A generalized Major index statistic

2008/07/02 by Assaf, Sami
#05A05 #05A15 #05A19 #05E05 #05E10 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.0807.0433

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 equidistributed 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.

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