You can access the full content of the following journals online, for free.

1. Applicable Analysis and Discrete Mathematics.

2. The Electronic Journal of Combinatorics.

3. Online Journal of Analytic Combinatorics.

4. Séminaire Lotharingien de Combinatoire.

5. International Journal of Mathematical Combinatorics.

6. INTEGERS. Electronic Journal of Combinatorial Number Theory.

7. International Journal on Applications of Graph Theory in Wireless Ad Hoc Networks and Sensor Networks (GRAPH-HOC).

8. International Journal of Mathematics and Soft Computing.

1. February 3rd, 2015. Introduction of the course. Evaluation system. Bibliography. Scope lecture on Set Theory & Combinatorics. Quick review on previous concepts. Homework 1 assignment, Due February 5th 2015.

2. February 5th, 2015. Quick review of basic concepts.

   1. Set theory: inclusion, equality, union, intersection, difference, symmetric difference, relative complement, disjoint sets. Partitions, identification between partitions and equivalence relations.

   2. Functions. Definition, domain, codomain, image, inverse image. Behavior of set operations under image and inverse image. Injective, Surjective and Bijective Functions.

   3. Statement of the Pigeonhole Principle and the Well Ordering Principle of the Natural Numbers.

3. February 10th, 2015. Characterization of cardinality with bijections. Using the pigeonhole principle in the proof. Several version of the Principle of Mathematical Induction: Weak, Strong and starting at a given integer. Examples of induction.

4. February 12th, 2015. Combinatorics, introduction to counting and the main aspects involved. Section, pages 1-3 Keller & Williamson . Section 1.1: Theorem 1.1, Theorem 1.2 (the multiplication principle), Definition 1.1 (Cartesian products), Example 1.2 Keller & Williamson . Characterization of lists and Cartesian products with functions. Homework 2 available.

5. February 17th, 2015. The Addition Principle, Theorem 1.3, Examples 1.3, 1.4 Keller & Williamson. Lists with no repetition, Theorem 1.4, Examples 1.8, 1.9 Keller & Williamson. Circular arrangements, Example 1.12 Keller & Williamson.

6. February 19th, 2015. Circular permutations, several examples with the placement technique and alternative solutions. A glimpse of potential mistakes in the combinatorial reasoning. Combinations, Theorem 1.6 Bender & Williamson, combinatorial proof of the number of combinations, pg 17 Enumerative Combinatorics, Volume I. Richard Stanley. Proposition 2.6 (pg 19), Example 2.12 (pg 20), Applied Combinatorics. Mitchel T. Kelller & William T. Trotter. Generating functions approach for the number of combinations, pg 17 Enumerative Combinatorics, Volume I. Richard Stanley.

7. February 24th, 2015. Recursion of combinatorics, starting paragraph section 1,4, pg 32 Bender & Williamson. Combinatorial interpretation of the triangular numbers. A generating function for the binomial coefficients: Example 1.14 Bender & Williamson and pg 17 Enumerative Combinatorics, Volume I. Richard Stanley. Theorem 1.7 , The Binomial Theorem, Bender & Williamson. Theorem 4.1 the Inclusion-Exclusion Principle, Bender & Williamson. Statement of of the Multisets Counting Problem. Problem Set 3 and Problem Set 4 available.

8. February 26th, 2015. Example 1.15 (pg 22), Example 1.16 (pg 23). Multisets, Theorem 1.8 (pg 37), Balls in Boxes, Example 1.28 (pg 37). The Multinomial Coefficient, Example 1.18 (pg 23), Bender & Williamson. Problem Set 5, First Draft and Homework 2 (due March 12th) available.

9. March 3th, 2015. The Multinomial Coefficient as permutations of multisets and as distribution of objects on labeled boxes with fixed quantities, pg 27 Enumerative Combinatorics, Volume I. Richard Stanley. Application of multisets to lattice paths, Proposition 1.21, pg 28 Enumerative Combinatorics, Volume I. Richard Stanley. The multinomial theorem, pg 28 Enumerative Combinatorics, Volume I. Richard Stanley.

10. March 5th, 2015. The multinomial theorem; mathematical analysis approach, pg 28 Enumerative Combinatorics, Volume I. Richard Stanley. Weak and strong compositions of integers, multisets approach, pgs 17-18 Applied Combinatorics. S.E. Payne. Partitions of sets and the Stirling Numbers of the Second Kind. pgs 33-34 Foundations of Combinatorics with Applications. Bender & Williamson.

11. March 10th, 2015. The Stirling Numbers of Second Kind Theorem 1.7.1, Bell Numbers Definition, pg 26, Theorem 1.7.2, pg 27 Applied Combinatorics. S.E. Payne. Counting the number of surjections from [n] onto [k], polynomial extension of x^n in terms of the Stirling Numbers pg 83, Enumerative Combinatorics, Volume I. Richard Stanley. First presentation of the Catalan numbers as divisions of convex polygons in triangles with no cutting diagonals, pgs 253, 254, Introductory Combinatorics (5th Edition). Richard A Brualdi.

12. March 12th, 2015. First Test. Find the solution of the first test HERE.

13. March 17th, 2015. The Catalan Numbers, different scenarios of presence and applications. Partitions of convex polygons by non-intersecting diagonals, Section 7.6 pgs 253-257, Introductory Combinatorics (5th Edition). Richard A Brualdi. Association of binary operations to a large computational operation, problem 33, pg 33 Applied Combinatorics. Mitchel T. Kelller & William T. Trotter. Diagram of binary trees and Computer´s Architecture design: parallel computing, weighted tasks assignment. Controlled walks on lattices, example 2.21, pg 24 Applied Combinatorics. Mitchel T. Kelller & William T. Trotter. Random Walks, The Gambler´s Ruin Problem and financial applications, Political Control Management Applications, Problem 33, pg 33 Applied Combinatorics. Mitchel T. Kelller & William T. Trotter and/or Section 1.3.1, pg 22, Combinatorics Through Guided Discovery. Kenneth P. Bogart. Problem Set 5 available.

14. March 19th, 2015. Permutations seen as linear orderings and bijection. The one-line notation, composition of permutations. Cycle factorization of permutations, existence, uniqueness of cycle decomposition through equivalence class arguments. Examples of cycle factorization and notation of permutations in terms of its cycle factorization, pgs 22-23, Enumerative Combinatorics, Volume I. Richard Stanley., alternatively see pgs 45-46, Foundations of Combinatorics with Applications. Bender & Williamson. alternatively see Sections 15.2.1 and 15.2.2 pgs 271-273, Applied Combinatorics. Mitchel T. Kelller & William T. Trotter.

15. March 24th, 2015. From now on, the lectures will be held in classroom 43-307. Definition of permutation type or cycle structure, pg 22. Counting of permutations of a given type, Proposition 1.3.2. The Stirling Numbers of the first kind, signless c(n, k) and regular s(n,k). Recursive relation for signless Stirling numbers of the first kind, Lemma 1.3.6. Statement of Proposition 1.3.7. Enumerative Combinatorics, Volume I. Richard Stanley.

16. March 26th, 2015. Proposition 1.3.7, Enumerative Combinatorics, Volume I. Richard Stanley. Theorem 1.5.5, Corollary 1.5.6, Applied Combinatorics. S.E. Payne. or, alternatively page 119, A walk through Combinatorics. Miklos Bona. Theorem 6.14, Lemma 6.15 and statement of Proposition 6.18, with an auxiliary proposition. A walk through Combinatorics. Miklos Bona. Homework 3 (April the 9th, 2015) available.

17. April 7th, 2015. Proposition 6.18, full proof. Lemma 6.19, informal schematics and comments about Lemma 6.20 and statement without proof of Theorem 6.24. Several examples to illustrate the action of the involved bijections in the permutations of analysis. A walk through Combinatorics. Miklos Bona.

18. April 9th, 2015. First lecture in English. The Sieve, also known as (AKA) The Principle of Inclusion Exclusion. Discussion of the basic cases: two sets, three sets. Main result: Theorem 6.1.1. Derangements: Theorem 6.3.1 Introductory Combinatorics (5th Edition). Richard A Brualdi. Surjections and Stirling Numbers of the Second Kind, Statement and discussion of Theorem 7.8 (proof to be presented next session). Applied Combinatorics. Mitchel T. Kelller & William T. Trotter.

19. April 14th, 2015. Proof of Theorem 7.8 Applied Combinatorics. Mitchel T. Kelller & William T. Trotter. Discussion on the Mobius inversion formula, Section 6.6, pgs 183, 184 and the combinatorial proof of the inclusion-exclusion principle, Theorem 6.1.1, Introductory Combinatorics (5th Edition). Richard A Brualdi. Second Test April the 23th from 6:00-9:00.

20. April 16th, 2015. Introductory group theory, definitions, examples, subgroups, cosets and the induced equivalence relation, Section 4.3, pages 111, 112, Foundations of Combinatorics with Applications. Bender & Williamson.

21. April 21st, 2015. Permutations as groups: The Symmetric Group of a Set. Definition of Stabilizer, Orbits and the set of elements fixed by a given permutation, Burnside's Lemma. Paragraph above Theorem 3.1.3, Theorem 3.1.3, Theorem 3.1.4 and Theorem 3.1.5, Applied Combinatorics. S.E. Payne.

22. April 23th, 2015. Second Test 6:00-9:00 room 43-110. Find the solution of the second test HERE. Also, Homework 4 available.

23. April 28th, 2015. Symmetries of a polygonal figure, definition. The group of permutation symmetries. Intuition of the Dihedral Group of a Regular Polygon, pages 546, 547 and 548 Introductory Combinatorics (5th Edition). Richard A Brualdi.

24. April 29th, 2015. The Dihedral Group of a Regular Polygon. Coloring Problems first exposure; Two by Two Chess Board Colorings: The Toy Problem, pages 92, 93, Counting: The Art of Enumerative Combinatorics, George E. Martin. Coloring of a linear n-string with four colors, page 558, intuitive discussion of the cycle structure of permutations and its Fixed set, page 563, Introductory Combinatorics (5th Edition). Richard A Brualdi.

25. May the 5th, 2015. Amend for the problem of coloring of a linear n-string with four colors, page 558 from previous class. Coloring the vertices of a Square, pags 563, 564, Introductory Combinatorics (5th Edition). Richard A Brualdi. Coloring of an eight by eight chessboard, pgs 259, 260, 261. Introduction to the symmetries of the tetrahedron, pgs 261, 262, How to Count: An Introduction to Combinatorics. Allenby & Slomson.

26. May the 7th, 2015. Coloring of the faces of a tetrahedron. The group os symmetries of the cube, coloring the faces of the cube, Problem 263, pg 119, Combinatorics Through Guided Discovery. Kenneth P. Bogart.

27. May 12th, 2015. Amends to the calculation of the cardinal for the groups of symmetries of the tetrahedron and the cube, and the presentation of a more general method. Comments on general contexts for solids such as the Rubik Cube and other structures. Comments on applications to other problems such as Crystallography, Genetics, Graph Theory. Closing remarks for the overall scope of the Combinatorial part of the class.

28. May 14th, 2015. Set theory. The Mayors' Paradox. Russel's Paradox, pg 7. The Axiom of Extension, pg 2. The Axiom of Specification pg 6. The no existence of the Universal Discourse, pg 7. The Axiom of Existence of Sets, and its consequence on the existence of the Empty Set, pg 8. The Axiom of Pairing pg 9. Naive Set Theory. Paul Richard Halmos.

29. May 19th, 2015. Set theory. The Axiom of Pairing and consequences pgs 9, 10. The Axiom of Union and consequence on Intersections pgs 12, 14. Complements and De Morgan's laws pgs 17, 20. The Power Set Axiom, pg 19. Ordered pairs and Cartesian Products, pgs 22, 23, 24, 25. Naive Set Theory. Paul Richard Halmos. The Axiom of Regularity, pg 117 Proofs and Fundamentals, a first course in abstract mathematics. Ethan D. Bloch.

30. May 21st, 2015. Set theory. Cartesian Products, the functions approach, pgs 34, 35, 36, 37. The Axiom of Infinity and definition of numbers, pags 43, 44. The Axiom of Choice pg 59. Comments on equivalence with Zermelo's Theorem and Zorn's lemma. Comments on consequences of accepting the Axiom of Choice, pgs 60, 61. Comments on Accepting the Axiom of Infinity. Naive Set Theory. Paul Richard Halmos.

31. May 26th, 2015. Set theory. Definition equipotence, definiton of cardinal. Examples of bijections between the Natural Numbers N and some of its proper subsets: when they differ by a finite number of elements, when they differ by an infinite number of elements. Bijections between the Natural Numbers N and N². Higlights for bijections between the Natural Numbers N and the rational Numbers Q. Comments on the Cantor-Bernstein-Schroeder theorem (proof postponed). Pages 90, 91, 92. Naive Set Theory. Paul Richard Halmos.

32. May 28th, 2015. Set theory. Countable sets. Existence of an Uncountable set {0,1}^N, Cantor's diagonal argument. Countable union of countable sets is countable. Bijections between the power set of N and {0,1}^N . Every infinite set contains a countable infinite set. Bijection between {0,1}^N and the interval [0,1], pgs 236, 237, 238 Proofs and Fundamentals, a first course in abstract mathematics. Ethan D. Bloch. Homework 5 Available, due June the 11th.

33. June 2nd, 2015. Set theory. Informal proof of the equipotence bewteen [0,1] and {0,1}^N . Cantor's Diagonal Argument Revisited illustrating the spirit of Cantor's Theorem. Cantor's Theorem, general form, pg 93 Naive Set Theory. Paul Richard Halmos. The hypothesis of the continuum and discusion on the indecidibles, pg 102 Naive Set Theory. Paul Richard Halmos. An example for the equipotence between [0,1] and [0,1), illustrating the spirit of the Cantor-Bernstein-Schroeder Theorem. Third Test June the 11th from 6:00-9:00.

34. June 4th, 2015. Final lecture of the course. Set theory. Cantor-Bernstein-Schroeder Theorem pgs 88, 89 Naive Set Theory. Paul Richard Halmos. Additional comments on Dynamical Systems for the construction of the proof. Closing remarks and examples. Third Test , June the 11th from 6:00-9:00. Find the solution of the third test HERE