# Research

## Publications

B. Baker Swart, M. Chen R. Florez, and A. Mukherjee. Getting high school students doing research. Submitted.

R. Higuita, R. Florez, M. R. Bacon, and C. K. Cook. Fibonacci-Theodorus Spiral and its properties. Submitted.

J-L. Baril, R. Florez, and J. L. Ramirez, Counting Symmetric and Asymmetric Peaks in Motzkin Paths with Air Pockets. Submitted.

R. Florez and T. Zaslavsky, Projective Rectangles. Arxiv Version. Submitted.

R. Florez, J. L. Ramirez, and D. Villamizar, Restricted Bargraphs and Unimodal Compositions. Submitted.

J-L. Baril, R. Florez, and J. L. Ramirez, Symmetries in Dyck paths with air pockets. Submitted.

R. Florez, J. F. Gonzalez, M. Matijasevick, C. J. Pardo, J. L. Ramirez, L. M.Simbaqueba, and F A. Velandia, Lattice Paths in Corridors and Cyclic Corridors. Submitted.

D. A. Narayan, L. Gomez, K. Rubi, J. Terrazas, and R.Florez, Extremal and Structural Methods for Obtaining Failed Zero Forcing Numbers of Graphs. Submitted.

R. Florez, T. Mansour, J. L. Ramirez, Fabio A. Velandia, and Diego Villamizar, Restricted Dyck Paths on Valleys Sequence. Submitted.

R. Florez, J. L. Ramirez, Fabio A. Velandia, and Diego Villamizar, Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions. Submitted.

M. Diaz, R. Florez, J. L. Ramirez, and M. Romero, Catalan identities for Generalized Fibonacci Polynomials. Accepted for publication by Fibonacci Quarterly.

E. Farnsworth, N. Gomez, H. Lino, R. Florez, B. Rooney, and D. Narayan, Realizing the asymmetric index of a graph. Accepted for publication by Journal of Combinatorial Mathematics and Combinatorial Computing.

R. Florez and D. Forge, Activity from matroids to rooted trees and beyond. J. Combin. Theory Ser. A 198 (2023), Paper No. 105755, 21 pp. Free version.

R. Florez, R. Higuita, and A. Mukherjee, The Geometry of some Fibonacci Identities in the Hosoya Triangle. Involve, 16 (2023), No. 2, 183–200.

R. Florez and J. L. Ramirez, Enumerating Symmetric Pyramids in Motzkin paths. Ars Math. Contemp. 23 (2023), #4.06, doi:10.26493/1855-3974.3061.5bf.

J. Brooks, A. Carbonero, J. Vargas, R. Florez, B. Rooney, and D. Narayan, Fixing Numbers of Point-Block Incidence Graphs. Mathematics (2023).

H. Ching, R. Florez, F. Luca, A. Mukherjee, and J. C. Saunders, Primes and composites in the determinant Hosoya triangle. The Fibonacci Quarterly, 60.5 (2022), 56-110. A version in FQ. Arxiv version.

R. Florez, J. L. Ramirez, Fabio A. Velandia, and Diego Villamizar. A refinement of Dyck paths: A combinatorial approach. Discrete Mathematics, Algorithms and Applications (DMAA), 14, No. 07, 2250026 (2022).

R. Florez and J. C. Saunders, Irreducibility of generalized Fibonacci polynomials. Integers 22 (2022). Arxiv Version.

M. Blair, R. Florez, and A. Mukherjee, Honeycombs in the Hosoya Triangle. Math Horizons, 29.3 (2022) 24-27. Axiv version.

R. Florez and J. L. Ramirez, Enumerations of rational non-decreasing Dyck paths with Integer Slope. Graphs and Combinatorics, 37 (2021), 2775-2801.

S.Elizalde, R. Florez, and J. L. Ramirez, Enumerating Symmetric Peaks in nondecreasing Dyck paths. In press for publication. ARS Mathematica Contemporanea, 21, 2021, 1-23.

M. Blair, R. Florez, and A. Mukherjee, Geometric patterns in the determinant Hosoya triangle. Integers 21 (2021), 1-24.

J. Brooks, A. Carbonero, J. Vargas, R. Florez, B. Rooney, and D. Narayan, Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs. Mathematics (2022).

R. Florez, L. Junes, and J. L. Ramirez, Counting asymmetric weighted pyramids in non-decreasing Dyck paths. Australasian Journal of Combinatorics, 79.1 (2021) 123-140.

H. Ching, R. Florez and A. Mukherjee, Families of Integral Cographs within a Triangular Array. Special Matrices, 8 (2020), 257-273.

S.Alzate, O.Correa, and R. Florez, Fibonacci identities from Jordan Identities, The Fibonacci Quarterly, 57.5 (2020) 2-20.

M.Blair, R. Florez, A. Mukherjee, and J. L. Ramirez, Matrices in the determinant Hosoya triangle. The Fibonacci Quarterly, 57.5 (2020), 34-54.

A. Brewer, R. Florez, A. Gregory, Q. Jones, D. Narayan, and L. Rodriguez, Infinite Families of Asymmetric Graphs. AKCE International Journal of Graphs and Combinatorics, 17.3 (2020) 1000-1009.

R. Florez, H. A. Newman, H. Miranda and D. Narayan, Uniform Edge Betweenness Centrality. Electronic Journal of Graph Theory and Applications (EJGTA), 8.2 (2020), 265-300.

R. Florez, and J. L. Ramirez, Enumerating Symmetric and Asymmetric Peaks in Dyck paths. Discrete Mathematics, 343 (2020) 112-118.

R. Florez, R. Higuita, Another proof for partial strong divisibility property of Lucas-type Polynomials. The Fibonacci Quarterly 58.1 (2020), 70-72. A version in FQ

R. Florez and T. Zaslavsky, Projective Planarity of Matroids of 3-Nets and Biased Graphs. 76.2 (2020), 299–338. A version in Australasian Journal of Combinatorics or in ArXiv.

R. Florez and A. Mukherjee, Classroom experiences in introducing students to conjectures, exploration and visual proofs using experiments on the mobius strip. PRIMUS. 30.1 50–66, 2020. A version in PRIMUS.

R. Florez and D. Narayan, Minimally Non-Asymmetric Graphs and Balanced Incomplete Block Designs. Congressus Numerantium v. 233 (2019), 245-254.

M. Blair, R. Florez, and A. Mukherjee, Matrices in the Hosoya Triangle. The Fibonacci Quarterly, 57.5 (2019), 15-28. A version in FQ or ArXiv.

R. Florez, L. Junes, and J. L. Ramirez, Enumerating several aspects of non-decreasing Dyck paths. Discrete Mathematics, 342.10 (2019), 3079-3097.

R. Florez, R. Higuita, and A. Ramirez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials. Journal of Integer Sequences, Vol. 22 (2019). Article 19.4.4.

R. Florez and T. Zaslavsky, Biased Graphs. VI. Synthetic Geometry. European J. Combin. 81 (2019), 119-141. A version in ArXiv.

R. Florez, E. Carter, B. Ek, D. A. Gonzalez, and D. Narayan, Intersecting Geodesics and Centrality Graphs. Involve-A journal of Mathematics, 12.1 (2019), 31-43.

E. Czabarka, R. Florez, L. Junes, and J. L. Ramirez, Enumerations of Peaks and Valleys on Non-decreasing Dyck Paths. Discrete Mathematics, 341.10 (2018), 2789-2807.

R. Florez and J. L. Ramirez, Some Enumerations on Non-Decreasing Motzkin Paths. Australasian Journal of Combinatorics, 72.1 (2018), 138-154. A free version Australasian Journal of Combinatorics

R. Florez, O. Haynes, D. Narayan and A. Strang, Enumerating shortest paths and determining edge betweenness centrality in Cartesian products of paths and cycles. Bulletin of ICA, Vol. 83 (2018), 29-51.

R. Florez, R. Higuita, and A. Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles. Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.

R. Florez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial. Integers, 18B (2018), Paper No. A2. A version in INTEGERS.

R. Florez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials. Integers, 18 (2018), Paper No. A14. A version in INTEGERS.

R. Florez, L. Junes, and J. L. Ramirez, Further Results on Paths in an n-Dimensional Cubic Lattice. Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.

R. Florez, J. Lind, and D. Narayan, Betweenness centrality in organizational Structures. Congr. Numer. 228 (2017), 281–288.

R. Florez D. Narayan, A. Sharma, R. Vargas, and A. Waldron, A Graph Theoretic Analysis of Leverage Centrality. AKCE Int. J. Graphs Comb. 14 (2017), no. 3, 295–306. Free version AKCE. A version in ArXiv.

R. Florez, and A. Mukherjees, Solving open problems with students as a first research experience. Teaching mathematics and its applications, 10 (2017), 1-7. Non-free version Teaching Mathematics and its Applications

R. Florez, R. Lopez, D. Narayan, H. Wickus, and J. Worrell, Towards a Characterization of Graphs with Distinct Betweenness Centrality. The Australasian Journal of Combinatorics, Volume 68 (part 2), 2017. Free version Australasian Journal of Combinatorics. MR3646041

R. Florez D. Narayan, A. Sharma, R. Vargas, and A. Waldron, Leverage centrality of knight’s graphs and Cartesian products of regular graphs and path powers. Involve. A journal of Mathematics. 10.4 (2017), 583-592. MR3630305

B. Baker Swart, R. Florez, D. Narayan, and G. Rudolph, Extrema Property of the k-Ranking of Directed Paths and Cycles. AKCE International Journal of Graphs and Combinatorics, 2.6 (2016), 1-16. MR3524756. Free version AKCE-Vol13-1 or AKCE-Vol 13-1. A version in ArXiv.

R. Florez and D. A. Narayan, Maximizing the number of edges in optimal k-rankings. AKCE International Journal of Graphs and Combinatorics, 12.1 (2015), 1-8. MR3410846. Free Version AKCE-Vol 10-1. A version in ArXiv.

E. Czabarka, R. Florez, and L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle. Journal of Integer Sequences, 18.1 (2015), 1-22. MR3303767.

E. Czabarka, R. Florez, and L. Junes, Some Enumerations on Non-decreasing Dyck Paths, The Electronics Journal of Combinatorics, 22.1 (2015) 1-22. MR3315445.

R. Florez, R. Higuita and L. Junes, 9-modularity and GCD Properties of Generalized Fibonacci Numbers, Integers, 14 (2014), 1-14. MR3268585.

R. Florez, R. Higuita and A. Mukherjees, Alternating sums in the Hosoya polynomial triangle, Journal of Integer Sequences, 17.9 (2014), 1-17. MR3261053. Free Version JIS-Vol 17-9

R. Florez, R. Higuita and L. Junes, GCD property of the Generalized Star of David in Generalized Hosoya's Triangle, Journal of Integer Sequences, 17.4 (2014). MR3168688. 1-17.

R. Florez and L. Junes, GCD properties in the Hosoya's Triangle.The Fibonacci Quarterly 50.2 (2012). MR2927935. Free version The Fibonacci Quaterly.

R. Florez and L. Junes, A Relation Between Triangular Numbers and Prime Numbers. Integers, 12 (2012), no. 1, 83-96. MR2955583.

R. Florez and D. A. Narayan, An optimal k-ranking characterization of oriented paths and cycles, Bulletin of the Institute of Combinatorics and Its Applications, 61 (2011) 97-108. MR2724560

R. Florez, Harmonic conjugation in harmonic matroids. Discrete Mathematics, 309 (2009), 2365-2372. MR2276967

R. Florez and D. Forge, Minimal non-orientable matroids in a projective plane, Journal of Combinatorial Theory Series A, 114 (2007), 175-183. MR2276967

R. Florez, Lindstrom's conjecture on a class of algebraically non-representable matroids. European Journal of Combinatorics, 27 (2006), 896-905. MR2226425

Florez, Rigoberto, Four studies in the geometry of biased graphs. Thesis (Ph.D.)-State University of New York at Binghamton. 2005. 78 pp. ISBN: 978-0542-19217-3. MR2707450

R. Florez and J. Velez, Failure of splitting from module-finite extension rings. Beitrage zur Algebra und Geometrie (contributions to algebra and geometry), 41 (2000), 345-357. MR1801426. Free version Beitr. Algebra Geom. Vol 41

## Talks

The strong divisibility property and the irreducibility of generalized Fibonacci polynomials. The Universidade Federal do ABC (UFABC) November 23, 2023, Sao Pablo, Brazil.

Conteo de Picos Simetricos de caminos de Dyck. Congreso colombiano de matematicas. June 7, 2023.

Identities and Irreducibility of generalized Fibonacci Polynomials. INTEGERS Conference 2023, University of Georgia, Athens, Georgia, May 18-20, 2023.

The strong divisibility property and the irreducibility of generalized Fibonacci polynomials. The 2022 BUGCAT November 5-6, 2022, Binghamton University. Slides

Divisibility properties of some generalized Fibonacci numbers. The Nineteenth International Conference on Fibonacci Numbers and Their Applications, 25-29 July, 2022 University of Sarajevo. Video

The strong divisibility property and the irreducibility of generalized Fibonacci polynomials. Jornadas de Algebra y Teoria de Numeros. Universidad del Cauca, Popayan, Colombia June 2, 2022.

Some enumerations on non-decreasing Dyck paths. New York Combinatorics Seminar, CUNY Graduate Center's Math Department and Computer Science Department, Feb 25, 2022. Virtual.

Some enumerations on non-decreasing Dyck paths. Encontro Latino-Americano de Matemática e Aplicações. Dec 02, 2021. Virtual.

Some enumerations on non-decreasing Dyck paths. 52nd Southeastern Lattice Paths, Combinatorics and Interactions, Lattice Paths, Combinatorics and Interactions, June 22, 2021. Virtual poster presentation.

Some enumerations on non-decreasing Dyck paths. Colloquia, Department of Mathematics, College of Charleston, April 19, 2021. Virtual.

A relation between non-decreasing Dyck paths and Riordan arrays. 52nd Southeastern International Conference on Combinatorics, Graph Theory and Computing, March 9, 2021. Virtual.

Some enumerations on non-decreasing Dyck paths. Rochester Institute of Technology (RIT), Nov 20, 2020. Virtual.

Some enumerations on non-decreasing Dyck paths. Discrete/Combinatorics, University of South Carolina Nov 13, 2020. Virtual.

Research with Undergraduate Students. The Geometry of Triangular Arrays Associated to Fibonacci Numbers. Departamento de Matem\'aticas Universidad del Cauca Aug 31, 2020. Virtual

A connection between Fibonacci Identities and Abstract Algebra. The Nineteenth International Conference on Fibonacci Numbers and Their Applications, 21 July 2020. Virtual

Enumerating Symmetric and Asymmetric Peaks in Dyck paths. Permutation, Patterns, The International Conference on Permutation Patterns, (June 30, 2020 Virtual). With Jose Luis Ramirez.

More about projective rectangles. Fall Eastern Sectional Meeting, Binghamton, NY Oct 12, 2019. Special Session on Algebraic Combinatorics on the Occasion of the 75th Birthday of Thomas Zaslavsky, 2019.

Projective Rectangles. Colloquia, Department of Mathematics, College of Charleston, Sept 20, 2019.

The strong divisibility property and the resultant of generalized Fibonacci polynomials. XXII Congreso Colombiano de Matematicas, Popayan (Colombia), June 11, 2019. A semi-plenary speaker.

Projective Rectangles. New York Combinatorics Seminar, CUNY Graduate Center's Math Department and Computer Science Department, March 15, 2019.

Projective Rectangles. Discrete/Combinatorics, University of South Carolina Feb 15, 2019.

The strong divisibility property and the resultant of generalized Fibonacci polynomials. Discrete/Combinatorics & Number Theory Crossover Seminar, University of South Carolina Oct 12, 2018.

The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials. INTEGERS Conference 2018, Augusta, Georgia October 6, 2018.

Projective Planarity of 3-Nets and Biased Graphs. 2018 SIAM Conference on Discrete Mathematics, Denver CO, June 6, 2018.

The Geometry of some Fibonacci identities in the Hosoya triangle. MAA Southeastern Section Spring 2018 Meeting, Clemson University, March 23-24, 2018.

Characterization of the strong divisibility property for generalized Fibonacci polynomials. Carolina Math Seminar, Benedict College, Columbia SC, Nov 3, 2017.

Projective Rectangles. AMS Fall Central Sectional Meeting University of North Texas, Denton, TX, Sep 9, 2017.

GCD Property of Generalized Fibonacci Polynomials and other identities. INTEGERS Conference 2016, University of West Georgia, October 8, 2016.

Una relacion entre los caminos de Dyck y los numeros de Fibonacci. Seminario de Redes Complejas y Combinatoria, Universidad Sergio Arboleda (Colombia), August 3, 2016.

Algunos Conteos de los caminos no decrecientes de Dyck. ALTENCOA 7-2016, Bucaramanga (Colombia), July 21, 2016.

Una relacion entre los caminos de Dyck y los numeros de Fibonacci. Universidad de Antioquia (Medellin), seminario institucional, July 18, 2016.

Non-decreasing Dyck paths and Fibonacci Numbers. The Combinatorics Seminar, Binghamton University, Binghamton NY, June 28, 2016.

Exploring Mathematics with Fibonacci Numbers and Hosoyas Triangle. Rochester Institute of Technology (RIT), June 23, 2016.

Projective Representation of Non-Representable Matroids (of Biased Graphs). Discrete Seminar, University of South Carolina Feb 12, 2016.

Projective Representation of Non-Representable Matroids (of Biased Graphs). Discrete Math Seminar, Texas State University, San Marcos, Nov 20, 2015.

Exploring Mathematics with Fibonacci Numbers and Hosoya's Triangle. Sigma XI brown-bag lunch presentation, The Citadel, October 16, 2015.

Experiments in Topology. Rochester Institute of Technology (RIT), June 25, 2015.

Explorando Matematicas con Relaciones Recursivas de Segundo Orden en Arreglos Triangulares. Foro departamento de Matematicas de Universidad del Valle (Cali). Seminario institutional, May 29, 2015.

Topologia Experimental. Universidad Sergio Arboleda (Bogota). Conductor of a workshop, May 23, 2015.

Explorando Matematicas en el Triangulo de Hosoya. Universidad de Antioquia (Medellin), seminario institucional, May 21, 2015.

Explorando Matematicas en el Triangulo de Hosoya. Seminario Universidad EAFIT (Medellin), May 15, 2015.

Some algebraic and geometric properties of Fibonacci polynomials in the Hosoya triangle. American Mathematical Society (AMS) Joint Mathematics Meetings, San Antonio TX, January 11, 2015.

Counting Non-Decreasing Dyck Paths. Discrete Seminar, University of South Carolina Nov 21, 2014.

A formal Power Series to Count non-decreasing Dyck paths. Colloquium, College of Charleston Sep 5, 2014.

Using Fibonacci Numbers to Count Dyck Paths. Sixteenth International Conference on Fibonacci Numbers and Their Applications, July 25, 2014.

Topics in the Hosoya's Triangle. Rochester Institute of Technology (RIT), June 25, 2014.

Seminario en Mathemática Elemental y Robótica. Universidad Sergio Arboleda (Bogotá). Conductor of a workshop, June 17-19, 2014.

Representacion Proyectiva de Matroids no representable (of Biased Graphs). Universidad de Antioquia (Medellin), seminario institucional, June 9, 2014.

A Relation Between Triangular Numbers and Prime Numbers. Rochester Institute of Technology (RIT), June 27, 2013.

Using Hosoya's Triangle to Count Some Dyck Paths. MAA, Southeastern Section, Winthrop University, March 16, 2013.

Representability of frame matroid and lift Matroid in a Projective Plane. Colloquium, College of Charleston Feb 15, 2013.

Convolutions on the Geometry of Hosoya's Triangle. American Mathematical Society (AMS) Joint Mathematics Meetings, San Diego CA, January 9, 2013.

GCD Properties in Hosoya's Triangle. MAA Southeastern Section Spring 2012 Meeting, Clayton State University, March 9-10, 2012.

Projective Representation of Non-Representable Matroids (of Biased Graphs). The Combinatorics Seminar, Binghamton University, Binghamton NY, May 23, 2011.

A representation of the bias matroid in a projective plane. Combinatorics seminar, Dept of Mathematics, University of South Carolina, Columbia, April 20 2011.

Some open questions. Regional Campuses Mathematics Seminar, University of South Carolina Sumter, Sep 17, 2010.

Harmonic Conjugation and Projective Rectangles. The Combinatorics Seminar, Binghamton University, Binghamton NY, May 25, 2010 .

A construction of a projective rectangle in a full algebraic matroid. Spring Southeastern Sectional Meeting, University of Kentucky, Lexington, KY, March 28, 2010.

Some topics in harmonic matroids. Regional Campuses Mathematics Seminar, University of South Carolina Lancaster, October 30, 2009.

A representation of the bias matroid in a projective plane. Fortieth Southeastern International Conference on Combinatorics, Graph Theory and Computing, March 2-6, 2009. Florida Atlantic University, Boca Raton.

Merging the Streams: Interdisciplinary Approaches to Improving Undergraduate Education and Civic Engagement. SENCER Summer Institute, San Jose CA, August 10, 2008.

A family of non-orientable matroids of rank three in a projective plane. American Mathematical Society (AMS) Joint Mathematics Meetings, San Diego CA, January 6, 2008.

What is a matroid?. The Mathematical Association of America (MAA) MAA/SIAM-SE conference at Georgia Southern University in Statesboro, Georgia. March 16, 2007.

A construction of a projective plane in a full algebraic matroids. Combinatorics seminar, Dept of Mathematics, University of South Carolina, February 28 2007.

Minimal non-orientable matroids in projective planes. The 28th Ohio State-Denison, Mathematics conference, Columbus Oh, May 21, 2006.

A Construction of a projective plane in harmonic matroids. American Mathematical Society (AMS) Joint Mathematics Meetings, San Antonio TX, January 13 2006.

Harmonic matroids as a tool to prove some conjectures in algebraic matroids. Combinatorics seminar, Dept of Mathematics, University of South Carolina, Columbia, September 22 2005.

Introduction to matroids II. Combinatorics seminar, Dept of Mathematics, University of South Carolina, Columbia, September 15 2005.

Introduction to matroids I. Combinatorics seminar, Dept of Mathematics, University of South Carolina, Columbia, September 8 2005

A conjecture of Lindstrom on a class of algebraically non-representable matroids. Discrete Geometry &Combinatorics Seminar, Dept of mathematics, Cornell University, March 9 2005.

Harmonic matroids and a conjecture of Lindstrom's. Mathematics Seminar, Scranton University, February 24 2005.

Harmonic Conjugation in Harmonic Matroids. Combinatorics and Number Theory Seminar, Dept. of Mathematical Sciences, Binghamton University, February 17 2005.

How the harmonic conjugate works in the algebraic matroids to construct algebraically nonrepresentable matroids. American Mathematical Society (AMS) Joint Mathematics Meetings, Atlanta GA, January 6 2005.

What is a matroid and what is it for? Mathematical Association of America (MAA) Seaway Section at Canisius College, Buffalo, NY, November 6 2004.

Lindstrom's Conjecture on a Class of Algebraically Non-Representable Matroids. Combinatorics and Number theory Seminar, Dept. of Mathematical Sciences, Binghamton University, October 1 2004.

Pseudomodularity and Full Algebraic Matroids. Algebra Seminar, Dept. of Mathematical Sciences, Binghamton University, March 25 2003.

The Relationship Between Full Algebraic Matroids and Lattice Properties: Pseudomodularity and Double Circuits. Combinatorics and Number Theory Seminar, Dept. of Mathematical Sciences, Binghamton University, March 24 2003.

Tutte-Grothendieck Invariance and the Chromatic and Characteristic Polynomials. Combinatorics and Number Theory Seminar, Dept. of Mathematical Sciences, Binghamton University, March 6 2002.