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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