1. Syllabus.

2. An update of the course content covered a session-by-session basis.

3. The course will be partially lectured in English. This is the Schedule for the Lectures in English. Since English is not your native language you are strongly encouraged to read in advance the corresponding material.

4. Textbook and Reading Materials.

5. Homework Assignments.

6. Test Dates and Solutions.

7. Link to the online Debate Forum of the Course (see Debate Activities) an other Open Debate Sites.

8. Related Software Links.

9. Related online Journals Links.

10. Links to online Free Courses Related to Discrete Mathematics and Combinatorics.

11. Inspirational and Confusing unrelated material intended to push out-of-the-box thinking and free association of ideas.

Debate Activities

The course includes an additional 2 hour Debate-Session per week and it is not mandatory. It is meant for the participants and the lecturer to discuss the main concepts of the course from the student´s perspective as well as exchanging problem-solving and learning strategies.

1. This is link for the Online Debate Forum. You have to be invited in order to participate.

2. This is a link to Mathematics Stack Exchange, which is a question and answer site for people studying math. It is free and open for everybody.

Notice that both platforms are in English and their interface uses LaTeX for equations editing.

1. Applied Combinatorics. Mitchel T. Kelller & William T. Trotter.

2. Combinatorics Through Guided Discovery. Kenneth P. Bogart.

3. Bijective Combinatorics. Nicholas Loehr.

4. Applied Combinatorics. S.E. Payne.

5. Introductory Combinatorics (5th Edition). Richard A Brualdi.

6. A Walk Through Combinatorics. Miklos Bona.

7. Counting: The Art of Enumerative Combinatorics, George E. Martin.

8. Enumerative Combinatorics, Volume I. Richard Stanley.

9. Principles and Techniques in Combinatorics. Chen Chuan-Chong & Koh Khee-Meng.

10. Naive Set Theory. Paul Richard Halmos.

11. Proofs and Fundamentals, a first course in abstract mathematics. Ethan D. Bloch.

12. Elementary Set Theory with Universal Sets. Randall Holmes.

13. An introduction to Set Theory. William A.R. Weiss.

1. COMBINATORICA written by Sriram Pemmaraju and Stephen Skiena, is a Mathematica collection of over 230 functions in combinatorics and graph theory.

2. Posets.m written in Mathematica by Curtis Greene, designed to generate, display, and explore partially ordered sets.

3. The Combinatorial Object Server maintained by Frank Ruskey, is an on-line device, which, on specifying a type of combinatorial object, together with specific parameter values, will return to you a list of all such objects.

4. WOLFRAM ALPHA can help you out with your online calculations (including combinatorial questions).

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. August 4th, 2015. Introduction of the course. Evaluation system. Bibliography. Scope lecture on Set Theory & Combinatorics. Quick review on previous concepts. Homework 1 assignment, Due August 18th 2015. Quick review of basic concepts.

   • Set theory: union, intersection, difference, symmetric difference, relative complement.

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

  2. Characterization of cardinality with bijections.

2. August 6th, 2015. Lecture in English. Quick review of basic concepts. Debate Sessions scheduled on Mondays 8:00 - 10:00, Room 43-307,

   • Set theory: disjoint sets and partitions and cardinals.

• 1. Statement of the Well Ordering Principle of the Natural Numbers.

  2. Principle of Mathematical Induction.

  3. Examples of induction worked rigorously.

3. August 10th, 2015. Debate Session 1. Discussion of problems 16, 32, 26 I-(i), (ii) and II-(i), (ii), Homework 1.

4. August 11th, 2015. Examples of Induction: inequalities, cardinal of the power set. Strong Principle of Mathematical Induction and examples. A central technique: identifying a particular element to give structure in a counting problem.

5. August 13th, 2015. Lecture in English. Induction: example with composed logical statements and associated intrinsic structures, gemotric example. Combinatorics, Basic Counting Principles. The Sum Rule, Theorem 1.2. The Product Rule, Theorem 1.5. Enumeration of Functions, Theorem 1.22. Enumeration of Bijections, Theorem 1.28 Bijective Combinatorics. Nicholas Loehr.

6. August 17th, 2015. Debate Session 2. Suspended due to holiday.

7. August 18th, 2015. Examples of the bijective technique. Definition of permutations, combinations, lists, words and anagrams. Comments on the role of repetitions and order. Counting the number of permutations with restriction on the last evaluation (no reference available). Counting the number of injections Theorem 1.42, Counting the number of combinations, Theorem 1.42. Bijection between the parts of a set X and all the possible binary words defined on the set X, Theorem 1.38. Counting the number of compositions of an integer, Theorem 1.41, Bijective Combinatorics. Nicholas Loehr. Homework 2 assignment, Due September 10th, 2015.

8. August 20th, 2015. Lecture in English. The Multinomial Coefficient as permutations of multisets, 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.

9. August 24th, 2015. Debate Session 3. Discussion of problems 3, 4, 15 and 16, Homework 2.

10. August 25th, 2015. Distributing k identical objects into n distinct boxes, Case (2), pg 42, Principles and Techniques in Combinatorics. Chen Chuan-Chong & Koh Khee-Meng. Circular arrangements, pgs 12-15 Principles and Techniques in Combinatorics. Chen Chuan-Chong & Koh Khee-Meng. The Binomial Theorem. Theorems 4.1, 4.2, 4.3 A Walk Through Combinatorics. Miklos Bona.

11. August 27th, 2015. Lecture in English. Theorem 4.4, Theorem 4.5, Theorem 4.6, Theorem 4.6, Theorem 4.7, A Walk Through Combinatorics. Miklos Bona. The President-Committee lemma (no reference). Comments on multisets identities with binomial sum format. Comments on Problem 33 (ii), (vi) Homework 1.. Starting from today the Online Debate Forum is available for all the participants.

12. August 31st, 2015. Debate Session 4. Discussion of problems 21, 22 and 47 Homework 2.

13. September 1st, 2015. Recursion results: Multisets Example 2.26, Lattice Paths Example 2.28, Dyck Paths Example 2.29 Bijective Combinatorics. Nicholas Loehr. Definition 5.1, Theorem 5.2, Corollary 5.3, Corollary 5.4, Definition 5.5, Example 5.6 A Walk Through Combinatorics. Miklos Bona.

14. September 3th, 2015. Lecture in English. Theorem 5.8, Corollary 5.9, Corollary 5.10, Definition 5.11, A Walk Through Combinatorics. Miklos Bona. A binomial recursive form of the the Sitriling numbers of the second kind Problem 11, page 79, Combinatorics Through Guided Discovery. Kenneth P. Bogart.

15. September 6th, 2015. Debate Session 5. Discussion of Problem 47-(iii) remains as an open question.

16. September 8th, 2015. Bell Numbers, Definition 5.11 and Theorem 5.12. Partitions, Definition 5.13, Theorem 5.17, Theorem 5.20, A Walk Through Combinatorics. Miklos Bona. Ferrers Boards and Rooks, Definition 2.60, Theorem 2.42, 2.43 and 2.44 on Integer partitions recursion. Definition 2.61, Non-attacking rook placement Definition 2.61, Theorem 2.63, rook-theoretic interpretation of Stirling Numbers, Bijective Combinatorics. Nicholas Loehr.

17. September 10th, 2015. Lecture in English. Closing comments on Ferrers boards and its equivalence with the Stirlilng numbers of second kind. Also, comments on the non-commutative relationship between rook placement and landing in the structure with recursion. An example of partitions and its "completion" in bijection. Definition 2.6 (conjugate partition) Bijective Combinatorics. Nicholas Loehr. Statement without proof of Theorem 5.18 A Walk Through Combinatorics. Miklos Bona. Permutations Structure, example 6.1 and introductory observations. Definition of cycles for permutations. Identification of cycles in permutations with equivalence classes. Lemma 6.5, Definition 6.5, Corollary 6.6. Definition of Cycle type. Statement without proof of Theorem 6.9, A Walk Through Combinatorics. Miklos Bona. Homework 3, First Draft Available.

18. September 14th, 2015. Debate Session 6. Suspended due to the University´s 2016-2018 Global Development Plan. Homework 3 assignment, Due October 6th, 2015.

19. September 15th, 2015. Review of cycle type and comments on cycle type and order of a permutation. Theorem 6.9 and simple version when no length has more than one cycle. Definition 6.11, Stirling numbers of the first type, signed and signless. Theorem 6.12 on recurrence for signless Stirling numbers of the first kind, A Walk Through Combinatorics. Miklos Bona.

20. September 17th, 2015. Lecture in English. Lemma 6.13 and the involvement of Stirling numbers of second kind s(n, k) in polynomials. Comments on Theorem 6.14 on the role of Sitrling numbers S(n, k), s(n,k) as coefficients between different basis of polynomials of degree less or equal than n. A combinatorial computation of c(n,k). Lemma 6.15 and highlights on the relationship between cylces, placement and monotonicity.

21. September 21st, 2015. Debate Session 7. Discussion of Problems 5, 6, 14 and 18. Discussion on the general strucutre of Problems 23 through 29 Homework 3.

22. September 22th, 2015. Inversions as a menas to study permutation structure. Definition 9.22 (Inversion and Sign of a permutation). Example 9.23 and class examples illustrating the involved concept of monotonicicty, determinant and orientation. Definition 9.24, transpositions and basic transpositions, discussion on the sign and inversions for transpositions. Lemma 9.25 Basic Transpositions and sorting. Lemma 9.26 Basic Transpositons and inversions. Bijective Combinatorics. Nicholas Loehr.

23. September 24th, 2015. First Test September 24th, from 6:00 am to 12:00 pm, Room 43-307.

24. September 28th, 2015. Debate Session 8. Solution of problems 26, 27, 28 and 29. Problems 23, 24 and 26 remain open and unsolved. All of them in Homework 3.

25. September 29th, 2015. Lemma 9.27, Lemma 9.28, Theorem 9.29, Theorem 9.31, Theorem 9.32, Theorem 9.33 Bijective Combinatorics. Nicholas Loehr. Graded tests returned. Due to the low attendance rate, the Debate Session is to be rescheduled.

26. October 1st, 2015. Lecture in English. Theorem 9.33, alternative proof Bijective Combinatorics. Nicholas Loehr. Definition of distance. Definition of basic transposition distance for permutations: d(p, q). Proof of good-definition of the function d(p,q) and the fact of being a metric. Definition of the Block Transposition distance: btd(p,q). Proof of good-definition of the function btd(p, q).

27. October the 5th, 2015. Debate Session 9. Canceled due to lack of quorum.

28. October the 6th, 2015. Closing comments on the Basic Transposition distance for permutations: d(p, q); as well as the Block Transposition distance: btd(p,q). Closing comments on the interaction genetics-combinatorics. Combinatorial definition of the determinant of a matrix 9.37. Comments on permutation matrices and brief discussion on the combinatorial structure of the determinant, Bijective Combinatorics. Nicholas Loehr. The inclusion-exclusion principle. Examples 7.1 and 7.2. Theorem 7.3, proof by induction; its combinatorial proof is left as an exercise. A Walk Through Combinatorics. Miklos Bona.

29. October 8th, 2015. Lecture in English. Example 7.4. Definition of fixed points and Derrangements. The Earth-covering example. Theorem 7.5, Theorem 7.6, A Walk Through Combinatorics. Miklos Bona. End of the Inclusion-Exclusion Principle. Homework 4 assignment, Due November 3th, 2015.

30. October the 12th, 2015. Debate Session 9. Canceled due to holiday.

31. October the 13th, 2015. Canceled due to lecturer's trip.

32. October 15th, 2015. Lecture in English. Opening Chapter for the Mobius Inversion Formula, Definitions of order relations: preorder, partial order, total order. Definitions of Partially Ordered Sets (POSETS) and Well-Ordered sets, Definition 16.1, Examples 16.2, 16.4, 16.5, 16.6 and 16.7. Definitions of Maximal and Maximum elements, definition of Chains, pg 383. Isomorphism of POSETS, Definition of Locally Finite POSETS, Definition 16.10, A Walk Through Combinatorics. Miklos Bona.

33. October 20th, 2015. Definition of antichains and size, definition of chain cover and size, page 385, failed attempt to prove Theorem 16.8 (the hard implication will be ommitted) A Walk Through Combinatorics. Miklos Bona.

34. October 22th, 2015. Lecture in English. Definition of Intervals in a POSET, Definiton of locally finite POSETS and aexamples, Definition of Ideals and Principal Ideals in a POSET. The Algebra A(X) of functions defined on the intervals of the POSET X, pg 387. Definition 16.10 (convolution of functions) and comments on matrix multiplication. The unit element of the Algebra A(X), Definition of Multichains pg 388. Definition 16.11 of the zeta function, Propostion 16.12, Theorem 16.15. A Walk Through Combinatorics. Miklos Bona. Second Test, October 31st, 2015, 6:00 am-12:00 pm, room 43-307.

35. October the 22th, 2015. Debate Session 10. Discussion of Problems 1, 2, 3, 5, 6, 7, 10, 11 and 12 Homework 4.

36. October 27th, 2015. Definition 16.14, Theorem 16.15, Corolllary 16.16, Example 16.17, Example 16.18, comments on Example 16.20. Review of principal ideals and compasison between locally finite POSETS and POSETS whose every principal ideal is finite, comments on the the Mobius Inversion Formula, Theorem 16.21, Corollary 16.22. A Walk Through Combinatorics. Miklos Bona.

37. October 29th, 2015. Lecture in English. Definition 16.23, Theorem 16.24. Definition of bounds, lower an upper, Definiton 16.25, Example 16.26, Example 16.27, Example of the Natural Numbers with both, the usual order and the division order, Proposition 16.29, Definition of Semilattice, Lemma 16.30 A Walk Through Combinatorics. Miklos Bona.

38. October the 29th, 2015. Debate Session 11. Discussion of Problems 17, 24, 25, 34 and 50 Homework 4.

39. November 3th, 2015. Last example, the POSET made by all possible partitions and proof of being a lattice using Lemma 16.30 A Walk Through Combinatorics. Miklos Bona. Closing comments on the Combinatorics part of the course. Opening of the Set Theory part of the course, introductory intuitive lecture. The majors' paradox, Russel's paradox, the hanged man's paradox and tribute to Don Miguel de Cervantes Saavedra. Axiom of Extensionality, Axiom of Specification, Axiom of the Existence of a set, in particular the existence of the empty set and the non-existence of the universal set. Naive Set Theory. Paul Richard Halmos.

40. November 5th, 2015. Lecture in English. This once lecture was made in Spanish, due to the complexity of the material of the lecture. Axiom of Pairing, Axiom of Union, Complements, Axiom of Powers, Axiom of Regularity, Ordered Pairs and Cartesian Product of Sets. Naive Set Theory. Paul Richard Halmos.

41. November the 5th, 2015. Debate Session 12. Suspended due to Lecturer's obligations with the Conference in Mathematical Biology.

42. November 10th, 2015. Cartesian Products, the functions approach and examples, of vector spaces, sequences and real functions of real variable. The Axiom of Infinity, its analogy with the Natural Numbers as the first inductive set. Construction of Natural Numbers as sets. The Axiom of Choice and its equivalence with Zorn's Lemma and Zermelo's Theorem. The Banach-Tarski Paradox as an illustration of the consequences of the Axiom of Choice. The consistence of the Axiom of Choice with the Zermelo-Frank Set Theory Axiomatic System. The independence of the Axiom of Choice with the Zermelo-Frank Axiomatic System. Naive Set Theory. Paul Richard Halmos. Graded tests returned. Homework 5 assignment, Due November 26th, 2015.

43. November 12th, 2015. Cardinality, introductory lecure: the equivalence of bijections and number of elements for finite sets. Discussion about the meaning and abstract nature of large numbers and infinity. Bijective definition of cardinality and consequences. Examples with the Natural Numbers with "finite head" as well as the equipotence of Natural Numbers with Even Numbers. Equipotence between the integers and the natural numbers. Proofs and Fundamentals, a first course in abstract mathematics. Ethan D. Bloch.

44. November 12th, 2015. Debate Session 13. Discussion of problems 1, 3, 7 and 8 Homework 5 .

45. November 17th, 2015. Examples of bijections between the Natural Numbers and the Natural Numbers multiplied by a finite set, between the Natural Numbers and the Natural Numbers multiplied by itself, between the Natural Numbers and the Rational Numbers. The sequences with values on 0,1 and the power set of the Natural Numbers. The Cantor's Theorem. Definition of order between cardinals, by means of injections. The Continuum Hypothesis and discussion about its relation with the Zermelo-Frank systems of set axioms, Godel's Incompleteness Theorem, Proofs and Fundamentals, a first course in abstract mathematics. Ethan D. Bloch.

46. November 19th, 2015. Lecture in English. Final lecture of the course. Definition of countable and uncountable sets. Construction of a countable set Y within an uncountable set X and stability of the cardinal ater its removal X - Y. Equipotence between the power set of the Natural Numbers and the interval (01,) as well as with any non-degenerate interval and the whole line of the Real Numbers. Countable union of countable sets. The "modulo Q" subset of the Real Numbers "R". The Cantor-Bernstein-Schroeder Theorem (without proof) for deriving equality of cardinals from two independent inequalities and associated comments, Proofs and Fundamentals, a first course in abstract mathematics. Ethan D. Bloch. & Naive Set Theory. Paul Richard Halmos. Third Test, November 28th, 2015, 6:00 am-12:00 pm, room 43-307.

47. November 19th, 2015. Debate Session 14. Discussion of problems 13, 15, 16, 17 and 24 Homework 5 .

48. November 26th, 2015. Debate Session 15. Final debate session of the course. Discussion of problems Homework 5 .