# Rafael S. González D'león

### 2010

Fall
MTH 161 DN. Calculus 1.
Spring
MTH 112. Calculus 2. (Tutoring session)

### 2009

Fall
MTH 110.Calculus 1. (Lab)

## Preprints & Publications

1. R. S. González D'León, M. L. Wachs. On the (co)homology of the poset of weighted partitions.
ArXiv:1309.5527
Abstract:
We consider the poset of weighted partitions $$\Pi_n^w$$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $$\Pi_n^w$$ provide a generalization of the lattice $$\Pi_n$$ of partitions, which we show possesses many of the well-known properties of $$\Pi_n$$. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted $$\mathfrak{S}_n$$-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of $$\Pi_n^w$$ has a nice factorization analogous to that of $$\Pi_n$$.

2. R. S. González D'León, M. L. Wachs. On the poset of weighted partitions.
DMTCS Proceedings 01 (2013): 1029-1040.
Abstract:
In this extended abstract we consider the poset of weighted partitions $$\Pi_n^w$$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $$\Pi_n^w$$ provide a generalization of the lattice $$\Pi_n$$ of partitions, which we show possesses many of the well-known properties of $$\Pi_n$$. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted $$\mathfrak{S}_n$$-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of $$\Pi_n^w$$ has a nice factorization analogous to that of $$\Pi_n$$.

3. R. S. González D'León. On the free Lie algebra with k linearly compatible brackets and the poset of weighted partitions (in preparation).

4. R. S. González D'León. Refinements of second-order Eulerian statistics, Stirling symmetric functions and k-Lyndon trees (in preparation).

5. P. Brändén, R. S. González D'León. On the half-plane property and the Tutte-group of a matroid.
J. Combin. Theory Ser. B. 100 - 2010 ArXiv: 0906.1071   MATLAB code
Abstract:
A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte-group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular.

6. R. S. González D'León. Master Thesis: Representing matroids by polynomials with the half-plane property.
Kungliga Tekniska Högskolan, May 2009. PDF   MATLAB code
Abstract:
A matroid M is said to have the weak half-plane property (wHPP) if there exists a stable multiaffine homogeneous complex polynomial f with support equal to the set of bases of M. This is a generalization of the halfplane property (HPP), where we require that all the coefficients of f are equal to zero or one. Both properties were recently treated by Choe, Oxley, Sokal and Wagner in [COSW04]. In [Ba07], Brändén proved that not every matroid is wHPP by showing that the Fano matroid F7 is not. We provide two new proofs of the fact that F7 is not a wHPP-matroid. We investigate and state conditions for when wHPP=HPP for M. We use concepts and techniques developed for the Tutte-group of a matroid and valuated matroids by Dress, Wenzel and Murota to prove that the projective geometry matroids PG(r-1; q) are not wHPP and that a binary matroid is a wHPP-matroid if and only if it is regular. This shows that there exist large families of matroids that are not wHPP. We answer questions posed by Choe et al., by proving that the coextensions AG(3; 2) and S8 of F7, and the matroids T8 and R9, are not wHPP, extending the answer given by [Ba07].

## Research Interests

Algebraic and Topological Combinatorics. See my Research Statement (pdf format).

See

## Education

Ph.D. in Mathematics, Current (05/2014 Expected). University of Miami, USA.

M.Sc. in Mathematics, May 2011. University of Miami, USA.

M.Sc. in Mathematics, June 2009. KTH - Royal Institute of Technology, Stockholm, Sweden.

Electrical Engineering, March 2006. UPB University, Medellín, Antioquia, Colombia

## Talks

The combinatorial structure behind the free Lie algebra. November 1, 2013. UM Math Graduate Students Seminar. University of Miami, Coral Gables, FL, USA.

On the free Lie algebra with k compatible brackets and poset topology. October 19, 2013. Special Session on Topological Combinatorics. AMS Sectional Meeting. Washington University, St. Louis, MO, USA.

(Poster) On the poset of weighted partitions. June 24, 2013. FPSAC'13 The 25th International Conference on Formal Power Series and Algebraic Combinatorics. Paris, France.

On the poset of weighted partitions. June 17, 2013. The 11th Nordic Combinatorial Conference (NORCOM). KTH, Stockholm, Sweden.

Introduction to graph coloring. March 1, 2013. UM Math Graduate Students Seminar. University of Miami, Coral Gables, FL, USA.

About algebras and operads. November 2, 2012. UM Math Graduate Students Seminar. University of Miami, Coral Gables, FL, USA.

On the (co)homology of the poset of weighted partitions. October 14, 2012. Special Session on Algebraic and Topological Combinatorics. AMS Sectional Meeting. Tulane University, New Orleans, LA, USA.

Weighted partition posets. October 2, 2010. Combinatorics Seminar. University of Miami, Coral Gables, FL, USA.

A cell structure for Grassmann Manifolds. February 17, 2012. UM Math Graduate Students Seminar. University of Miami, Coral Gables, FL, USA.

Generatingfunctionology (or looking for a closed formula for the Fibonacci sequence). October 20, 2011. UMMU University of Miami Mathematics Union. University of Miami, Coral Gables, FL, USA.

What is ... a Matroid?. September 9, 2011. UM Math Graduate Students Seminar. University of Miami, Coral Gables, FL, USA.

About the Potts model and some of its combinatorial relations. September 17, 2010. UM Math Graduate Students Seminar. University of Miami, Coral Gables, FL, USA.

On the Half-plane Property and the Tutte-group of a Matroid. March 2, 2010. Combinatorics seminar. University of Miami, Coral Gables, FL, USA.

Representing matroids by polynomials with the half-plane property. May 27, 2009. Combinatorics seminar. KTH-The Royal Institute of Technology, Stockholm, Sweden.

See

## My Curriculum Vitae

Curriculum Vitae (pdf format)

## Rafael S. González D'león

Ph.D. Student in Mathematics
Department of Mathematics
University of Miami