Bijective Combinatorics (Math 206A, Winter 2026)

Instructor: Igor Pak

Class schedule: MWF 3:00 - 3:50 pm, MS 5137.

Office Hours: M 3-3:50.

Grading: The grade will be based on attendance and class participation.

Content

Textbook:

• R.P. Stanley, Enumerative Combinatorics (EC), vol. 1 and vol. 2 (second edition).

Additional reading

1. MacMahon's Master Theorem
   • MMT and Dixon's identity WP pages
   • 100 years of Dixon's identity by James Ward
   • Noncommutative extensions... by Konvalinka–Pak
   • Commutation and Rearrangements by Cartier–Foata (in French, a combinatorial proof of MMT)

2. Lindstrom–Gessel–Viennot lemma
   • LGV lemma WP page
   • original 1989 preprint by Gessel–Viennot
   • A combinatorial approach to matrix algebra by Doron Zeilberger (determinant of a product and the Cayley–Hamilton theorem)

3. Spanning trees, LERW and Wilson's algorithm
   • Simulation of LERW on a square grid and LERW WP article
   • original article by David Wilson
   • concise treatment of Wilson's algorithm (and hypergraph generalization) by Gorodezky–Pak
   • random walk constructions by Aldous and by Broder
   • Random spanning tree by Alain Guenoche (trivial det-based algorithm, in French)

4. Total positivity of planar networks
   • original paper by Curtis–Ingerman–Morrow
   • Loop-erased walks and total positivity by Sergey Fomin

5. Linear extensions and log-concavity, with applications to random walks
   • LE WP page (see also this detailed survey on the number of LE by Chan–Pak)
   • original paper by Chung–Fishburn–Graham (log-concavity conjecture proved by an explicit injection for posets of width two)
   • Log-concavity in planar random walks by Chan–Pak–Panova

6. Heaps of pieces
   • original article by Xavier Viennot
   • Viennot's course on heaps (videos and slides)
   • application to diameter of U(n,q) by Jordan Ellenberg
   • Random Walks and Heaps of Cycles by Philippe Marchal (heaps give Wilson's LERW algorithm)
   • application to rectangular polyomino enumeration, a nice exposition by Christian Krattenthaler

7. Computing determinant by a straight line program
   • straight-line program WP article
   • Les Valiant's original article and modern overview by Peter Burgisser
   • original article by Mahajan–Vinay (introducing the involution on clows)
   • Division-Free Algorithms ..., a readable overview by Gunter Rote
   • Berkowitz's Algorithm and Clow Sequences by Michael Soltys (complete explanation of clows in the context)
   • Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees by Fomin–Grigoriev–Koshevoy
   • Some Exact Complexity Results for Straight-Line Computations over Semirings by Jerrum–Snir

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Last updated 2/10/2026.