arXiv subjects

Jerónimo Valencia-Porras

Publications and source records attributed to Jerónimo Valencia-Porras.

5 recordsLinked to original sources

arXiv2026

Order and Gorenstein properties for lattice path matroid polytopes

We characterize matroids whose polytopes are order polytopes as a special class of lattice path matroids, called snakes. This shows that snakes are exactly the objects lying in the intersection of the Neggers-Stanley conjecture and a conjecture of de Loera, Haws, and Köppe. We then characterize Gorenstein lattice path matroid polytopes, yielding a new class of matroids satisfying the latter conjecture. Finally, we also show that lattice path matroids are exactly the class of positroids in which the coarsest subdivision of the matroid base polytope into translates of order polytopes is a (finest) matroidal subdivision.

math.CO↗

arXiv2026

Type $C$ multiline queues and the open-boundary TASEP

The totally asymmetric simple exclusion process (TASEP) with open boundaries is a finite Markov chain describing particles hopping between adjacent sites on a one-dimensional lattice with left and right boundary transitions governed by parameters $α$ and $β$. The multispecies TASEP is a higher-rank generalization in which particles have different species. Multiline queues were introduced by Ferrari and Martin (2007) to compute the stationary distribution of the multispecies TASEP on a circle. It has remained an open problem to find a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP. Using Kirillov--Reshetikhin crystals of type $C$, we construct type $C$ multiline queues and a corresponding Ferrari--Martin pairing algorithm that projects them to TASEP configurations. This yields a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP for the $α=β=1$ specialization.

math.CO↗

arXiv2025

Macdonald polynomials at t = 0 through twisted multiline queues

Multiline queues are versatile combinatorial objects that play a key role in understanding the remarkable connection between the asymmetric simple exclusion process (ASEP) on a circle and Macdonald polynomials. Specializing the results of Corteel--Mandelshtam--Williams (2018) to the $t=0$ case yields a formula for the $q$-Whittaker polynomials through the Ferrari--Martin (2007) algorithm with a major index ($\texttt{maj}$) statistic. In this paper, we reinterpret the $\texttt{maj}$ statistic as a $\texttt{charge}$ statistic on reading words, thereby bypassing the Ferrari--Martin algorithm to obtain an elegant formula for the $q$-Whittaker polynomials. Our methods naturally extend to the case of bosonic multiline queues, with which we obtain analogous results for the modified Hall--Littlewood polynomials using a $\texttt{cocharge}$ statistic on reading words. Twisted multiline queues (GMLQs) are obtained from the action of the symmetric group on the rows of a multiline queue. The Ferrari--Martin algorithm was extended to GMLQs by Arita--Ayyer--Mallick--Prolhac (2011), and Aas--Grinberg--Scrimshaw (2020) showed it is preserved under this action. We extend these results by defining a $\texttt{maj}$ statistic on GMLQs that is also preserved under this action. This yields a novel family of formulas, indexed by compositions, for the $q$-Whittaker polynomials. Additionally, we define a procedure on both GMLQs and bosonic multiline queues that we call collapsing, which can can be realized via the Kashiwara (crystal) operators on type-A Kirillov--Reshetikhin crystals. As an application, we naturally recover the Lascoux--Schützenberger $\texttt{charge}$ formula for the $q$-Whittaker and modified Hall--Littlewood polynomials, and the classical and dual Cauchy identities for Schur functions.

math.CO↗

arXiv2025

A combinatorial proof of an identity involving Eulerian numbers

We give a combinatorial proof of an identity that involves Eulerian numbers and was obtained algebraically by Brenti and Welker (2009). To do so, we study alcoved triangulations of dilated hypersimplices. As a byproduct, we describe the dual graph of the triangulation in the case of the standard simplex, conjecture its structure for general hypersimplices, and prove combinatorially that the Eulerian numbers coincide with the normalized volumes of the hypersimplices.

math.CO↗

arXiv2022

Inequalities for $f^*$-vectors of Lattice Polytopes

The Ehrhart polynomial $\text{ehr}_P(n)$ of a lattice polytope $P$ counts the number of integer points in the $n$-th integral dilate of $P$. The $f^*$-vector of $P$, introduced by Felix Breuer in 2012, is the vector of coefficients of $\text{ehr}_P(n)$ with respect to the binomial coefficient basis $ \left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}$, where $d = \dim P$. Similarly to $h/h^*$-vectors, the $f^*$-vector of $P$ coincides with the $f$-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of $f^*$-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of $f$-vectors of simplicial polytopes; e.g., the first half of the $f^*$-coefficients increases and the last quarter decreases. Even though $f^*$-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart $h^*$-vector, there is a polytope with the same $h^*$-vector whose $f^*$-vector is unimodal.

math.CO↗