Algebraic combinatorics on words /M. Lothaire.
- Cambridge ; New York : Cambridge University Press, (c)2002.
- 1 online resource (xiii, 504 pages) : illustrations
- Encyclopedia of mathematics and its applications ; v. 90 .
Includes bibliographies and index.
Finite and Infinite Words -- Semigroups -- Words -- Automata -- Generating series -- Symbolic dynamical systems -- Unavoidable sets -- Sturmian Words -- Equivalent definitions -- Standard words -- Sturmian morphisms -- Unavoidable Patterns -- Definitions and basic properties -- Deciding avoidability: the Zimin algorithm -- Avoidability on a fixed alphabet -- Sesquipowers -- Bi-ideal sequences -- Canonical factorizations -- Sesquipowers and recurrence -- Extensions of a theorem of Shirshov -- Finiteness conditions for semigroups -- The Plactic Monoid -- Schensted's algorithm -- Greene's invariants and the plactic monoid -- The Robinson--Schensted--Knuth correspondence -- Schur functions and the Littlewood--Richardson rule -- Coplactic operations -- Cyclage and canonical embeddings -- Codes -- X-factorizations -- Defect -- More defect -- A theorem of Schutzenberger -- Numeration Systems -- Standard representation of numbers -- Beta-expansions -- U-representations -- Representation of complex numbers -- Periodicity -- Periods in a finite word -- Local versus global periodicity -- Infinite words -- Centralizers of Noncommutative Series and Polynomials -- Cohn's centralizer theorem -- Euclidean division and principal right ideals -- Integral closure of the centralizer -- Homomorphisms into k[t] -- Bergman's centralizer theorem -- Free subalgebras and the defect theorem -- Appendix: some commutative algebra -- Transformations on Words and q-Calculus -- The q-binomial coefficients -- The MacMahon Verfahren.
Combinatorics on words has arisen independently within several branches of mathematics, for instance number theory, group theory and probability, and appears frequently in problems related to theoretical computer science. The first unified treatment of the area was given in Lothaire's book Combinatorics on Words. Originally published in 2002, this book presents several more topics and provides deeper insights into subjects discussed in the previous volume. An introductory chapter provides the reader with all the necessary background material. There are numerous examples, full proofs whenever possible and a notes section discussing further developments in the area. This book is both a comprehensive introduction to the subject and a valuable reference source for researchers.