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New Euler–Mahonian Statistics on Permutations and Words

✍ Scribed by Robert J Clarke; Einar Steingrı́msson; Jiang Zeng

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
338 KB
Volume
18
Category
Article
ISSN
0196-8858

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✦ Synopsis

We define new Mahonian statistics, called MAD, MAK, and ENV, on words. Of these, ENV is shown to equal the classical INV, that is, the number of inversions, while for permutations MAK has been already defined by Foata and Zeilberger. It Ž . Ž . is shown that the triple statistics des, MAK, MAD and exc, DEN, ENV are equidistributed over the rearrangement class of an arbitrary word. Here, exc is the number of excedances and DEN is Denert's statistic. In particular, this implies the Ž . Ž . equidistribution of exc, INV and des, MAD . These bistatistics are not equidis-Ž . tributed with the classical Euler᎐Mahonian statistic des, MAJ . The proof of the main result is by means of a bijection which, in the case of permutations, is Ž . essentially equivalent to several bijections in the literature or inverses of these .

These include bijections defined by Foata and Zeilberger, by Franc ¸on and Viennot and by Biane, between the symmetric group and sets of weighted Motzkin paths.

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