Key
[AC] = Stanley, Richard P. Algebraic Combinatorics: Walks, Trees, Tableaux, and More. Springer, 2018. ISBN: 9783319771724. Online 2013 version.
[EC1] = ———. Enumerative Combinatorics. Vol. 1. Cambridge, UK: Cambridge University Press, 1997. ISBN: 9780521553094. Online version.
[EC2] = ———. Enumerative Combinatorics. Vol. 2. Cambridge, UK: Cambridge University Press, 2001. ISBN: 9780521789875.
Lecture sessions with reading assignments are listed below.
LEC #  TOPICS  READINGS 

2 
Catalan numbers (cont.): formula for C_{n}, reflection principle, necklaces, triangulations of polygons, plane binary trees, parenthesizations. 
Stanley, Richard P. “Exercises on Catalan and Related Numbers” (PDF). (excerpted from [EC2]) Stanley, Richard P. “Catalan Addendum” (PDF). 
3 
Pattern avoidance in permutations. Stack and queuesortable permutations. Young diagrams and Young tableaux. Hooklength formula. 
[AC] “Chapter 8: A Glimpse of Young Tableaux” (PDF  1.2MB). If you want to learn more details about the links between combinatorics of Young tableaux and representation theory, see Sagan, Bruce E. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Springer, 2001. ISBN: 9780387950679. 
4 
FrobeniusYoung identity. Schensted correspondence. Longest increasing and decreasing subsequences in permutations. 
[AC] “Appendix 1 to Chapter 8: The RSK Algorithm” (PDF  1.2MB). Schensted, C. “Longest Increasing and Decreasing Subsequences.” Canadian Journal of Mathemetics 13 (1961), 179191. Knuth, Donald E. “Permutations, Matrices, and Generalized Young Tableaux” (PDF). Pacific Journal of Mathematics 34 (1970), 709727. Greene, Curtis. “An Extension of Schensted’s Theorem” (PDF). Advances in Mathematics 14 (1974), 254265. 
5 
Proof of the hooklength formula based on a random hook walk. 
Greene, Curtis, Albert Nijenhuis, and Herbert Wilf. “A Probabilistic Proof of a Formula for the Number of Young Tableaux of a Given Shape” (PDF). Advances in Mathematics 31 (1979), no. 1. 
6 
Hook walks (cont.). Linear extensions of posets. Hooklengthtype formulas for shifted shapes and trees. 
Knuth, Donald E. The Art of Computer Programming, Volume 3: Sorting and Searching. AddisonWesley Professional, 1998. Section 5.1.4. ISBN: 9780201896855. 
7 
qfactorials and qbinomial coefficients. 
[AC] “Chapter 6: Young Diagrams and qbinomial Coefficients” (PDF  1.2MB). 
8 
Grassmannians over finite fields: Gaussian elimination and rowreduced echelon form. 
[EC1] “Section 1.7: Permutations of Multisets” (PDF  4.4MB). (see Propositions 1.7.2 and 1.7.3) 
9 
Sets and multisets. Statistics on permutations: inversions, cycles, descents. 
[EC1] “1.2 Sets and Multisets” (PDF  4.4MB), “1.3 Cycles and Inversions” (PDF  4.4MB), and “1.4 Descents” (PDF  4.4MB). 
15 
Posets and lattices. Boolean lattice. Partition lattice. Young’s lattice. 
[EC1] Chapter 3: Partially Ordered Sets: “3.1 Basic Concepts” (PDF  4.4MB), “3.2 New Posets from Old” (PDF  4.4MB), and “3.3 Lattices” (PDF  4.4MB). 
16 
Distributive lattices. Birkhoff’s fundamental theorem for finite distributive lattices. 

17 
Sperner’s property. Symmetric chain decompositions. Sperner’s and Dilworth’s theorems. Greene’s theorem. 

18 
Greene’s theorem vs Schensted correspondence. Up and down operators. Differential posets. 

19 
Differential posets (cont.). Fibonacci lattice. Unimodality of Gaussian coefficients. 
[AC] “Chapter 6: Young Diagrams and qbinomial Coefficients” (PDF  1.2MB). (see Corollary 6.10) 
20 
Proof of unimodality of Gaussian coefficients (cont.). Theory of partitions. Euler’s pentagonal number theorem. 

23 
Two combinatorial proofs of Cayley’s formula. 
[AC] Chapter 9. “Appendix: Three Elegant Combinatorial Proofs” (PDF  1.2MB). Egecioglu, Ömer and Jeffrey B. Remmel. “Bijections for Cayley Trees, Spanning Trees, and Their qanalogues.” J. Combinatorial Theory, Ser. A, 42 (1986), 1530. 