Instructor: Alexander Postnikov
Time: Monday, Wednesday, Friday 1:002:00 pm
Place: MIT, Room 1390
Office hours: By appointment
Description:
Applications of algebra to combinatorics and vise versa.
We will discuss enumeration methods, permutations, partitions,
partially ordered sets and lattices, Young tableaux,
graph theory, matrix tree theorem, electrical networks,
random walks, convex polytopes, and other topics.
Course Level: Advanced Undergraduate.
Keywords:
Catalan numbers, Dyck paths, triangulations, noncrossing set partitions,
symmetric group, statistics on permutations, inversions and major index,
partially ordered sets and lattices, Sperner's and Dilworth's theorems,
Young diagrams, Young's lattice, Gaussian qbinomial coefficients,
standard Young tableaux (STY),
RobinsonSchenstedKnuth (RSK) correspondence,
partitions, Euler's pentagonal theorem, Jacobi triple product,
noncrossing paths, LindstromGesselViennot lemma,
spanning trees, parking functions, Prufer codes,
matrixtree theorem, electrical networks, random walks on graphs,
graph colorings, chromatic polynomial, Mobius function,
continued fractions,
enumeration under group action, Burnside's lemma, Polya theory,
transportation and Birkhoff polytopes, cyclic polytopes, permutohedra,
domino tilings, matching enumeration, Pfaffians, Ising model.
Grading: Based on several Problems Sets.
Problem Sets:
Lectures:
(We include links to the lecture notes from previous years.)

Mon, Feb 6. The Catalan numbers: Dyck paths, recurrence relation,
and exact formula.
Notes

Wed, Feb 8. The Catalan numbers (cont'd):
reflection method and cyclic shifts.
Notes

Fri, Feb 10. The Catalan numbers (cont'd): combinatorial interpretations
(binary trees, plane trees, triangulations of polygons,
noncrossing and nonnesting matchings, etc).
Notes

Mon, Feb 13.
The Narayana numbers.
Integer partitions. Young diagrams and standard Young tableaux.
The hook length formula.
Notes
(pages 17) and the last page of
these notes
 Wed, Feb 15.
Probabilistic "hook walk" proof of the hook length formula.
Notes
(pages 817)
 Fri, Feb 17. Other hooklengthtype formulas:
Tree hook length formula and "broken leg" hook length for shifted shapes.
Bijection between increasing binary trees and permutations.
Alternating permutations.
Notes (pages
1 and 79)
 Tue, Feb 21.
Set partitions. Arc diagrams and rook placements.
Noncrossing and nonnesting set partitions. Peaks and valleys.
Notes
(pages 516)

Wed, Feb 22.
Cycles in permutations.
The Stirling numbers of the first kind. Two Stirling triangles.
The Stirling numbers (of both kinds) as matrix coefficients in
change of bases matrices.
Notes (pages 1221),
and these
Notes (pages 915)

Fri, Feb 24.
The Stirling numbers of both kinds (cont'd).
Statistics on permutations.

Mon, Feb 27.
Statistics on permutations: inversions, descents, cycles, major index,
records, exceedances.
Notes
(pages 28)

Wed, Mar 1.
The Eulerian numbers and the Eulerian triangle. Sperner's Theorem. Posets.
Chains and antichains.
Notes (page 16)
and these
Notes (pages 15)

Fri, Mar 3. Ranked posets. Rank unimodality and Sperner property of posets.
Symmetric chain decompositions (SCD). Products of posets.
Notes (pages 612)

Mon, Mar 6. Construction of a SCD for a product of chains.
Dilworth's, Mirsky's, and Greene's theorems for posets.
Posets associated to permutations. Increasing and decreasing subsequences
in permutations.
Notes
(pages 14)

Wed, Mar 8.
Ramsey's and ErdosSzekeres theorems. Graphs vs posets vs permutations.
Lattices: Boolean lattice, partition lattice, Young's lattice.
Notes
(pages 49)

Fri, Mar 10. The lattice of linear subspaces in a vector space over
a finite field.
qanalogs: qnumbers, qfactorials, Gaussian qbinomial coefficients.
qPascal's triangle.
Notes
(pages 15)

Mon, Mar 13. qbinomial coefficients (cont'd).
Grassmannians over finite fields and Gaussian elimination.
Young diagrams
as reduced rowechelon forms of matrices.
Notes
(pages 610)

Wed, Mar 15.
The RobinsonSchensted(Knuth) correspondence (RSK).
Increasing and descreasing subsequences in permutations.
Notes
(pages 17)
 Fri, Mar 17. RSK (cont'd).
Symmetry of RSK. Involutions. Pattern avoidance in permutations.
321avoiding permutations and Dyck paths.
Up and down operators on Young's lattice.
Notes
(pages 716)

Mon, Mar 20.
Differential posets. Up and down operators.
Notes
(pages 114)

Wed, Mar 22.
Fibonacci lattice.
Notes (pages 1419).
Partition theory: generating functions, partitions with
odd and distinct parts.
Notes
(pages 15)

Fri, Mar 24.
Partition theory (cont'd).
Partitions with odd distinct parts. Partition identities.
Notes
(pages 68)

Mon, April 3.
Partition theory (cont'd).
Euler's Pentagonal Theorem.
Notes
(pages 815)
 Wed, Apr 5.
Partition theory (cont'd).
Jacobi's triple product formula.
Notes
(pages 19)
 Fri, Apr 7.
Counting labelled trees.
Cayley's formula.
Notes
(pages 110)
 Mon, Apr 10.
Two bijective proofs of Cayley's formula:
Prufer's coding and
EgeciogluRemmel bijection.
Notes
(pages 1014) and
Notes
(pages 17)
 Wed, Apr 12.
Spanning trees in graphs.
Adjacency matrix. Incidence matrix. Laplacian matrix.
Matrix Tree Theorem.
Notes
(pages 815)
 Fri, Apr 14.
Proof of the Matrix Tree Theorem. Oriented incidence matrix.
CauchyBinet formula.
Notes
(pages 110).
 Wed, Apr 19.
Matrixtree theorem (cont'd). Examples: The complete graph and
the hypercube graph.
Notes
(pages 1112) and
Notes
(pages 19)
 Fri, Apr 21. Electrical (resistor) networks.
Kirchhoff's and Ohm's laws. Kirchhoff matrix = weighted Laplacian
matrix.
Notes
(pages 110)
 Mon, Apr 24.
Electrical networks (cont'd). Effective resistance. Random walks
on graphs.
Notes
(pages 1116)
and
Notes
(pages 611).
 Wed, Apr 26.
Chip firing game.
Notes
 Fri, Apr 28.
Parking functions.
Notes
(pages 110)

Mon, May 1.
Chip firing game.
Notes

Wed, May 3.
Lindstrom's lemma (aka the GesselViennot method).
Notes

Fri, May 5.
Extreme points of subsets in R^n. The Birkhoff polytope.

Mon. May 8.
The ArnoldEulerBernoulli triangle.
Alternating permutations.
Andre's theorem. Tangent and secant numbers.
Notes
 Fri, May 12.
Eulerian cycles. BEST Theorem.
Notes
Mon, May 15.
Rhomus tilings and domino tilings.
Notes and
these
Notes.
Additional reading materials:
(The students are not required to buy these books.)
[AC]
Algebraic Combinatorics: Walks, Trees, Tableaux, and More
by R. P. Stanley, Springer:
PDF
[EC1]
Enumerative Combinatorics Vol 1 by R. P. Stanley,
Cambridge University Press.
Last updated: May 15, 2023 