Resúmenes

• Nicolás Ávila

The µ-Polynomials of Nestohedra 

The goal of this talk is to present the µ-polynomial associated to building sets. These are an extension of the µ-polynomials of graph associahedra, introduced by Wasch and Gonzáles D'León in the setting of weighted posets. In fact, building sets are combinatorial objects enclosing the concept of connectedness and generalizing the notion of a graph. As it turns out, the µ-polynomials are related with h-polynomials of Nestohedra, well-known objects in the literature. We shall see how to compute one from another for a given building set. In fact, our formulas will prove that both polynomials, up to a sign, coincide for trees graphs. Finally, we present the notion of µ-trees, which are rooted trees allowing to recover the µ-polynomials by counting descents. This in particular will show that µ-polynomials of graphs have integer coefficients of the same sign. 

• Sergio Carrillo

The µ-polynomial of cyclic graphs and interlacing sequences 

The aim of this talk is to study the µ-polynomial of cyclic graphs as introduced for general graphs by Wasch and Gonzáles D'León in the setting of weighted posets. After recalling the notions of interlacing sequences of polynomials and real-rootedness, we will show that our family of polynomials satisfy such properties. To this end we provide a four-term recurrence for them with the aid of Legendre polynomials. These results support a general conjecture for mu-polynomials of graphs G, namely, these are real rooted. Moreover, we can always contract an edge of G such that the µ-polynomial associated to the resulting graph interlaces the original µ-polynomial. We will also provide further examples that illustrate this phenomenon. 

• Luis Contreras

       Cover monoidal categories and Free cover monoids

We study a new product on the category of combinatorial species, which we call the cover Cauchy product. This product is defined as a relaxed version of the classical Cauchy product on species. We prove that the category of species together with the cover Cauchy product has the structure of a monoidal category. We define a cover Cauchy product on exponential generating functions and show that the exponential generating function of the cover Cauchy product of two species is precisely the cover Cauchy product of their exponential generating functions. Finally, we show that the free cover Cauchy monoid in one generator is given by the species of packed words, which are in bijective correspondence with surjections.  

• Rafael Díaz

    Análogos continuos de objetos combinatorios

En el año 2015 – en trabajo conjunto con Leonardo Cano – propusimos un método para construir análogos continuos de objetos combinatorios. El método procede en cuatro etapas: 1) seleccionar un objeto combinatorio, 2) reformularlo en términos de caminos en reticulados (existencia y dependencia se estudian caso por caso), 3) reemplazar caminos reticulares por caminos dirigidos, 4) reemplazar conteo de caminos reticulares por computación de volúmenes de caminos dirigidos (este ultimo pazo es el más sutil). El objetivo de la charla es revisar estas ideas, y plantear nuevas rutas de avance en el área.

[1] R. Díaz, L. Cano, Indirect Influences on Directed Manifolds, Advanced Studies in Contemporary Mathematics 28 (2018), No. 1, pp. 93 - 114.

[2] R. Díaz, L. Cano, Continuous analogues for the binomial coefficients and the Catalan numbers, Applicable Analysis and Discrete Mathematics 2019, Volume 13, Issue 2, Pages: 542-568.

[3] Q.-N. Le, S. Robins, C. Vignat, T. Wakhare, A Continuous Analogue of Lattice Path Enumeration, The Electronic Journal of Combinatorics, Volume 26, Issue 3 (2019).

[4] T. Wakhare, C. Vignat, A Continuous Analogue of Lattice Path Enumeration Part II, Online Journal of Analytic Combinatorics, Issue 14 (2019), 04.

[5] L. Cano, S. Carrillo, ¿Podemos detectar la curvatura gaussiana contando caminos y midiendo sus longitudes?, Vol. 38 Núm. 1 (2020): Revista Integración, temas de matemáticas.

• Kenny Fernández

      La construcción de Grothendieck en el contexto de acciones de grupos

Es un resultado clásico que la categoría de representaciones conjuntistas de un grupo topológico constituye un topos de Grothendieck. Cuando el grupo es finito, la construcción de Grothendieck toma el nombre de categoría transportadora y permite construir una variedad de sitios cuyas categorías de haces son equivalentes a la categoría de representaciones. Nuestra intención en esta charla es dar a conocer y ejemplificar los sitios en cuestión. 

• Rafael González

Flow polytopes as a unifying framework for some familiar combinatorial objects 

Flow polytopes are a family of beautiful geometric objects that have connections to many areas in mathematics including optimization and representation theory. Computing their volumes and enumerating lattice points of some particular flow polytopes turn out to be combinatorially interesting problems that involve beautiful enumeration formulas and many familiar combinatorial objects. Baldoni and Vergne found a series of formulas for both of these purposes, which they call Lidskii formulas, that are combinatorially powerful and pleasant. A later proof of the Lidskii formulas has been achieved by Mészáros and Morales, following the ideas of Postnikov and Stanley, using polytopal subdivisions. For a smaller class of flow polytopes, these subdivisions are triangulations that coincide with a family of framed triangulations defined by Danilov, Karzanov, and Koshevoy. These triangulations have interesting hidden combinatorial structure. We will give an introduction to flow polytopes and these formulas, including some recent applications, and a series of open problems and conjectures on which we are currently working on. 

• Pedro Hernández

Sobre una generalización de la categoría de módulos sobre el álgebra repetitiva 

El álgebra repetitiva y, principalmente, su categoría de módulos finitamente generados han jugado (¡y aún juegan!) un papel destacado en la teoría de representaciones de álgebras, en categorías derivadas, geometría algebraica, entre otras áreas. En esta charla presentaremos una generalización de la categoría de módulos sobre un álgebra repetitiva y algunos de los nuevos desafíos que resultan con este abordaje. Este es un trabajo conjunto con Germán Benitez (UFAM, AM, Brasil). 

• Fernando Morales

EXPECTED PERFORMANCE AND WORST CASE SCENARIO ANALYSIS OF THE DIVIDE-AND-CONQUER METHOD FOR THE 0-1 KNAPSACK PROBLEM

In this paper we furnish quality certificates for the Divide-and-Conquer method solving the 0-1 Knapsack Problem: the worst case scenario and an estimate for the expected performance. The probabilistic setting is given and the main random variables are defined for the analysis of the expected performance. The performance is accurately approximated for one iteration of the method then, these values are used to derive analytic estimates for the performance of a general Divide-and-Conquer tree. 

• Eddy Pariguan

Categorificación. Ejemplos y Reflexiones

Presentaremos una breve introducción al concepto de categorificación, y veremos diversos ejemplos. Haremos énfasis en la categorificación de anillos y presentaremos un modelo de categorificación para el espacio afín haciendo uso de la teoría de especies.  

• José Luis Ramírez

Pattern Avoidance in Restricted Growth Words 

Restricted growth words w=w_1w_2... w_n  are words defined over the set of non-negative integers by w_1=0 and 0≤w_i ≤st(w_1... w_i-1)+1, where st is an integer statistic. Classical examples of statistics are the maximum entry, the last symbol, the number of ascents or the number of weak ascents. For these statistics we obtain interesting combinatorial objects such as the restricted growth functions, the Catalan words, the ascent and weak ascent sequences, respectively.  These families of words provide a rich source of combinatorial ideas. They have been studied in connection with other discrete structures such as set partitions, Dyck paths,  (2+2)-free posets, Fishburn permutations, upper-triangular matrices of non-negative integers having neither columns nor rows of only zeros, among other. In this talk, we present several  enumerative results for restricted growth words that avoid a given pattern.   This notion is analogous to the concept of permutation pattern introduced by Donald Knuth. The  case of permutation patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns and restricted growth words have served as a unifying interpretation that relates a vast array of combinatorial structures.   This talk is based on joint work with Jean-Luc Baril, Beáta Benyi, and Toufik Mansour.

References

[1] J.-L. Baril, J.L. Ramírez, Descent distribution on Catalan words avoiding ordered pairs of relations, Adv. in Appl. Math. 149 (2023), 102551.

[2] B. Benyi, T. Mansour, and J. L. Ramírez. Pattern avoidance in weak ascent sequences, arXiv:2309.06518 (2023).

[3] M. Bousquet-Mélou, A. Claesson, M. Dukes, and S. Kitaev. (2+2)-free posets, ascent sequences and pattern avoiding permutations, J. Combin. Theory Ser. A 117 (2010), 884–909.

[4] S. Corteel, M. Martinez, C. Savage, and M. Weselcouch. Patterns in inversion sequences I. Discrete Math. Theor. Comput. Sci. 18 (2016), p#2.

[5] J. L. Ramírez and A. Rojas-Osorio. Consecutive patterns in Catalan words and the descent distribution. Bol. Soc. Mat. Mex. 29 (2023).

• Ricardo Vallejos

Sobre la teoría de representaciones de racks y quandles

Los racks y quandles son estructuras algebraicas que surgieron en el estudio de la teoría de nudos.Fueron introducidos por David Joyce en 1980 para abordar el problema de clasificación de nudos,  desde entonces las estructuras de rack y quandle han aparecido naturalmente en varios contextos y se ha convertido en un área de especial interés en el álgebra y la topología. Por otra parte, el concepto de representación de una estructura algebraica ha demostrado ser una valiosa herramienta en la solución de problemas en matemáticas, siendo la teoría de representaciones de grupos finitos las más antigua y ampliamente estudiada. Recientemente, se ha empezado a estudiar la teoría de representaciones de racks. En esta charla se dará una introducción al concepto de representación de rack y se mostrará una conexión con la teoría de representaciones de grupos. 

• María Ronco 

Loday’s triangulation for graph associahedra

(Joint work with S. Fishel) 

In [3], J.-L. Loday describes a way to triangulate Stasheff polytope. The number of simplices appearing in the triangulation of the Stasheff polytope of dimension n coincides with the number of parking functions on {1, . . . , n}. In [4], we described a simple way to define a partial order of the faces of any graph-associahedra, as introduced by M. Carr and S. Devadoss in [1]. We use the algebraic description of graph associahedra given in [2] to give a more conceptual description of these partial orders. From them, we obtain a way to triangulate all graph-associahedra, as well as to asign certain families of integers functions to the these triangulations.

    [1] M. Carr , S. Devadoss, Coxeter complexes and graph-associahedra. Topology Appl., 153(12):2155–2168, 2006.

    [2] S. Forcey, M. Ronco, Algebraic structures on graph associahedra, J. of the London Math. Soc., Vol.106, Issue                      (2022)1189–1231.

[3] J.-L. Loday, Parking functions and triangulation of the associahedron, Proceedings of the Street’s fest 2006, Contemp. Math. AMS 431 (2007), 327–340.

[4] M. Ronco, Generalized Tamari order, in .Associahedra, Tamari Lattices and Related Structures”, Tamari Memorial Festschrift, eds. F. M ̈uller-Hoissen, J. Pallo and J. Stasheff, Progress in Mathematics 299 (Birkhäuser,2012) 337–348..