Séminaire Lotharingien de Combinatoire, B60c (2008), 13 pp.
Sami H. Assaf
A Generalized Major Index Statistic
Inspired by the k-inversion statistic for LLT polynomials, we
define a k-inversion number and k-descent set for words. Using
these, we define a new statistic on words, called the k-major
index, that interpolates between the major index and inversion
number. We give a bijective proof that the k-major index is
equi-distributed with the major index, generalizing a classical
result of Foata and rediscovering a result of Kadell. Inspired by
recent work of Haglund and Stevens, we give a partial extension of
these definitions and constructions to standard Young
tableaux. Finally, we give an application to Macdonald polynomials
made possible through connections with LLT polynomials.
Received: April 14, 2008.
Accepted: July 2, 2008.
Final Version: July 2, 2008.
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