Alessio Sgubin - WebPage

What to find on this page

Here you can find my research projects in Mathematics.

A comprehensive list of the articles I am working in:

• M. D'Adderio, A.S. - A Loehr-Remmel bijection in the $n \times kn$ grid and sandpiles (Article, preprint)
• M. D'Adderio, A.S. - On a bijection of Loehr and Remmel (Expository, preprint)

On the side menu you can find an abstract version of the papers listed above.

M. D'Adderio, A.S. - On a bijection of Loehr and Remmel

Abstract

In this expository article we review a remarkable bijection $\phi_n$ due to Loehr and Remmel from the set of parking functions of size n into itself, which sends the bistatistic (dinv, area) into the bistatistic (area, pmaj). The only novelty of the present work is our definition of $\phi_n$, which is more direct than the original one, hence easier to compute and to work with.

arXiv preprint

M. D'Adderio, A.S. - A Loehr-Remmel bijection in the $n \times kn$ grid and sandpiles

Abstract

We extend the $\mathsf{pmaj}$ statistic of Loehr and Remmel to labelled Dyck paths in the $n \times kn$ grid, and generalize their bijection sending the bistatistic $(\mathsf{dinv},\mathsf{area})$ to $(\mathsf{area}, \mathsf{pmaj})$, proving in this way a new combinatorial formula for $\nabla^k e_n$ ($k \geq 1$). At $k = 1$ we recover the original statistic and the original bijection. Moreover, we provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs $G_{\mu, \nu}^{(k)}$, indexed by an integer $k \geq 1$ and two compositions $\mu$ and $\nu$: at $k = 1$ these are the clique-independent graphs of D'Adderio et al. Finally, we define a $\mathsf{delay}$ statistic on these configurations, and we show that, together with the usual level statistic, it can be used to provide a new combinatorial interpretation of the polynomials $\langle \nabla^k e_n,e_\mu h_\nu \rangle$ from the $(n,kn)$-shuffle theorem. At $k = 1$ we recover the main results of D'Adderio et al.

arXiv preprint