A Fundamental Transformation on the Rearrangement of Words

Abstract D. Foata and D. Zeilberger (1990, Stud. Appl. Math. 83 , 31-59) proved that the Denert Statistic "den" when associated with the exceedance number "exe" is Euler-Mahonian on the symmetric group. This result is extended to the case of arbitrary words (with repetitions). A fundamental transformation on the rearrangement classes is explicitly constructed and has the property that the bistatistic "descent number-major index" of the transformed word has the same value as the bistatistic (exe, den) of the initial word. This new transformation can be viewed as the q -analog of the fundamental transformation given by P. Cartier and D. Foata (1969, "Problemes combinatoires de permutations et rearrangements," Springer-Verlag, Berlin) that associated the univariable exceedance number and descent number statistics in a one-to-one manner.