As detailed on the syllabus, your content grade in this course will be determined by your proficiency on a variety of standards. (This is known as Standards Based Grading.)

The list of standards will grow throughout the semester.

## Standards

**Standard 1. Basic Counting.** Can you apply basic counting techniques involving words and sets with and without repetition? Can you apply the sum and product principles?

**Standard 2. Counting the Complement.** Can you apply the counting technique of counting the complement? Can you determine and count the universe and the complement in order to count a set?

**Standard 3. Finding a Bijection.** Given two sets A and B, can you determine a rule/function that pairs the elements of A and B in such a way that this rule/function generalizes to an infinite family of sets?

**Standard 4. Proving a Bijection.** Given a function between sets A and B, can you prove that the function is a bijection?

**Standard 5. The Equivalence Principle.** Can you prove that a relation is an equivalence relation? Do you know when the equivalence principle applies? Can you apply the equivalence principle?

**Standard 6. Combinatorial Proof.** Can you prove an identity using a combinatorial proof? This includes the square-domino interpretations of the Fibonacci numbers.

**Standard 7. Counting Distributions.** Can you answer a question involving distributing objects into boxes and/or equivalent counting questions?

**Standard 8. Principle of Inclusion / Exclusion.** Can you apply the principle of inclusion / exclusion? Given a problem, can you determine the sets that are being intersected?

**Standard 9. Generating Function Determination.** Given a situation involving combinatorial objects of various sizes, can you determine a generating function that counts the objects? Can you determine the compact form for such a generating function?

**Standard 10. Coefficient Extraction.** Given a generating function in its compact form, can you extract the necessary coefficient by hand?

**Standard 11. Combinatorics of Generating Functions.** Are you able to provide combinatorial interpretations of multiplication, powers, and compositions of generating functions as sequences of combinatorial objects? Given a combinatorial question involving sequences of combinatorial objects, can you determine the correct product, power, or composition of generating function that counts that question?

**Standard 12. Partition Generating Functions.** Can you correctly interpret the generating function for partitions and restricted partitions? Can you use these generating functions to prove partition identities?

**Standard 13. Ferrers/Young Diagrams for Partitions.** Do you understand how Ferrers diagrams and Young diagrams represent partitions? Can you use them in combinatorial proofs related to partitions?

**Standard 13. Catalan Numbers.** Do you know the Catalan numbers? Can you apply the bijections between the combinatorial interpretations we discussed in class?

**Standard 14. The Generating Function of Catalan Numbers.** Do you know the generating function *C(x)* for Catalan numbers? Can you explain how it arises **combinatorially** from a combinatorial interpretation of the Catalan numbers?

**Standard 15. Combinatorial Statistics.** Can you apply the descent statistic, inversion statistic, and major statistic to a given permutation? Can you compute a combinatorial statistic on a discrete object?

**Standard 16. q-analogs.** Do you understand the concept of a

*q*-analog? Do you know [

*n*]

_{q}, the

*q*-factorial, and the

*q*-binomial coefficients?