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Séminaire Lotharingien de Combinatoire, B60c (2008), 13 pp.

# Sami H. Assaf

# A Generalized Major Index Statistic

**Abstract.**
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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