Math 264B: Combinatorics II
Winter 2019

Instructor: Brendon Rhoades
Instructor's Email: bprhoades (at) math.ucsd.edu
Instructor's Office: 7250 APM
Instructor's Office Hours: By appointment
Lecture Time: 12:00-12:50pm MWF
Lecture Room: B402A APM


Grading:

Hmm....

Textbook: We will loosely follow Enumerative Combinatorics, Volume 2, by Richard Stanley.

Lectures:

1/7: Administrivia. Review of formal power series/exponential generating functions. Multiplication of egfs. Product rule and examples. Convergence in K[[x]]. Composition of formal power series. Exponential Formula. Examples (Bell numbers, Touchard's Theorem).
1/9: More on the Exponential Formula. Counting symmetric matrices. Terms in a generic symmetric determinant. Combinatorial composition of generating functions; example. Forests and trees.
1/11: Generating functions for (rooted) trees and (planted) forests. "Infinite exponent tower" for rooted trees. Prufer code and tree enumeration.
1/14: Functional inverses of formal power series: existence and uniqueness. The ring K((x)) of formal Laurent series. The Lagrange Inversion Formula.
1/16: Using Lagrange Inversion to count trees and forests. Using Lagrange Inversion to count partition trees.
1/18: Three proofs that Dyck paths are counted by Catalan numbers: Lagrange Inversion, Reflection Principle, Cyclic Lemma. Rational Catalan numbers. Rational Dyck paths. Rational (inhomogeneous) noncrossing partitions and cyclic sieving.
1/28: Homogeneous rational noncrossing partitions and cyclic sieving. The associahedron. Flag simplicial complexes. Two models for the rational associahedron. Collapses and combinatorial deformation retraction.
1/30: Rational duality. Alexander duality between rationally dual associahedra. Rim hooks and k-cores. Uniqueness of k-cores using abaci.
2/1: a,b-cores. Jacqueline Andersen bijection with a,b-Dyck paths. Olsson-Stanton Theorem. Armstrong's Ex-Conjecture (Johnson's Theorem, Thiel-Williams 24). a,b-parking functions and the Cyclic Lemma. a,b-Parking functions.
2/4: Symmetric polynomials. Structure of Q[x1,...,xn]^{S_n}. Symmetric functions. m-basis. e-basis.
2/6: Homogeneous symmetric functions. Omega involution. h-basis. Schur functions. Proof of symmetry: Bender-Knuth operators.
2/8: Jacobi-Trudi formula. Dual Jacobi-Trudi. Action of omega on s-basis.
2/11: Bialternant formula for Schur polynomials. (Isobaric) divided difference operators. Braid relations. Schur polynomials, Demazure characters, and Schubert polynomials.
2/13: Cauchy kernel. Cauchy identities: m/h, p/p, s/s. Dual Cauchy Kernel. Dual Cauchy identities: m/e, p/p, s/s. Omega on s. RSK.
2/15: Hall Inner Product. Dual bases. Transitioning between p and s: character theory of symmetric groups.
2/20: Polynomial GL-modules. Weyl characters. Schur functors. Schur-positivity. Boolean product polynomials (Open problem).
2/22: Dual Pieri Rule/Pieri Rule. Proof with bialternant formula. h-to-s and e-to-s transitions.
2/25: Murnaghan-Nakayama rule. Proof with bialternant formula. Rim-hook tableaux. Connection with S_n representation theory.
2/27: Ballot sequences. Reading words. Littlewood-Richardson Rule. Examples. LR positivity via geometry and representation theory.
3/1: Jeu-de-taquin rectification. Sketch of proof of Littlewood-Richardson. Kronecker problem. Plethysm problem.


Lecture Notes: Batch 1


Homework Assignments:

Hmmm...