arXiv subjects
Publications and source records attributed to Mercedes Rosas.
We present bijections connecting tree records, the girth of a connected endofunction, and the genesis sequence (the first sequence in OEIS). Using these, we derive generating functions for tree and forest record numbers in terms of Cayley's tree function and give a new proof of Cayley's forest formula.
We introduce the priority lattice, a structure arising from the priority search algorithm on rooted trees and forests. We prove bijectively that its maximal chains are labeled by parking functions, and that the maximal chains of its principal ideals are labeled by partial parking functions. We establish that it is a graded lattice and compute its M\"obius function and characteristic polynomials.
In this work, we present a series of bijections that reveal the deep connections between the concepts of tree records, the girth of a connected endofunction, and the genesis sequence, the first sequence in the OEIS. We use these results to derive the generating functions for the tree and forest record numbers, expressing them in terms of the Cayley's tree function. Finally, we provide a new proof for Cayley's forest formula.
A record of a rooted Cayley tree is a node whose label is the largest along the unique path to the root. In this work, we find elegant functional equations relating the generating functions for records of rooted Cayley trees and for records of forests of rooted trees with the Cayley tree function, and explore the consequences of our results.
This work builds on the notion of record of rooted trees. We provide an alternative definition of parking functions, derive from it a record-preserving bijection between rooted trees and parking functions, and establish a join equidistribution result between a 5-tuple of statistics on rooted trees and a corresponding 5-tuple of statistics on parking functions. Some enumerative questions are also considered.
We present the first combinatorial proof of the Graham-Pollak Formula for the determinant of the distance matrix of a tree, via sign-reversing involutions and the Lindström-Gessel-Viennot Lemma. Our approach provides a cohesive and unified framework for the understanding of the existing generalizations and $q$-analogues of the Graham-Pollak Formula, and facilitates the derivation of a natural simultaneous generalizations for them.
The $\mathit{SU}(3)$ tensor multiplicities are piecewise polynomial of degree $1$ in their labels. The pieces are the chambers of a complex of cones. We describe in detail this chamber complex and determine the group of all linear symmetries (of order $144$) for these tensor multiplicities. We represent the cells by diagrams showing clearly the inclusions as well as the actions of the group of symmetries and of its remarkable subgroups.
We compute with SageMath the group of all linear symmetries for the Littlewood-Richardson associated to the representations of $SL_3$. We find that there are 144 symmetries, more than the 12 symmetries known for the Littlewood-Richardson coefficients in general.
The Kronecker coefficients are the structure constants for the restriction of irreducible representations of the general linear group $GL(n m)$ into irreducibles for the subgroup $GL(n)\times GL(m)$. In this work we study the quasipolynomial nature of the Kronecker function using elementary tools from polyhedral geometry. We write the Kronecker function in terms of coefficients of a vector partition function. This allows us to define a new family of coefficients, the atomic Kronecker coefficients. Our derivation is explicit and self-contained, and gives a new exact formula and an upper bound for the Kronecker coefficients in the first nontrivial case.
We investigate the combinatorics of the general formulas for the powers of the operator $h \partial^k$, where $h$ is a central element of a ring and $\partial$ is a differential operator. This generalizes previous work on the powers of operators $h \partial$. New formulas for the generalized Stirling numbers are obtained.
We study the commutation relations and normal ordering between families of operators on symmetric functions. These operators can be naturally defined by the operations of multiplication, Kronecker product, and their adjoints. As applications we give a new proof of the skew Littlewood-Richardson rule and prove an identity about the Kronecker product with a skew Schur function.
This text is an appendix to our work "On the growth of Kronecker coefficients", arXiv:1607.02887. Here, we provide some complementary theorems, remarks, and calculations that for the sake of space are not going to appear into the final version of our paper. We follow the same terminology and notation. External references to numbered equations, theorems, etc. are pointers to arXiv:1607.02887.
We study the rate of growth experienced by the Kronecker coefficients as we add cells to the rows and columns indexing partitions. We do this by moving to the setting of the reduced Kronecker coefficients.
The reduced Kronecker coefficients are particular instances of Kronecker coefficients that contain enough information to recover them. In this notes we compute the generating function of a family of reduced Kronecker coefficients. We also gives its connection to the plane partitions, which allows us to check that this family satisfies the saturation conjecture for reduced Kronecker coefficients, and that they are weakly increasing. Thanks to its generating function we can describe our family by a quasipolynomial, specifying its degree and period.
We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka--Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles.
En esta serie de tres articulos, damos una exposicion de varios resultados y problemas abiertos en tres areas de la combinatoria algebraica y geometrica: las matrices totalmente no negativas, las representaciones del grupo simetrico, y los arreglos de hiperplanos. Esta primera parte presenta una introduccion a las matrices totalmente no negativas, y su relacion con las funciones simetricas. In this series of three articles, we give an exposition of various results and open problems in three areas of algebraic and geometric combinatorics: totally non-negative matrices, representations of the symmetric group, and hyperplane arrangements. This first part presents an introduction to totally non-negative matrices and their relationship with symmetric functions.
En esta serie de tres articulos, damos una exposicion de varios resultados y problemas abiertos en tres areas de la combinatoria algebraica y geometrica: las matrices totalmente no negativas, las representaciones del grupo simetrico, y los arreglos de hiperplanos. Esta segunda parte trata la coneccion entre las funciones simetricas y la teoria de representaciones. In this series of three articles, we give an exposition of various results and open problems in three areas of algebraic and geometric combinatorics: totally non-negative matrices, representations of the symmetric group, and hyperplane arrangements. This second part treats the connection between symmetric functions and representation theory.
In this series of three articles, we give an exposition of various results and open problems in three areas of algebraic and geometric combinatorics: totally non-negative matrices, representations of the symmetric group, and hyperplane arrangements. This first part is an introduction to hyperplane arrangements from a combinatorial point of view. ----- En esta serie de tres articulos, damos una exposicion de varios resultados y problemas abiertos en tres areas de la combinatoria algebraica y geometrica: las matrices totalmente no negativas, las representaciones del grupo simetrico, y los arreglos de hiperplanos. Esta tercera parte presenta una introduccion a los arreglos de hiperplanos desde un punto de vista combinatorio.