Learn · Algebraic Combinatorics
Lecture 69 of 85Week 9
Labelled abaci and alternants
Prof. Amritanshu Prasad Prof. Sankaran Viswanath · IMSc
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- Algebraic Combinatorics, IMSc (at NPTEL)
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All the lectures
- Examples of Mobius Inversion
- Partially Ordered Sets
- Hasse Diagrams
- Isomorphsms of Posets
- Maximal, Minimal, Greatest, Least
- Induced Subposets
- Incidence Algebras
- Inversion in Incidence Algebras
- Mobius Inversion
- Examples of Mobius Functions
- Product Posets and their Mobius Functions
- Opposite of a Poset
- The Poset of Set Partitions
- Connected Structures
- Lattices
- Weisner's Theorem
- The Lattice of Non-Crossing Partitions
- The Canonical Product Decoposition for Intervals of Non-Crossing Partitions
- The Mobius Function for Non-Crossing Partitions
- Ideals in a Poset
- Mobius Function of J(P)
- Young's Lattice
- Distributive Lattices
- Formal Power Series
- The Necklace Problem
- Combinatorial Classes
- Sums, Products, and Sequences of Combinatorial Classes
- Power Set, Multisets, and Sequences
- A Little Dendrology
- Super Catalan/Little Schroeder numbers
- Regular Languages
- Finite Automata
- The Pumping Lemma
- The Dyck Language
- Permutations and their cycles
- Permutation Groups
- Orbits, fixed points, stabilizers
- The orbit counting theorem
- The Polya Enumeration Theorem
- The Cycle Index Polynomials
- Cycle Index of the Octahedral Group
- Cycle Index of the Full Permutation Group
- Combinatorial Species
- Generating Series of a Species
- Cycle Index Series of a Species
- Isomorphism of Species
- Visualization of Species
- Sum of Species
- Product of Species
- Sums and Products: More Examples
- Substitution of Species
- Derivative of a Species
- Powers and Seqeunces of Binomial Type
- Pointing and Cayley's Theorem
- R-enriched Trees
- R-enriched Endofunctions
- Lagrange Inversion Forumla
- Motivation for the LGV Lemma
- Statement of the LGV Lemma
- Nice Applications of the LGV Lemma
- Sign-Reversing Involutions
- Proof of the LGV Lemma
- The Cauchy-Binet Formula
- Symmetric polynomials: definition and examples
- Monomial symmetric polynomials
- Elementary and Complete symmetric polynomials: part 1
- Elementary and Complete symmetric polynomials: part 2
- Alternating polynomials
- Labelled abaci and alternants
- Schur polynomials
- Pieri rule - statement and examples
- Pieri rule - proof
- The second Pieri rule
- Semi-standard tableaux
- Triangularity of Kostka matrix
- Monomial expansion of Schur
- The RSK correspondence
- Jacobi Trudi identities via LGV lemma
- Formal ring of symmetric functions in infinitely many variables
- Generating functions for e, h
- The power sum symmetric functions
- The inner product and Cauchy identity
- Monomial expansions and RSK
- Skew Schur functions and the LR rule
- Live Session 04-02-2022