José Aliste-Prieto

Proper caterpillars are distinguished by their chromatic symmetric function

We show that the symmetric function generalization of the chromatic polynomial, or equivalently, the U-polynomial, distinguishes among a large class of caterpillar trees that we call proper, thus improving previous results by Martin, Morin and Wagner.

Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line

In this paper, we study translation sets for non-decreasing maps of the real line with a pattern-equivariant displacement with respect to a quasicrystal. First, we establish a correspondence between these maps and self maps of the continuous hull associated with the quasicrystal that are homotopic to the identity and preserve orientation. Then, by using first-return times and induced maps, we provide a partial description for the translation set of the latter maps in the case where they have fixed points and obtain the existence of a unique translation number in the case where they do not have fixed points. Finally, we investigate the existence of a semiconjugacy from a fixed-point-free map homotopic to the identity on the hull to the translation given by its translation number. We concentrate on semiconjugacies that are also homotopic to the identity and, under a boundedness condition, we prove a generalization of Poincare’s theorem, finding a sufficient condition for such a semiconjugacy to exist depending on the translation number of the given map.

Rapid Convergence to Frequency for Substitution Tilings of the Plane

This paper concerns self-similar tilings of the Euclidean plane. We consider the number of occurrences of a given tile in any domain bounded by a Jordan curve. For a large class of self-similar tilings, including many well-known examples, we give estimates of the oscillation of this number of occurrences around its average frequency times the total number of tiles in the domain, which depend only on the Jordan curve.

On trees with the same restricted U-polynomial and the Prouhet–Tarry–Escott problem

Abstract This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U -polynomial (or, equivalently, the same chromatic symmetric function). We consider the U k -polynomial, which is a restricted version of U -polynomial, and construct, for any given k , non-isomorphic trees with the same U k -polynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are distinguished by the U -polynomial up to isomorphism.

On graphs with the same restricted U -polynomial and the U -polynomial for rooted graphs

Abstract In this abstract, we construct explicitly, for every k, pairs of non-isomorphic trees with the same restricted U-polynomial; by this we mean that the polynomials agree on terms with degree at most k. The construction is done purely in algebraic terms, after introducing and studying a generalization of the U-polynomial to rooted graphs.