Séminaire Lotharingien de Combinatoire, B46d (2001), 19 pp.
An Eulerian Partner for Inversions
A number of researchers studying permutation statistics on
the symmetric group Sn have considered pairs
(x, Y), where x is an Eulerian statistic and Y is a Mahonian statistic.
Of special interest are pairs
such as (des, maj), whose joint distribution on Sn is
given by Carlitz's q-Eulerian polynomials.
We present a natural Eulerian statistic stc such that
the pair (stc, inv) is equally distributed
with (des, maj) on Sn, and provide a simple
bijective proof of this fact. This result solves the problem
of finding an Eulerian partner for the Mahonian statistic inv.
We conjecture several properties of the joint distributions of stc
with the statistics des and maj.
Received: April 30, 2001; Accepted: June 15, 2001.
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