Thek-Extensions of some New Mahonian Statistics

In previous work by the authors, new Mahonian statistics ENV, MAD and MAK were defined on words, and it was shown that ENV is equal to the classical statistic INV and that the triple statistics (des, MAK, MAD) and (exc, DEN, ENV) are equidistributed over any rearrangement class of words. Here, exc and des are the classical Eulerian statistics, while DEN is Denertʹs statistic. In addition, a bijection between the symmetric group and sets of weighted Motzkin paths was used to give a continued fraction expression for the generating function of (exc, INV) or (des, MAD) on the symmetric group. These results are extended to the case in which the letters of the alphabet used are divided into two classes—large and small—with corresponding changes to the definitions of the above statistics.