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Course11110615885 lectures

Algebraic Combinatorics

Prof. Amritanshu Prasad Prof. Sankaran Viswanath · IMSc

Start with lecture 1 At NPTEL

Lectures

Week 0

  1. Examples of Mobius Inversion Video
  2. Partially Ordered Sets Video
  3. Hasse Diagrams Video
  4. Isomorphsms of Posets Video
  5. Maximal, Minimal, Greatest, Least Video
  6. Induced Subposets Video
  7. Incidence Algebras Video
  8. Inversion in Incidence Algebras Video
  9. Mobius Inversion Video
  10. Examples of Mobius Functions Video
  11. Product Posets and their Mobius Functions Video
  12. Opposite of a Poset Video
  13. The Poset of Set Partitions Video
  14. Connected Structures Video

Week 2

  1. Lattices Video
  2. Weisner's Theorem Video
  3. The Lattice of Non-Crossing Partitions Video
  4. The Canonical Product Decoposition for Intervals of Non-Crossing Partitions Video
  5. The Mobius Function for Non-Crossing Partitions Video
  6. Ideals in a Poset Video
  7. Mobius Function of J(P) Video
  8. Young's Lattice Video
  9. Distributive Lattices Video

Week 3

  1. Formal Power Series Video
  2. The Necklace Problem Video
  3. Combinatorial Classes Video
  4. Sums, Products, and Sequences of Combinatorial Classes Video
  5. Power Set, Multisets, and Sequences Video
  6. A Little Dendrology Video

Week 4

  1. Super Catalan/Little Schroeder numbers Video
  2. Regular Languages Video
  3. Finite Automata Video
  4. The Pumping Lemma Video
  5. The Dyck Language Video

Week 5

  1. Permutations and their cycles Video
  2. Permutation Groups Video
  3. Orbits, fixed points, stabilizers Video
  4. The orbit counting theorem Video
  5. The Polya Enumeration Theorem Video
  6. The Cycle Index Polynomials Video
  7. Cycle Index of the Octahedral Group Video
  8. Cycle Index of the Full Permutation Group Video

Week 6

  1. Combinatorial Species Video
  2. Generating Series of a Species Video
  3. Cycle Index Series of a Species Video
  4. Isomorphism of Species Video
  5. Visualization of Species Video
  6. Sum of Species Video
  7. Product of Species Video

Week 7

  1. Sums and Products: More Examples Video
  2. Substitution of Species Video
  3. Derivative of a Species Video
  4. Powers and Seqeunces of Binomial Type Video
  5. Pointing and Cayley's Theorem Video
  6. R-enriched Trees Video
  7. R-enriched Endofunctions Video
  8. Lagrange Inversion Forumla Video

Week 8

  1. Motivation for the LGV Lemma Video
  2. Statement of the LGV Lemma Video
  3. Nice Applications of the LGV Lemma Video
  4. Sign-Reversing Involutions Video
  5. Proof of the LGV Lemma Video
  6. The Cauchy-Binet Formula Video
  7. Symmetric polynomials: definition and examples Video

Week 9

  1. Monomial symmetric polynomials Video
  2. Elementary and Complete symmetric polynomials: part 1 Video
  3. Elementary and Complete symmetric polynomials: part 2 Video
  4. Alternating polynomials Video
  5. Labelled abaci and alternants Video
  6. Schur polynomials Video

Week 10

  1. Pieri rule - statement and examples Video
  2. Pieri rule - proof Video
  3. The second Pieri rule Video
  4. Semi-standard tableaux Video
  5. Triangularity of Kostka matrix Video

Week 11

  1. Monomial expansion of Schur Video
  2. The RSK correspondence Video
  3. Jacobi Trudi identities via LGV lemma Video
  4. Formal ring of symmetric functions in infinitely many variables Video

Week 12

  1. Generating functions for e, h Video
  2. The power sum symmetric functions Video
  3. The inner product and Cauchy identity Video
  4. Monomial expansions and RSK Video
  5. Skew Schur functions and the LR rule Video

Live Session

  1. Live Session 04-02-2022 Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IMSc
Department
Mathematics
Teacher
Prof. Amritanshu Prasad Prof. Sankaran Viswanath
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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