Combinatorics of Words (Math 206, Fall 2014)

Instructor: Igor Pak
pak@math.ucla (add .edu at the end)

Class schedule: MWF 3:00 - 3:50 pm, MS 5138

Office Hours: M 4-5, MS 6125

Main Reading Sources:

• M. Lothaire (L1), Combinatorics on Words, Cambridge University Press, 1997.
• M. Lothaire (L2), Algebraic Combinatorics on Words, 2002; available here.
• Jean Berstel, Aaron Lauve, Christophe Reutenauer and Franco Saliola (BLRS), Combinatorics on Words: Christoffel Words and Repetitions in Words, CRM Monograph, 2008.
  Draft version is available here.
• Richard Stanley, Enumerative Combinatorics, vol 1 (EC1, second edition), and vol. 2 (EC2).
  Download vol 1 from the author's website.
• Philippe Flajolet and Robert Sedgewick (FS), Analytic Combinatorics
  Download the book here.

Sections by topic, and additional reading:

Warning: some of these links require UCLA subscription; whenever possible I tried to include free links.

1. Thue-Morse sequence:
   • Arto Salomaa, Jewels of Formal Language Theory, 1981, §1.1-4; download here.
   • L1, §2.1-3; available here.
   • BLRS, Ch. 2, §1.1-2, 2.1-4.
   • Mark Sapir, Combinatorics on Words with Applications, course notes, §2.1-3.
     This is a part of the Non-commutative combinatorial algebra, 2014; available here.
   • J.P. Allouche and J. Shallit, The ubiquitous Prouhet-Thue-Morse sequence (connections to chess, differential geometry, etc.); download here.
   • M. Morse and G.A. Hedlund, Unending chess, symbolic dynamics and a problem in semigroups, Duke Math J. (1944); available here.
   • M. Gardner, The Magic Numbers of Dr. Matrix, 1985; Commentaries on the Los Angeles section (popular introduction).
2. Prouhet-Tarry-Escott problem:
   • BLRS, Ch. II, §1.3.
   • E.M. Wright, Prouhet's 1851 Solution of the Tarry-Escott Problem of 1910, Amer. Math. Monthly (1959); available here.
   • T.N. Sinha, On the Tarry-Escott Problem, Amer. Math. Monthly (1966); available here (lower bound on the size of the solution).
   • P. Borwein and C. Ingalls, The Prouhet-Tarry-Escott problem revisited, Enseign. Math. (1994); available here (note an easy proof of existence of a solution of size at most square the degree).
   • P. Borwein, Prouhet–Tarry–Escott problem, 2002; download TE section here (recent computational results).
3. Tower of Hanoi:
   • BLRS, Ch. II, §2.6.
   • A.M. Hinz, S. Klav˛ar, U. Milutinovic and C. Petr, The Tower of Hanoi – Myths and Maths, 2013 (UCLA access here)
   • B.M. Stewart and J.S. Frame, Solutions: 3918, Amer. Math. Monthly (1941); available here.
   • M. Szegedy, In How Many Steps the k Peg Version of the Towers of Hanoi Game Can Be Solved?, in Proc. STACS 1999; available here.
   • R. Grigorchuk and Z. Šunić, Asymptotic aspects of Schreier graphs and Hanoi Towers groups, Comptes Rendus Math. (2006); available here.
4. Sturmian words:
   • L2, §2.1-2.
   • P. Arnoux, Sturmian sequences, in P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, 2001, §6.1; available here.
5. Quasideterminants and MMT:
   • I. Gelfand, S. Gelfand, V. Retakh and R. Wilson, Quasideterminants, Advances in Math (2004), main reference.
   • M. Konvalinka and I. Pak, Non-commutative extensions of the MacMahon Master Theorem, Advances in Math. (2007).
   • MacMahon Master theorem, Wikipedia page; simple proof of Dixon's theorem.
   • I.J. Good, A short proof of MacMahon's ‘Master Theorem, Proc. Cambridge Philos. Soc. (1962).
6. Plactic Monoid, Knuth equivalence, RSK and Jeu-de-taquin:
   • Stanley's EC2, Appendix to Ch 7.
   • L2 Ch 6.

Homeworks:

• HW1 is available here, due Nov 7, 2014.

Course notes:

Scans of three substitute lectures by Alejandro: one, two, three.

Click here to return to Igor Pak Home Page.

Last updated 10/21/2014.