Algorithm Clonal Selection in the Application of the Cloud's Resource Scheduling

In this paper, thinking over characteristics of ant colony optimization Algorithm, taking into account the characteristics of cloud computing, combined with clonal selection algorithm (CSA) global optimum advantage of the convergence of the clonal selection algorithm (CSA) into every ACO iteration, speeding up the convergence rate, and the introduction of reverse mutation strategy, ant colony optimization algorithm avoids local optimum. Depth study of the cloud environment ant colony clonal selection algorithm resource scheduling policy, clonal selection algorithm converges to solve optimization problems when sufficient condition for global optimal solution based on clonal selection algorithm for various applications such as BCA and CLONALG algorithm, using these sufficient condition to meet and simulation platform CloudSim achieve a simulation by extending the cloud. Experimental results show that this task can be shortened fusion algorithm running time cloud environment, improve resource utilization. Demonstrate the effectiveness of the method.

cs.DC↗